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Demystifying Einstein’s Field Equations on General Relativity

Disclaimer: This post will make sense only to those interested in physics, more precisely to those who know general relativity and cosmology, and want to have a deeper insight into the mathematics of general relativity without having to do any actual Math.

Nice to see you continue reading this post, Welcome aboard

This is a slightly non technical version of my article on Einstein’s field equations intended to explain what the field equations of general relativity actually mean.

Mathematics of General relativity is really complicated and is full of tensors, and many science enthusiasts and students find it really difficult to understand. So did I a long time back when I first came across it. So here I try to explain in simple english the meaning of Einstein’s field equations.

Let’s start with a bang!!

Ricci tensor * (Ricci Scalar) * (Metric Tensor) = ((8 * PI * G) / c4) * (Stress Energy Tensor)

The above set of equations (Yes, it is not a single equation!!) are called Einstein’s Field Equations. These equations describe the way Matter tells space-time how to curve!!

We all know that according to general relativity mass curves space time. For instance, the mass of sun has curved the space time around it and hence all the planets of the solar system move around the Sun in this curvature. Note that the planets do not know that Sun exists out there in the middle. All they do is to move in their local curved path which takes them around the sun.NOTE: We can simplify the above equations further as

Einstein Tensor = Stress Energy Tensor

Where, Einstein Tensor = Ricci tensor * (Ricci Scalar) * (Metric Tensor)
And the units are taken such that c=8 * PI * G =1

Now, let me give a brief overview here. The LHS of this equation describes the space-time geometry and the RHS describes the associated mass-energy responsible for that curvature. In other words, field equations relate mass-energy and the space-time curvature at every point in space-time!
To be more simplistic, say if RHS is about the mass of our Sun, then the LHS would be the space-time curvature caused by this mass.
Einstein’s field equations were originally written to describe a 4 dimensional universe. But we can also easily describe any n-dimensional universe using these equations!!

So far so good. Now, let’s get a bit deeper into the mathematics.

Physical Constants – Old Friends in the new equations!

The quantities PI, G and c in the equation are well known mathematical and physical constants. PI is our old school friend in Mathematics; G is again our old friend in physics called Newton’s gravitational constant and c is what relativity talks a lot about, the speed of light.

Tensor Calculus

Field equations are tensor equations and completely rely on tensors to tell what they want to. This is because tensors are a unique way of expressing values independent of the frame of reference. So in the field equations, tensors are used to express physical quantities independent of the reference frames. Tensors are expressed as multi dimensional arrays. In case of 4D space-time the tensors of the field equations are a bunch of 4X4 matrices! But, please do not confuse tensors with Matrices. A matrix can be a tensor only if it obeys tensor transformation rules.

Why Tensors ?

General theory of relativity has its foundation in the principle of general covariance, which states that laws of physics take same mathematical form in all the frames of reference. In other words, the laws of physics remain the same throughout the universe. (Of course, except at a blackhole singularity where all physical laws break down!)

Tensors are a mathematical way of expressing the above-mentioned principle. Irrespective of the frames of reference used, the mathematical formulations used to express the physical laws remain the same while using tensors. Once we have these tensors, it becomes just a lengthy complicated mathematical activity to formulate the core mathematics of general relativity, which is what Einstein did with the help of his good mathematician friend Marcel Grossman.

A Small Mistake – Missing out the Metric

Originally when Einstein formulated the field equations he thought that the equations were

Ricci Tensor = Stress-Energy Tensor

He thought that it was the right solution because this very well explained the age old problem of the perihelion precession of mercury!! But he soon realized that without the metric tensor and Ricci scalar, local conservation of energy and momentum would not be possible unless universe had equal density of matter everywhere!! In other words mountains, empty space, stars all should have same density for the above equation to be true, which is obviously not the case.

Einstein then went back to his mathematical investigations and finally published the field equations in its current form.

Solutions to Field Equations. One or Many?

When expanded for 4 dimensions, the field equations result in a set of 10 non-linear partial differential equations and have to be solved for the metric tensor!! As any mathematician knows non linear equations are very difficult to solve without doing suitable approximations. However there have been cases where solutions to the field equations have been provided completely, and are called exact solutions. Exact? Well, yes!

A great difficulty in solving the field equations is its non-linear nature. In quantum mechanics the Schrodinger’s equation is linear in the wave function and hence it is relatively easy to solve it compared to the field equations.

Linear equations mean that the system being defined is just a direct sum of its parts or their multiples. A non-linear system is the one which is not so!! To be slightly more technical, linear systems obey the principle of superposition while non-linear systems do not! Principle of superposition simply means that a linear combination of the solutions to a system is also a solution to the system. This principle does not hold for non linear systems and field equations being non-linear in nature are most complicated to solve.

Solutions to the field equations are called metrics. Yes, metrics define the space-time geometry based on the given input values. There are also hypothetical solutions that arise while solving the field equations. For instance, the worm-hole metrics solution defines space-time shortcuts within or across universes, provided the matter defined in stress-energy tensor of the equation is exotic. Exotic matter is matter with negative energy density.

Einstein’s field equations also describe the different evolution models of the universe. Depending on the energy density and the expansion rates they describe whether the universe will continue to expand forever or whether the universe will collapse back in a big crunch, etc.

Components of General Relativistic Field Equation

Let us now talk about each of the field equation components:

Ricci Tensor

Ricci Tensor in the field equation defines the deviation of the n-dimensional volume of the space in a curved space-time from the flat Euclidean space. For instance in a flat space time, Pythagoras theorem holds good for a right angled triangle, whereas on the surface of a sphere the relationship between hypotenuse and the other two sides of a right angled triangle do not obey the Pythagoras theorem. Ricci tensor defines this amount of deviation in terms of volume in a curved space from that of flat space.

Ricci Scalar

Ricci Scalar is just a number that defines the curvature of space-time. Every point in the space-time has this number and it defines the intrinsic (meaning as observed from within) curvature at that point in space-time.

If this number is zero then the space is same as a Euclidean flat space.

If this number is positive then the space has lesser volume compared to similar Euclidean space! Imagine a soccer ball whose internal volume as measured from inside the soccer ball is smaller than its volume measured from outside the soccer ball!!

If this number is negative then the space has more volume compared to similar Euclidean space! Imagine a soccer ball whose internal volume as measured from inside the soccer ball is larger than its volume measured from outside the soccer ball!!

Metric Tensor

Metric tensor is used to measure the geometry of space-time. Note that since we are also talking about time (when we say space-time), the geometry also talks about the causal structure of space-time aka: past, present and future.
In other words, metric tensor is used to all space-time geometry related quantities like distance between two points, volume of a given section, evolution of the structure i.e. future, past, present etc.

Mathematically, in 4D the metric tensor is a collection of 10 numbers. By carefully observing as to how these numbers are varying for adjacent points in space-time we can determine whether the space-time is curved or flat.

Ricci Scalar in terms of Ricci Tensor and Metric Tensor

Since all the components in the LHS of the field equations talk about the curvature aspect which is space-time geometry, they all are related to each other as follows:

Ricci Scalar is the result of the contraction of Ricci tensor and Metric tensor.

Contraction is a mathematical process where two tensors are summed up resulting in a third tensor which is two ranks less than the original ones. Since Ricci tensor and Metric tensor are rank 2 tensors, the contraction between the two of them results in a rank zero tensor which is actually a scalar. Yes, scalars are rank zero tensors, vectors are rank 1 tensors

Stress-Energy Tensor

This tensor is the source of the space-time curvature. It describes the energy density and the momentum at the given point in space-time. The value of this tensor is zero at points where there is no energy density.

Just like the the metric tensor, the stress energy tensor is just a set of 10 numbers in 4D space-time.

One number defines how much mass-energy density is there at the point.

Three numbers define the momentum of the matter at that point.

Next three numbers define the pressure in each of the three spatial directions at that point.

Last three numbers define the stress in the matter at that point.

Schwarzschild Metric

After the flat space-time metric, Schwarzschild Metric is the simplest metric in general relativity. It is used to describe the space-time geometry outside a non-charged, perfectly spherical, non-rotating mass. Please note that perfectly spherical is an ideal condition for normal objects like planets, normal stars etc. So is the condition non rotating! In mathematical terms, non-rotating means zero angular momentum! By the way, Schwarzschild metric was the first exact solution of the field equations. It is extensively used to study non-rotating black holes.

Note that the only information available about a black hole are its angular momentum, charge and mass. So for Schwarzschild black holes, two black holes can be distinguished ONLY based on their mass!

Any non rotating, non charged, spherical mass with a radius less then Schwarzschild radius ends up as a black hole. Since there is no lower limit for the amount of mass, theoretically any mass can be reduced into a black hole, including sub-atomic particles!!

Vacuum Field Equations

Please note that, Schwarzschild metric is a solution for vacuum field equations. Vacuum field equations are those field equations where the measurement of space-time geometry is done only outside the mass in question. For instance if there is a spherical mass of radius 5 kilometers and if we take the center of the sphere as the origin for our reference frames, vacuum field equations describe space-time geometry only for those values where distance from origin is greater than 5 kilometers!

Since vacuum field equations talk about the geometry of space time outside a mass, the stress energy tensor is zero in these equations!! More simplification, easy mathematics

We often hear that the mathematics of general relativity breaks down at singularity. Singularity is the infinitely small point at which the entire mass of the object lies. This gives rise to infinities in mathematical formulations and the general relativity stops making any sense here.

Event horizon of a black hole lies exactly at the Schwarzschild radius. For normal objects like planets and stars, their actual radius is greater than their Schwarzschild radius. For black holes, their Schwarzschild radius is greater than their actual radius. Now here, general relativity vacuum field equations still continue to make sense since the condition is still satisfied i.e. space-time points where we take measurements be outside the mass in question.

Once we reach the singularity point in a black hole, the math tells us that the curvature is infinite!! But are singularities possible?? General relativity predicts singularities because it imposes no limits on minimum size of a particle or mass. But is it the truth?? because at this microscopic level where field equations break down, quantum mechanics enters the picture. There is an uncertainty of Planck’s length scale! Something is missing there. That is the reason we hear physicists saying that a unification of relativity and quantum mechanics should complete the picture!

Kerr Metric – Answer to rotation

The above said details were about non-rotating black holes. In rotating black holes (whose metric is much more complicated than Schwarzschild metric), called Kerr black holes, we don’t have a point singularity. This is because a point cannot support rotation, and we cannot make rotation zero as angular momentum has to be conserved! Hence in Kerr black holes we have a ring singularity instead of a point singularity. This ring rotates with an angular momentum as earlier. However this ring has zero thickness, even though the radius is non zero!

Kerr Metric and Time Travel

In Schwarzschild black holes one cannot avoid singularity after crossing event horizon because here beyond the event horizon all world lines end only at the singularity!

But, this is not the case for Kerr black holes. A Kerr black hole has two event horizons! Once we cross into the outer event horizon, it is still like a Schwarzschild event horizon where all ways lead towards the inner event horizon!! We have no way out! But once we cross the inner event horizon, we are free till we hit the singularity. In other words, the mathematics tells us that, beyond the inner event horizon before we hit the ring singularity, it is possible to move to escape routes which lead to some other inner event horizon leading to some other outer event horizon which can take us out to some other space-time point in this universe or even in a parallel universe!!

So effectively, Kerr black holes can serve as worm holes. Probably a future newspaper headlines would read like, Kerr Travels: 2 Minutes travel to Alpha Centauri in our (ISO) Intergalatic Standards Organization certified Worm Bus safely!. Then there might be some other news headlines too like Kerr Disaster: Worm Bus hits ring singularity at NGC 364 Kerr Hole!

But the problem is that the inner event horizon of Kerr blackholes is highly unstable due to infinite blue shifting of infalling radiation.

Ergosphere – A Cosmic Just Miss!!

A Kerr black hole has a region outside its event horizon called Ergosphere where the space itself is being dragged at the speed of light due to the space dragging effect caused by the angular momentum of the black hole. This is a high source of energy since we can have particles which enter this zone and come out with more energy that can be used to drive a electricity generator!! This is possible because the particles in Ergosphere are still outside event horizon and can hence escape singularity!

This process of extracting energy from rotating black holes is called Penrose process and could be a great and cheap source of energy for advanced civilizations. The way ancient civilizations on earth were located near river basins, we can search for existence of advanced civilizations near rotating black holes!

Naked! Oh My Gawd! Yet Single (Singularity)?

As mentioned earlier, a Kerr black hole has two event horizons! And as the spin increases the two event horizons move towards each other and then towards singularity!! At this point, the black hole has no event horizon at all!!!! There is just a singularity and this is called naked singularity, all exposed to the outside world!

Naked singularities mean a lot. During the formation of Non rotating Schwarzschild black holes, we cannot observe the collapse of the star beyond the event horizon!! Where as in case of rotating Kerr black holes, naked singularity means, it may also be possible to observe the collapse of a star all the way till it hits singularity!!

Cosmic Censorship!

We have a problem with naked singularities. Before the formulation of naked singularities we were sure that an observer who sees a singularity cannot avoid falling in it! In other words, only an observer who crosses the event horizon can see a singularity and since he has crossed then horizon already, he cannot avoid singularity now! This is referred to as weak cosmic censorship. Note that here there is no time frame for an observer to reach the singularity once he has seen it!

With the formulation of naked singularities, weak cosmic censorship is violated. Hence Penrose came up with the strong cosmic censorship hypothesis, which says, one cannot observe singularity at all!!! In other words, naked singularities cannot exist!

The problem with naked singularities is that, physics loses its deterministic power when there are naked singularities. For singularities that are hidden beyond the event horizon, evolution of the universe is still predictable, while this is not the case with naked singularities since we don’t have a boundary like an event horizon in case of a naked singularity.

The only naked singularity allowed for universe to be deterministic is the big bang singularity itself!

The Great Cosmological Constant

Oh Yes! One more thing. The cosmological constant.

The field equations mentioned so far describe a dynamic (expanding or contracting) universe. But Einstein initially thought that the universe was static. So he decided to add a compensating factor (called the Cosmological Constant) to the field equations so that the field equations describe a static universe, which Einstein himself later termed as the biggest blunder of his life!!
The field equation with the cosmological constant looks as below:
Ricci tensor * (Ricci Scalar) * (Metric Tensor) + (Cosmological Constant) * (Metric Tensor) = ((8 * PI * G) / c4) * (Stress Energy Tensor)

Anything else? Yes, Geodesics please..

Yes, the mathematics of general relativity is not just about Einstein field equations. We also have the geodesic equation. Field equations and the geodesic equation together describe the core mathematics of the general theory of relativity.
Geodesic equations describe the world line (path traveled in space-time) of a particle free from all forces. Please note that in the general theory of relativity gravity is not a force, instead it is just a geometry of space-time!

In other words geodesic equations describe the way Space-time tells matter how to move. To solve a geodesic equation, we need to have a knowledge about the space-time geometry to define the world line, in other words we need the metric tensor, hence it is required that we first solve the field equations before moving on to solve the geodesic equations.

Finally, The Secret Relationships between Matter, Space-Time and the Equations

So we have a complete picture now.

Field equations tell the way Matter tells space-time how to curve
Geodesic equations tell the way Space-time tells matter how to move.

Hope this article helps in understanding the mathematics of general relativity without having to resort to solving any of those complicated equations.

A word of caution, field equations are really very complicated to solve and need a lot of imagination to solve them even with a relatively good approximation. Now that we have supercomputers at our disposal, solving field equations is a job better handled by computers

Finally, let me quote Einstein himself

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality

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Gurudev is the developer of Gurunudi AI Platform. This is his official website where he pens his thoughts on a wide range of topics, answers queries, shares resources and tools developed by him.

89 COMMENTS

  1. I recognize that the LHS of the field eq. is well understood and the differential non-Euclidean geometries of Riemann and Gauss are more than 125 years old(I just don’t know them except very very superficially). But the RHS of the equation is what I have absolutely no idea about, What is a “stress-momentum-energy” array [T_ij( )] for matter? Is it related to the motion(its acceleration) of the matter in any way? If It is then I can see that the spinning disk thought experiment might imply that mass energy cause the geometry of spacetime to curve(the g_ij( ) to be affected).

    • You are right, the stress energy tensor is the actual source of the gravitational field. It has a non-zero value only INSIDE the matter in question, it deals with matter density, momentum, stress – all of which play an active role in the resulting gravitational field which in turn cause the space-time curvature.

  2. Dear Mr.Gurudev
    I wrote to you some last year when I was looking at Wheelers book “Journey into Gravity and Spacetime” along with E.F.Taylors new book on “Exploring Black Holes” but I found your article “Demystifying the Einstein Field Equations” easier to follow. Since last year I have been looking into Buddhism and find that all of science pales in comparison to Buddhas Dharma(4 Noble TRUTHS and no-Self). Yhe idea of emptiness as the lack of identity seems to go to far for me(but who knows!). Well in any case I will try to get up to speed about GR again by reading your article and what I asked you and your answers.
    I will try to put what you say in your article and what you say in your answers into some kind of picture for myself and then send it to you.
    Yours truly
    Francis Ferrara

    • Nice to hear back from you. Yes, a lot of the core universal truth requires a more of an analytical mind to understand it rather than math equations. The core of Buddhist preachings and the Advaita philosophy have a lot in common with quantum world. Good luck for your knowledge exploration. Sure, will try to answer you questions based on whatever knowledge I have on the subject.

  3. Dear Mr. Gurudev
    I have really enjoyed this disscussion with you about Einsteins fiels eq. and GR in its overall structure, but I think that the theory is more about the human creativity of Mr.Einstein(his philosophy,experiences,distrust of authority and that inexplicable thing that artists and scientists all have) rather that the theory being inferred and distilled out of observations. Observations may have given him clues but Putting them together was a work of art by an artist.
    Francis

    • Yes I agree with you on this. Einstein was a great genius who could think beyond dimensions and it was as if he understood the mind of God during creation! This is evident by the facts that even though he himself didnt believe it initially, his theory predicted Big bang, black holes etc. Einstein stands apart from rest of the mass in that experiments were done later (infact are being done even today) to test the predictions of his theory where as the theory itself was created mostly out of his own thoughts (thought experiments). Almost all other physicists based their theories largely on already available experimental data.

      Also almost all theories are for that matter creative ordering of observational data and human predictions, because nature itself doesnt have any boundaries in terms of its laws. Nature doesnt differentiate between physics, chemistry, DNA, black holes – it is we for our understanding have created all these braches, we have physics, we have chemistry, we have biology – but then since nature itself draws no such hard boundaries, we also then have bio-physics, bio-chemistry, physio-chemistry to also study the laws sitting in the boundaries drawn by us. And yes discovery of new science is definitely an art which requires creativity else we could have programmed Robots to discover new laws of science :)

  4. Dear Mr. Gurudev
    In regard your first answer of Dec 27,2010 at 8;46 am–

    How do you know that the existence of the “spacetime” continuum and of the “mass-energy” continuum are dependent on each other? Why do you say that there can be no “spacetime” without “mass-energy”?
    I can see that space and time form a unity called “spacetime”,Special Relativity shows this; and it also shows that mass and energy form a unity called “mass-energy”(isn’t this related to a continuum called momenergy to?). BUT why do you say the “spacetime” continuum is dependent and meaningless without the “mass-energy” continuum? And that in particular there is a relation between the two of the curvature(geometry structure) of the spacetime of the spacetime continuum to the “mass-energy” of the momenergy continuum?
    Francis

    • space-time and mass-energy are related because they are different manifestations of the same entity – the pre big-bang soup from which space-time as well as mass-energy were created. Imagine an empty space-time without mass-energy, well you cant imagine that, because there is nothing that ticks (time runs inside mass ie inside quantum particles), and without mass-energy there is nothing to measure in space, to measure distance you need reference points ie mass. If we have a universe where there is only space-time and no mass-energy, then where did the mass-energy of that universe which existed in its big bang singularity disappear? My very own thought is that space-time is just another manifestation of mass-energy and is probably even the source of the mysterious dark-energy. After all everything is finally an outcome of energy, and so should be space-time I guess. I guess our understanding of the actual nature of space-time is still limited and is prejudiced by what we observe in the observable universe.

      For instance, how can we be sure that there isn’t an even larger portion of our observable universe which is moving away from us beyond the speed of light? If regions beyond the observable universe are expanding at a rate greater than the speed of light, then no light (ie information) will ever reach us from such regions of the universe, nor will there be any observable impacts of gravity from such regions.

  5. This whole artucle tries to explain this array relation known as the Einstein Field eq. which is a relation between the “curvature” of the spacetime continuum and the “mass-energy”(momenergy) of another continuum where the same EVENTS are coordinatized by momentum and energy coordinates instead of by space and time like the LHS resulting in the spacetime continuum. BUT

    Why did Einstein even ever consider the possibility that the non-Euclidean geometric structure of the spacetime continuum(it’s curvature) should be related and caused by the “mass-energy”–momentum flow distribution in this spacetime(all over it) So that the body doing the moving along the geodesic of the curved spacetime should also be seen as contributing to the very “curvature” which guides it along the local geodesic,but that the most curvature comes from the “mass-energy” all over the cosmos. True his thought experiment of the spinning(accelerating) disk did show that the acceleration,and hence by the POE that the gravitational field, could be see as the “curvature of the spacetime”. So that in essence the acceleration can also be seen as equivalent to the geometric “curvature” . BUT WHY should the acceleration(the “curvature” or non-Euclidean geometry of spacetime)” ever be related to the “mass-energy” distribution in the spacetime?
    yours truly
    Francis Ferrara

    • Yes, even the mass which is following a geodesic gives its own contribution in manipulating the curvature because of its mass, and hence to keep equations simple we consider “negligible” mass or point mass whose effects are so small that they can be ignored while studying the effects of a given mass like say Sun or Earth. But yes, in practice every mass makes its contribution to the curvature. If we study the effect of Earth on moon, its not just that of earth, but also of the mass of moon itself, which is not negligible compared to that of Earth!

      Acceleration is related to mass-energy distribution because it is in the first place CAUSED by the mass-energy distribution. If there was no mass-energy, then there would be no curvature, and hence no acceleration at all. Consider an magnet which deviates the path of charged particles moving in its vicinity causing their velocity to change ie causing acceleration. Now if this magnet (similar to mass-energy) was absent, then the charged particle would never have had deviated from its flight path (ie it would not have had acceleration)!

  6. Dear Mr.Gurudev
    Motion along a curved trajectory in a 3 D space does make us feel a force(called a ficticious force by Newtons lights) BUT I am not sure that this curved motion causes the force, no to me it is inertia which causes the force. This force is the resistance of the mass to the change in velocity along the curved trajectory(if there were no resistance I don’t think there would be any so called ficticious force). When there is so called inertial motion that is motion in a st line(on a geodesic of extreme(min in this case) of distance) we have no force.So when we have natural motion along a 4 D world line which is a geodesic(max of time in this case) of spacetime we have no force.
    either
    Francis

    • Yes you are absolutely right that this fictitious force is actually due to inertia and is caused because of the resistance to natural trajectory. This is also the reason as you said that we dont feel this force in geodesic motion. Many a times it looks like physicists unnecessarily complicate things by creating different avatars of the same effect. Things which could be explained in simple terms are given complex names and complex definitions :)

  7. Dear Mr.Gurudev
    I think the basic problem I have with relativity is that it is a whole new way to look at the world,and I am trying to get to it from the little I know about Newtonian classical mechanics,throwing out some old concepts because relativity explains some observations and reconciles some discrepancies that Einstein realized could be reconciled IF certain other concepts were accepted. The trouble is that I am no Einstein and the jumps he made I can’t make. I have been told that the idea of Newton that force causes acceleration is really just as mysterious as curvature causing aceleration,it is only that we have gotten use to Newtons claim but not to Einsteins. Real science is Just as much about human creativity as it is about logical connections(probably even more so).
    Francis

    • Yes you are right. What relativity or newtonian mechanics tells is about what is observed. It does not tell us HOW that what is observed takes place. Like Newton said Sun attracts Earth, but HOW does Sun attract Earth, what exactly is the medium of this action was not described by Newton. He only reported what he observed. Similarly Einstein further refined it and came to the conclusion that space-time is curved. But he could not define WHAT EXACTLY is the medium for this, what exactly is curved? Humanity as a whole is trying to solve the mystery of Universe, one step at a time, first report observations, create a law, see if it passes or fails, apply the law (semiconductors, electronics, computers etc) and in parallel try to understand HOW it works (string theory?)

  8. Dear Mr. Gurudev
    IF “curvature” causes change in velocity THEN I agree with you that “it” also causes acceleration, BUT I don’t see why “curvature”(the non-Euclidean geometric structure of spacetime) should have anything to do with velocity no less cause it to change.
    Francis

    • Well, what is velocity? Distance traveled (space) divided by Time taken (time). And what is curvature? The geometry of space and time. So curvature is what defines velocity of bodies in free motion ie bodies moving about freely in the universe like galaxies, stars, falling stone, apple, satellites etc

  9. Dear Mr.Gurudev
    I understand how the spinning disk thought experiment relates acceleration to “curvature”,and that from this spinning disk experiment that acceleration implies and causes “curvature” of the spacetime which the accelerating frame coordinatizes,BUT I can not see how “curvature” causes acceleration? I guess this is the “grip” that J.Wheeler talks about in his explanation of the two sides of GR
    1.spacetime “grips” mass and tells it how to move.
    2.mass “grips” spacetime telling it how to curve.
    Wheeker also talks about “the boundry of a boundry being zero” as the principle which explains both but I dont even know what this principle is suppose to mean.
    yours truly
    Francis

    • How curvature causes acceleration? Well, that one should be quite easy. How do we FEEL curvature or REALIZE curvature? planets moving around Sun, ball falling back to earth, stars moving around galactic center etc. All these involve change in velocity (which is a vector) – which is nothing but acceleration. In ideal flat space-time velocity remains constant. Curved space-time causes velocity to change, the steeper the curvature, the faster the change in velocity – and this change in velocity is nothing but acceleration.

  10. Dear Mr.Gurudev
    Are the accelerational(non-inertial) terms which seem to be created by the acceleration of a frame really due to the “curvature” of the spacetime when coordinatized by the frame that is accelerating(is the non-inertial frame)??
    I had always thought that these accelerational(non-inertial) terms came about simply because the mass, which has its motion referred to the accelerating frame, possesses inertia and therefore to move in non-inertial motion(with the acceleration of the reflected frame a force MUST(according to Newton) come into existence. But I guess that according to Einstein “curvature ” will do this according to
    var[ds^2]=0, and NO force is necessary.

    PS Alot of what I write I write for myself to clarify things in my mind.
    Francis

    • My understanding of this is as follows:

      Acceleration and the force we feel depend on the source of the acceleration.

      No force will be felt if we follow the natural space-time tracks ie curvature – like in a freefall, even though we will be accelerating wrt the source of the curvature (say Earth).
      Force will be felt in the frame only when we resist the space-time track and not follow it, like live on the surface of the earth instead of free falling towards its center makes us feel the force in the form of our weight, similarly artificially inducing acceleration like say in a space-ship makes us feel the same force. So in my view, the entire business of inertia is a result of offering resistance to the natural space-time tracks.

      Of course, then again, if we are in free fall towards a blackhole, we will still feel the force (even though in freefall) simply because the curvature difference between our feet and head itself is so high that it will stretch us out like a noodle. But the reason here to feel this force is totally different, which is that the curvature is so high near a blackhole that the falling body is required to accelerate at different rates at different parts of the body’s length, as a result of which the body stretches out.

  11. Dear Mr.Gurudev
    While the Einstein Field Eq. is all about the “curvature”(non-Euclidean geometry) of the spacetime continuum,How does the “curvature” relate to and implement the basic fact of GR that the form of the laws should remain the same and unchanged even when accelerational(non-inertial)terms are taken into account, when the laws are referred to the accelerating frames?
    It would seem that the accelerational terms(which come into existence when we refer the laws to the accelerational frames) would go to necessarily change the forms of the laws when referred to the non-inertial(accelerating)reference frames.
    The only answer I have been abke to come up with is that the principle of Equivalence is able to have the gravitational field cancell out these accelerationhal terms that otherwise would change the form of the law. BUT what does this have to do with “curvature”???
    yours truly
    Francis

    • The principle of equivalence was what Einstein started off with because without that no science would be effective. If the laws change in different frames of reference depending on their motion or gravity, then science would become a complete mess. Now gravity causes motion, and curvature causes acceleration. If gravity is very weak and can be ignored, then in such frames curvature has no impact and we can simplify the frame to SR. In non-accelerating frames, the component of acceleration becomes zero, the laws do not change however. The effects of acceleration and gravitation are similar, because the very means by which gravitations displays its effect is via acceleration – remember g – acceleration due to gravity. An Einstein’s thought experiment would make this clear. Suppose your have a space ship slightly above the surface of the earth, and if it has to be stationary there wrt earth’s surface, its rockets would have to fire out exactly to cause an acceleration of 9.8ms-2 so that it cancels out earth’s downward pull. Now from within this spaceship if all its doors and windows are closed, there is no way for one to tell whether the space ship has its rockets fired or is it standing still on earth. If you have a interstellar trip, and cause your space ship to be accelerating at the rate of 9.8ms-2 then you will feel the same effects as on earth. So in summary it is the curvature which causes acceleration, and it is the acceleration which can give you the effects of curvature. If your space ship accelerates at the speed of light, then you can have a black hole formed at its rear end!

  12. Dear Mr.Guredev
    I should also start looking at the second edition of Exploring Black Holes whic E.f.Taylor and Berschwingwer are writing(see Taylors web site)
    Francis

  13. Dear Mr. Gurudev
    I have some speculation about this last question ,and then I,ll try to get back to the subject of the Field Equation in your article(especially the differential geometry that these tensor arrays discribe about the curvature of the mannifolds)
    ——-
    Maybe I could get away from the fields having to depend on mass by changing GR from being about accelerating (non-inertial frames which give rise to non-inertial accelerational so called ficticious terms and instead having the form of the law be invariant to something other than motion,
    (that is NOT have it extended from uniform motion of SR to accelerated non-inertial frames for GR) since SR is a totally ficticious world that does not exist. So why should I try to extend a non existent world to the reak world of GR(non-inertial accelerated frames.
    Maybe the thing which we should try to say the form of the laws are invariant(co-variant )to should be types of spacetime transformations,and the prin of EQUIVALENCE should NOT be between accelerational(non-inertial) terms and the gravitational field terms so that they can cancell out the accelerational terms that otherwise would change the form of the law so that now it remains unchanged. BUT maybe the EQUIVALENCE should be between something all fields possess and these spacetime transformations that keep the spacetime interval(given interms of the metric functions) the same in all frames not just motional ones.
    Let me stop here since i really don;t know what I am talking about. But do you think this is all wet?

    I think I should start looking at Edwin Taylors and J Wheelers first edition of Exploring Black Holes mwhich is on line at E.F.Taylors web site(eftaylor.com)
    yours truly
    Francis

    • Yes, to start with the world of SR does not exists out there, unless and until we are ready to ignore minor effects of gravity, say in deep space far away from any massive bodies and if we are ready to confine to an area where effects of gravity are very small. The very reason we can have SR as an approximation is because of all the four fundamental forces of nature, gravity is the weakest and can be successfully ignored in many situations. SR exists because, one would not want to solve complex tensor equations for simple problems. Same holds good about Newton’s law of gravity too, which holds good for most situations on earth, and we dont want to calculate complex tensor equations just to get negligible differences in results in some 10th or 12th decimal place.

      However, principle of equivalence is an universal principle and is the very reason for the existence of physical sciences. Its an obvious assumption we always make in all cases, but was explicitly states in relativity by Einstein to make sure it holds good. When some physicist discovers some new law in his lab, we never question whether that law applies in a space ship traveling at half the speed of light, instead we ASSUME that it holds good there too. That is because we believe that the universe does not have different set of laws in different places. But that does not mean we are right. So the principle of equivalence only tries to ensure that our assumptions are correct.

      Consider this hypothetical scenario. Tomorrow if we happen to find a frame where principle of equivalence does not hold good, then we would require another new theory which would describe about WHEN this deviation takes place and by HOW MUCH. And then we would further complicate our tensor equations by introducing additional components to include this deviation. And then we would be probably saying that in most cases if we ignore the difference in equivalence, then all our present laws hold good, and in other cases we would require this new theory.

    • Yes you are correct in saying that the principle of equivalence holds good for all frames and is independent of SR or GR and is a universal law without which all our existing scientific understanding about physical laws would break down.

  14. Dear Mr. Gurudev
    I have some speculation about this last question ,and then I,ll try to get back to the subject of the Field Equation in your article(especially the differential geometry that these tensor arrays discribe about the curvature of the mannifolds)
    ——-
    Maybe I could get away from the fields havung to depend on mass by changing GR frome being about accelerating (no0n-inertial frames which give rise to non-inertial accelerational spo called ficticious terms and instead havung the form of the law be invariant to something other than motionDear Mr. Gurudev
    I don’t think the following is based on your article but it is a question which has always bothered me. How come “the time dilation“ of SR seems to be totally relative,while the gravitational time dilation of GR seems to be REAL(in the sense that a clock in a strong grav field clocks out time slow while one in a weak field seems to run fast, when even in GR the laws will keep the same form in the accelerating frame AS LONG AS we take into account the gravitational field terms.
    There are things in relativity that I don’t understand,and I realize that this always will be with me, But there also are things which I just don’t feel right about and these bother me a lot more(feeling wins hands down against reason all the time,and I think it should)–one of these has to do with gravitational time dilation as opposed to SR “time dilation”. While the first looks like its real, that in SR is totally relative(just like uniform motion is), that is in SR “there is no really about it”. The “dilated “ time is just as real,good,correct and right as the so called proper time where the clock that clocks out this time is “at rest” relative to the frame this time is referred to wherears the “dilated” time is clocked out by a clock that is moving(uniformly) relative to the frame the time is referred to.
    ———————-
    The main problem I have with General Relativity is that it sees general motion(in particular accelerated motion) as relative ONLY when we also take into consideration the gravitational field so that its terms, when taken as equivalent to the accelerational non-inertial terms, can be considered to “cancel out”(in a local region) these accelerational terms which otherwise would change the form of the laws when they are referred to the accelerating frame, and taken into account in the form of the law. So that the ONLY way we can extend the principle of SR to the accelerating frames is to ALSO consider the gravitational field terms AS EQUIVALENT in every way.
    With SR I have no such problem because uniform motion(to which SR is limited) is truly totally relative and means something ONLY in relation to some other frame,otherwise it is MEANINGLESS.
    Could my solution be NOT to believe so strongly in SR(but uniform motion seems so nice) but to realize that the whole system of SR is just a ficticious world that does NOT exist and that I should NOT think that I can get to the real world of GR by simply extending this fiction.
    Maybe the basic principle of GR should NOT be “that the laws form be the same and unchanged in accelerating(non-inertial) systems” as was true in the ficticious case of SR.
    For the principle of EQUIVALENCE can do this ONLY for the gravitational field(which is mass dependent) and in the real world their are other fields which DON”T have a principle of EQUIVALENCE with accelerational(non-inertial,mass dependent) terms in them.So maybe what we need is some Equivalence which is good for all fields(not just the gravitational). But the accelerational terms of non-inertial frames seems to require that the field be dependent on mass like the gravitational field is. How could I get away from this dependence on mass in both the accelerational non-inertial frame and the field called gravity?

    Maybe the fact that accelerational motion “really” slows down the time(clocked out by a clock) in a strong gravitational field(large curvature) as opposed to the time running fast(clocked out by a clock in a weak field(small curvature)) should not be questionable simply because in SR there is no “really” about time dilation since all uniform motion is totally relative and has no meaning otherwise.
    yours truly
    (that is NOT have it extended from uniform motipon of SR to accelerated non-inertial frames for GR) since SR is a totally ficticious world that does not exist. so why should I try to extend a non existent world to the reak world of GR(non-inertial accelerated frames.
    Maybe the thing which we should try to say the form of the laws are invariant(co-variant )to should be types of spacetime transformations,and the prin of EQUIVALENCE should NOT be between accelerational(non-inertial) terms and the gravitational field terms so that they can cancell out the accelerational terms that otherwise would change the form of the law so that now it remains unchanged. BUT maybe the EQUIVALENCEn should be between something all fields possess and these spacetime transformations that keep the spacetime interval(given interms of the metric functions) the same in all frames not just motional ones.
    Let me stop here since i really don;t know what I am talking about. But do you think this is all wet?

    I think I should start looking at Edwin Taylors and J Wheelers first edition of Exploring Black Holes mwhich is on line at E.F.Taylors web site(eftaylor.com)
    yours truly
    Francis

  15. Dear Mr. Gurudev
    Didn’t realize you were away,hope all is well in the new year.
    —————
    I don’t think the following is based on your article but it is a question which has always bothered me. How come “the time dilation“ of SR seems to be totally relative,while the gravitational time dilation of GR seems to be REAL(in the sense that a clock in a strong grav field clocks out time slow while one in a weak field seems to run fast, when even in GR the laws will keep the same form in the accelerating frame AS LONG AS we take into account the gravitational field terms.
    There are things in relativity that I don’t understand,and I realize that this always will be with me, But there also are things which I just don’t feel right about and these bother me a lot more(feeling wins hands down against reason all the time,and I think it should)–one of these has to do with gravitational time dilation as opposed to SR “time dilation”. While the first looks like its real, that in SR is totally relative(just like uniform motion is), that is in SR “there is no really about it”. The “dilated “ time is just as real,good,correct and right as the so called proper time where the clock that clocks out this time is “at rest” relative to the frame this time is referred to wherears the “dilated” time is clocked out by a clock that is moving(uniformly) relative to the frame the time is referred to.
    ———————-
    The main problem I have with General Relativity is that it sees general motion(in particular accelerated motion) as relative ONLY when we also take into consideration the gravitational field so that its terms, when taken as equivalent to the accelerational non-inertial terms, can be considered to “cancel out”(in a local region) these accelerational terms which otherwise would change the form of the laws when they are referred to the accelerating frame, and taken into account in the form of the law. So that the ONLY way we can extend the principle of SR to the accelerating frames is to ALSO consider the gravitational field terms AS EQUIVALENT in every way.
    With SR I have no such problem because uniform motion(to which SR is limited) is truly totally relative and means something ONLY in relation to some other frame,otherwise it is MEANINGLESS.
    Could my solution be NOT to believe so strongly in SR(but uniform motion seems so nice) but to realize that the whole system of SR is just a ficticious world that does NOT exist and that I should NOT think that I can get to the real world of GR by simply extending this fiction.
    Maybe the basic principle of GR should NOT be “that the laws form be the same and unchanged in accelerating(non-inertial) systems” as was true in the ficticious case of SR.
    For the principle of EQUIVALENCE can do this ONLY for the gravitational field(which is mass dependent) and in the real world their are other fields which DON”T have a principle of EQUIVALENCE with accelerational(non-inertial,mass dependent) terms in them.So maybe what we need is some Equivalence which is good for all fields(not just the gravitational). But the accelerational terms of non-inertial frames seems to require that the field be dependent on mass like the gravitational field is. How could I get away from this dependence on mass in both the accelerational non-inertial frame and the field called gravity?

    Maybe the fact that accelerational motion “really” slows down the time(clocked out by a clock) in a strong gravitational field(large curvature) as opposed to the time running fast(clocked out by a clock in a weak field(small curvature)) should not be questionable simply because in SR there is no “really” about time dilation since all uniform motion is totally relative and has no meaning otherwise.
    yours truly
    Francis

  16. Dear Mr. Gurudev
    1 Why is the “volume” as measured from outside the soccer ball the “volume” element of the flat Euclidean space?
    2 I don’t understand why the scalar defining the intrinsic curvature of the curved mannifold shoul be considered to be measured from within the soccer ball element,to me all thay matters is that it is measured in the curved mannifold(on the spheres surface in the 2D case)
    your truly
    francis

  17. Dear Mr. Gurudev
    I now see that for a given radius(distance)the area of a circle in a positively curved manifold like a sphers suface embedded in a flat Euclidean space is less than the area of a circle(of the same radius) in the flat Euclidean embedding space. We can see this because to flatten out the sperical circle(of 2D on the spheres surface) onto a flat plane(2D)of the embedding space we can only do this if we add some small triangular areas and also some circumference,since the circumference of the circle on the sphers surface is is also less than that of the circles circumference when drawn in the flat Euclidean space. This is true because for the same circumference(of the same radius circle either on the sphers surface or in the flat Euclidean space),the ratio of Cir/2.radius will be less than Pi for the circle drawn on the positively curved surface than in the flat space since the radius in this case will be greater for the sphere than for the flat Euclideasn space.
    yours truly
    Francis

  18. Dear Mr.Gurudev
    Could I say that the Ricci tensor in the field eqs, defines the deviation of the n=2 dimensional area of the curved space away from the area of a flat Euclidean space(it is embedded in)?
    Does the Ricci tensor array define the amount of departure from the Euclidean flat geometry in terms of the are in a curved spacetimeaway from that in a flat space?
    I want to drop from n=4 dimensions of spacetime to n=2 dimensions of a bent spherical surface embedded in a flat 3 space since I can visualize this. But I really don’t know what Gauss and Riemann did for two dimensions let alone 4. yours truly
    Francis

  19. Dear Mr.Gurudev
    What does the stress-energy array describe? Does this array describe the way “mass(momentum)-energy” is distributed in the spacetime continuum,OR should we say that this array describes the way momentum-energy is distributed to form the momenergy continuum cooresponding to the spacetime? Could we say that the “curved spacetime” is equivalent(proportional) to the momenergy continuum of the events?
    Francis

  20. The physical reason that “mass-energy” curves spacetime(that is causes the geometric structure of the ST to become non-Euclidean and not flat) is because this “curvature” can be seen as due(identical) to the quantization of the mass-energy of the so called graviton(gravitational field) quanta.
    Can the electromagnetic field be seen as the “curvature” of something other than spacetime ,or maybe as a property of spacetime other than “curvature”(geometry)? And is there somekind of causal relation between the quantization called the photon in the e7m field and this something(or property)?
    yours truly
    Francis

  21. What I am trying to say in all these comments about the sperical surface(which I can visualize) is that the geometry on its surface is Not Euclidean since the ratio of the circumference of a circle on it to the diameter(2.radius) on it is not Pi but is less than Pi since on the surface the radius is grater than the radius in flat space. But in this case I find that A>Pi R^2 even though I was told that the curvature of a sphere is positive. So that in analogy to what you said in your article I would have expected that
    A<Pi R^2
    As you can see I'm all mixed up even in the new year.
    Francis

  22. Dear Mr. Gurudev One of the messages I wrote you even I did not understand when I re read it so I tried to reword it a little.Here it is

    Dear Prof. Gurudev
    In the section on the Ricci tensor array while you talked in terms of an n=3 Dimensional mannifold where the Ricci tensor describes the deviation of the geometric structures geometry of the curved spacetime away from the Euclidean geometry interms of the volume of the curved spacetime(and where a pos deviation means a volume less than a Euclidean volume.
    I have seen curvature talked about only for a 2 D surface which I could picture as a spherical shape embedded in a flat space with non-Euclidean geometry upon its “surface”. In this case would the Ricci tensor then describe the deviation of the area in the curved spacetime away from the area of a flat Euclidean mannifold? For the case of the 2 D sphere embedded in a flat spacetime I can see that in this curved “surface” that the ratio of Circumference to Diameter is not equal to Pi but is less than Pi since the diameter on the sphere is greater than the Euclidean diameter.,But I am NOT capable of translating this to what you talk about(namely that the volume is less for a pos value of curvature.
    For the 2 D sphere embedded in a flat 3 space which I can visualize as curved(having non-Euclidean geometry on its “surface”–in this mannifold). Where I can see that C/DD_E, and I was told that the curvature for a such a sperica; surface was poss,so to keep with what I think your analogy should be taken as I would get A<Pi.R^2 and this doesn't jibe for on the spherical surface for C/DPi R^2. I really don’t have much faith in what I have tried to say and it may be all wet and almost certainly is.
    Can you tell me a good,simple ,elementary explanation about curvature which Gauss and Reimann studied. in 2 D it all comes down to DEFINITIONS I think that are given in terms of the osculating circles(in particular thei radius of curvature 1/R.
    Francis

  23. A geodesic path in a curved spacetime is a path(world line) of an extreme of “interval” between two events on the path through the spacetime continuum. In 3 D ut is an minimum of distance but in a 4 D spacetime it is a path of maximum ageing(time).
    Francis

  24. Dear Mr.Gurudev
    In the picture that I sent the grav.field on the left is the new true gravitational field which is the old so called Tidal field(of relative accelerations of freely falling bodies.By the “Big Idea” this Tidal field is seen as the “curvature” of spacetime. Am I right in saying that the motion of two test bodies in a large spacetime region of a falling elevator will have the bodies move along geodesics of the region and while the old Newtonian force will be transformed away,the bodies will have small accelerational terms(Tidal terms).
    Francis

  25. Dear Mr. Gurudev
    The idea that in GR the gravitational field intensity bends time and slows its flow is very troubling to me. It brings me back to the idea of time dilation in SR but here in SR time dilation is TOTALLY relative since uniform motion is indistinguishable and so both observers see the other observers clock clocking out time differently but neither is more correct ,real or right than the other. Both see(measure) the others time to be clocked out slower(and dilated)BUT this is just my problem because in GR the observer who is in a stronger gravitational field has a clokc which goes slower while the other observer who is in a weaker field has a clock whic really goes faster. If one sees the clock in the strong field going slow the other sees the clock in the weak field going fast,But this bothers me becaus arnt SR and GR based on the same principle namely that the form of the laws are the same in all motional frames weather uniformly moving OR accelerated???
    In SR I can see that the time dilation is reciprocal since the motion is TOTALLY relative and it is MEANINGLESS to claim one observer is “at rest” while the other is “moving”. Both have equal right to claim they are in motion and its the other fellow who is “at rest” ,(uniform motion is totally relative)and its the relation of this relative motion to the Minkowskian(flat) “spacetime” which gives rise to the difference in the way clocks “clock out” time with regard to different uniformly moving frames. In GR the spacetime continuum geometry may be non-Euclidean but is this a powerful enough change as to make the structure of time to REALLY(and physically to slow down in its flow?? Shiould we even consider time AS a flow??
    Francis

  26. Dear Mr. Gurudev
    Hope you got the email with the picture.

    Could the idea that the gravitational field is caused by the exchange of quantized bosons called virtual gravitons be the reason why mass-energy causes the spacetime to curve so that the test body will then move through this curved spacetime along a geodesic according to var[int[ds^2]]=0 where the g_ij( ) solutions to the 16PDE are contained in the interval so that the test body moves according to
    var[int[g_ij()dx_i dx_j]-0?
    If so I don;t know enough physics or have enough smarts to even glimps how this happens, does anyone?
    I have heard that out of superstring theory of the graviton mode of resonance of the string exchange at low energy,come the equations of the curvature of spacetime that GR talks about. Maybe the idea of how photon exchange causes e&m light waves in the electromagnetic field will show us how the exchange of graviton string bosons will cause gravity waves and maybe gravity itself–the curvature of the spacetime seen as the gravitational field.
    Francis

  27. Dear Mr. Gurudev
    I really appreciate your taking time to explain all these things about GR. Your respons is really quite out of the ordinary,and you should take pride in your effort which I am sad to say may not be fully realized in me,but I do appreciate it very much.
    I will try to scan and send you what I think is a picture of GR.
    Francis

  28. Dear Prof. Gurudev
    In the section on the Ricci tensor array while you talked in terms of an n=3 Dimensional mannifold where the Ricci tensor describes the deviation of the geometric structures geometry)of the curved spacetime away from the Euclidean geometry interms of the volume of the curved spacetime(and where a pos deviation means a volume less than a Euclidean volume.
    I have seen curvature talked about only for a 2 D surface I could picture as a spherical shape with non-Euclidean geometry upon its surface. In this case would the Ricci tensor then describe the deviation of the area in the curved spacetime away from the area of a flat Euclidean mannifold? For the case of the 2 D sphere embedded in a flat spacetime I can see that in this curved “surface” that the ratio of Circumference to Diameter is not equal to Pi but is less than Pi since the diameter on the sprere is greater than the Euclidean diameter.,But I am NOT capable of translating this to what you talk about.

    For the 2 D sphere embedded in a flat 3 space which I can visualize as curved(having non-Euclidean geometry on its “surface”–in this mannifold). Where I can see that C/DPi.R^2 and to keep with what I think your analogy should be taken as I would get A<Pi.R^2 and this doesn't jibe. I really don't have much faith in what I have tried to say and it may be all wet and almost certainly is.

    Can you tell me a good,simple ,elementary explanation about curvature which Gauss and Reimann studied. in 2 D it all comes down to DEFINITIONS I think that are given in terms of the osculating circles(in particular thei radius of curvature 1/R.
    Francis

  29. Dear Prof. Gurudev

    Why do we say that the tidal accelerational terms in a freely falling(accelerating) frame–that these relative accelerations of the gravitational field– show(reflect)the deformation(curvature) of the spacetime continuum? I thik we say this because the gravitational field can be seen as the curvature of spacetime.

    But a problem I have is that whiloe spacetime is 4 Dimensional(and I can’t visualize this) the relative tidal accelerationjs are only 3 Dimensional and I dont really know how we can have a correspondence between things with unlike dimensionalities.
    Francis

    • Well, I am not a Prof as such, just a lot more curious about our universe trying to find answers to questions like who am I, what is this universe, where did all this come from, what is our purpose etc :)

      Regarding the time part of the space-time question, it is one of the most fundamental aspect of relativity which most students miss out initially. Time also gets curved and bends just like space, and in reality its space-time out there, even though we are used only to space being affected by gravity. The way gravity or mass affects time is that time slows down as the space-time curvature increases. So the time part of the curvature during free fall is as the freely falling frame approaches a massive object (say Sun or Black hole or even Earth) the rate of flow of time slows down. For instance, suppose we consider that at a distance very far away from a massive star if we count 10 seconds in the local frame, then during the same time period only 9 seconds would have gone by at a frame which is on the surface of that star. This is just an example. Gravity not only curves space, but also time – so its space-time together that needs to be taken into consideration.

  30. Dear Mr. Gurudev
    I will accept thae fact that Einstein simply postulated that mass-energy ‘curves’ spacetime and gives it a non-Euclidean geometric structure,and I realize that we can never know the reason of an intuitive insight(guess), but do we even have a smidgen of a guess of what led him to this insight?
    How could he have thought that mass-energy was in any way connected to spacetime? True his 1905 paper on the photon as “quantized energy” that was quantized at a definite location in space for very high frequency was, as he said, revolutionary. So that it did have something to do with space and time(and Einstein never liked this relationship even when it was given the name of entanglement and was well defined by Schroedinger).
    The idea of the “lack of separability” as the reason for entanglement may lead people who understand these things to the reason WHY and HOW mass-energy curves spacetime.
    yours truly
    Francis

    • Einstein did not postulate simply that mass-energy curves spacetime. Infact he was first not at all dealing with gravity when he started with special relativity. Once he wanted to extend special relativity to include gravitation, it meant investigating into the effects of gravity on special relativity which is the effects of gravity on space-time. And he already had Newton’s laws in place about gravitation. So it was obvious that gravity caused the flat space-time of special relativity to become curved, but the exact nature of this complex relation had to be worked out and only Einstein could resolve it. With a brilliant thought he discovered that gravity was the geometry of space-time and not some force, which is were the picture of person in freefall not feeling the force of acceleration comes in. The rule is its not that whether you are accelerating or not that matters, it is whether you are following the space-time curvature or not. We on earth are always resisting the space-time curvature here and so always feel the resistance as our weight which is the pull by earth which we always keep resisting by pushing our feet against the earth.

      Infact even Einstein got the first set of equations of general relativity (which he published in 1911) wrong since he had left out the impact of gravity on space! Only in the second attempt he got them right! Isn’t it strange that initially in 1911 Einstein only talked about the impact of gravity on time and had observed that gravity will slow down time, but he got the numbers only half correct. He had to think for another four years to get the equations right.

  31. Dear Mr. Gurudev
    I have a “big picture” of GR expressed in the form of a diagrame of sorts but I don’t know how to draw it on the computer,would you be able to give me an address where I can send it so you can tell me what is right and what is wrong about it?
    It contains the the idea that 1.spacetime is a structure that can become curved(described by tensor arrays) by the mass-energy density, and vise-versa in a “back and forth” interchange subject to the 2.POE. This shows that acceleration is the connector between the gravitational field on one side(POE)and the curvature(non-Euclidean geometry) on the other(the spinning disk). So maybe what I should be concerned with is not the physical reason of HOW mass-energy curves spacetime or HOW it cause accelerastion but HOW “mass-energy” is connected to the “gravitational field” in some kind of relation that exists for some reason. But what this reason is you feel is contained more in Quantum theory than Relativity.
    Francis

    • Well, I dont consider myself an expert in the subject, just share my thoughts and knowledge :) You can probably get it scanned and mail it to me at itzguru(AT)gmail.com
      When we say that mass-energy curves space-time, we have to see what in the mass causes this curvature. Finally it comes down to atoms and molecules within the mass and that is the jurisdiction of quantum mechanics. Finally the curvature is because of the nucleus inside an atom because that is where most of the mass-energy lies. In case of neutron stars it is be the closely packed neutrons, and in case of blackholes it is because of a immensely dense nucleus called the singularity. So if we can figure out how nucleus curves space-time around it, probably which is also the reason why electrons move around it, and I guess this is where we can also unify gravity with other forces like electro-magnetic, weak and strong nuclear forces. Nobody knows if electrons experience force of acceleration while revolving at high speed around nucleus. Also at this micro level, space-time in my opinion gets quantized ie is available in chunks and unlike the GR continuum and I guess resolving that would gives the grand unified theory. The space-time continuum at larger scales of GR is because of the relatively large scale of observational field compared to the fluctuation scale which is in terms of planck’s length is what I guess.

  32. How does the Quantum theory illuminate the physical nature of spacetime for us. What principle in Quantum theory will show us HOW mass-energy curves the spacetime continuum when above the {lanck scale but destroyes this continuity when at or below the Planck scale. Its not that the uncertainty doesnt exist untill we get below the Planck scale for it is seen well in advance (at the scale of atoms even)
    Francis

    • There is this attempt in the form of Quantum Gravity theory which tries to unite GR with Quantum Mechanics which is basically an attempt of trying to explain GR from a quantum mechanical point of view. And here Gravity is said to be caused due to the exchange of quantum particles called gravitons, just like the way virtual photon exchange causes electromagnetic force. However this is largely still a work under progress. Graviton is not yet detected, but is at the core of effort to unify GR and quantum mechanics, and if true implies that gravity itself is quantized!

  33. Dear Mr. Gurudev
    I have re read your answer to me of Dec.27.2010 at 8;46 am,and am not sure what you meant in the third paragraph from the end.
    ———————–
    Coming back to the HOW question, if we observe the evolution of GR we find that Einstein set out with the need to do away with the requirement for the existence of ether when he started with special relativity and succeeded in explaining the constancy of the speed of light. But finally when GR was completed we see that ether is nothing but the space-time itself! There is nothing like pure vacuum because space-time exists everywhere. And it cannot be curved unless and until it is physical is what is my interpretation. But the physical nature of the space-time is extremely flexible and has more to do with quantum mechanics than relativity is what I guess. If we are able to resolve how the space-time structure can be bent by mass-energy, I guess that would be the unifying principle for GR and QM.,
    HitXP » Demystifying Einstein’s Field Equations – by Gurudev

    What do you mean by the phrase, ”unless and until it is physical”? I think you mean that its curvature is meaningless UNLESS it is operationally DEFINED by some physical process. Is this so?
    Yours truly
    Francis

    • Yes, I mean space-time has to be something like what water is to fish. Fish swims in water, similarly mass-energy moves in spacetime. It will be illogical if we say that space is empty and at the same time say that it can be bent, we can’t bend something if it has nothing in it :)
      If earth has to follow a curve space-time around Sun, then that curved track should be present physically there, because earth is also a physical entity. So space-time HAS to be physical. How can time be physical? Time is the rate of flow of events at different points in space, and space cannot exist without the flow of events, and the flow of events are equally affected by the way space is affected and hence we have space-time.

  34. Dear Mr.Gurudev
    What is the tensor array that occurs on the RHS of the Einstein field eq. which you call the
    “stress(momentum)-energy” density array [T_ij( )],

    ([R_ij()-1/2.R[g_ij()])=(8piG/c^4)[T_ij()]

    as you say the array[T_ij()] represents the
    “momentum(mass)-energy” at each point distributed over all the ij combinations(IS THIS THE CASE?)

    Isn;t there some kind of relation between the [R_ij()] components and the components of the metric tensor array,[g_ij()]? would it be something like R_ij()
    6 g_ij( )/6( )??

    • In the simplest of the cases and in most of the cases the momentum-energy component will be zero. The RHS comes into picture only when you are trying to solve the equation at a point say inside the core of a star, or say below the Earth’s surface. So it gets related to metric tensor array only in such scenarios.

  35. Dear Mr. Gurudev
    Maybe, as you say,the physical reason of HOW
    “mass-energy” curves “spacetime”(bring about its non-Euclidean geometric structure),will be explained only in the future as the principle that unifies GR with Quantum theory. Maybe I should not worry about HOW(physically) mass-energy causes spacetime to curve, maybe what I should consider is an equivalent problem, namely the relationship between “mass-energy” and that of accelerated motion, since acceleration is Equivalent to the “curvature” by the principle of Equivalence.
    My question then becomes HOW is acceleration related to(and caused by) “mass-energy”

    • We cannot consider the relation between mass-energy and acceleration without considering the impact on spacetime. The way acceleration becomes related to massenergy is as follows.
      1. mass-energy curves space-time
      2. The curvature increases as we approach a massive body
      3. The curvature is nearly flat in space-time far away from any massive body
      4. Which means that there is acceleration as you move nearer to a massive body because of the constantly increasing curvature
      For instance in case of free fall towards earth this is the standard g ie acceleration due to gravity which indicates that as we approach the surface of the earth the local spacetime curvature is increasing at a rate of 9.8ms-2

  36. From your article we see that the complete picture of General Relativity is this “back and forth non-linear dance” between “mass-energy”(of the momenergy continuum) and the “curvature”(non-Euclidean geometry) of the spacetime continuums structure. I was once told that Einstein and Infeld showed that contained within the Field Equations were the geodesic equations that determined the way mass moved(along a geodesic) in the curved spacetime(the gravitational field).
    This self referential “back and forth-flow” may be the glory of GR but I think it is also its downfall(I really don’t know what I mean but feel that its non-linearity is trouble!
    Francis

    • Yes, this is exactly the reason why GR breaks down at singularity, because here curvature becomes infinite, time stops and space ceases to exist and GR doesn’t knows what is happening there :)

  37. Dear Mr. Gurudev
    IF the “curvature” at a point, in general, is given by the “mass-energy” all over the cosmos(the distribution array),HOW COME the radius R from which e&m light waves cannot move outward from because of the curvature AT this radius, has a radius value which is ONLY dependent on the mass M of the star within this radius and not by all the other “mass-energy” distributed all over the cosmos?
    R=2GM/c^2

    Francis

    • Now that is a very good question. The reason the equation does not include other mass-energy impact is because they are assumed to be negligible, if you see the derivation of this equation, the very assumption starts with only considering the impact of the mass M in question. In other words, it is assuming that impact of any other mass-energy is negligible. If we were to apply this equation to say a binary pulsar system which have almost collapsed into each other, then we need to consider the mass M as the sum of the masses of both the stars in the binary system because neither of them is negligible in this case!

  38. Dear Mr.Gurudev
    I don’t think this “claim” was mentioned in your article but for the last two years, off and on, I have been looking at the book “Exploring Black Holes” that E.F.Taylor and Prof. Bertschwanger are writing as a second edition to the one Taylor co-authored with J.Wheeler(1st edition)–see the website eftaylor.com on General Relativity.–
    I can’t even say that this “claim of mine” is right since I have gotten it out of what I think was said by Prof.Taylor but I am sure I twisted it. Well in any case when I wrote him I was always putting emphasis on General Relativity being all about acceleration. I think he asked me one time why I was so obcessed with acceleration(I had not even realize that it was linear acceleration only). I think he told me that the POE and linear acceleration did go together but that General Relativity was even more general than simply acceleration. But somehow I got the idea that in its most essential elements GR was really all about “spacetime transformations” that kept the laws of physics co-variant(the same in all frames of reference) weather they were motional(accelerated) frames or not.So that we COULD use frame other than motional ones,ones where the spacetime twisted and turned stretched and contracted BUT that the spacetime could NOT tear and cut or be pasted back together as in topology. Is this the realm of quantum gravity that Wheller and Prof.Brian Greene writes about in his “Fabric of The Cosmos”??

    write you next year
    Francis

    • Yes, at a technical level GR is all about porting from one frame to the other, but when we start attempting to experience general relativity we cannot miss out acceleration. Yes space-time at the quantum level can literally be cut, pasted twisted and turned, many a times it sounds really scary, but I guess it is constantly happening even inside our bodies all the time, after all almost our entire body gets completely recycled (at a chemical level) every few months or so :)
      The question is, still we remain who we are, so are we just the information pattern stored in our brains, and in that case do we still continue to exist or say continue to live, if all our information snapshot in the brain is ported to say a similarly structured neural system simulated by computer hardware? Dont think so, probably this is where soul and mind comes into picture. After all modern science is NOT COMPLETE SCIENCE, nor can we guarantee that whatever we think as science today is actually ‘true’. Even ptolemy said their science was true, so did the proponents of earlier static universe, so did Newton about absolute space and absolute time. 100 years from now, parts of our today’s scientific facts may enter these ‘old science’ pages, who knows!

  39. Dear Mr. Gurudev
    Is the spatial curvature AT a given point caused by the mass-energy AT that point,and also by all the
    mass-energy in the universe,that is by its distribution all over the cosmos? So that there are two sources of the non-Euclideanism(curvature) the first being the mass-energy AT the local region of spacetime in question,and the second being the distribution all over the cosmos at other spacetime “POINTS”.
    The twist of GR(it is a background independent theory) is that the very mass-energy that follows the geodesic of the curved spacetime causes the very curvature that determines the motion by var{int[g_ij dx_i dx_j]=0,and so does the mass-energy distributioin all over the cosmos.
    Why does the matter(mass-energy) of the stress(momentum)-energy tensor array distribution] tell spacetime how to curve”[take on a non-Euclidean geometric structure] according to the co-variant–(same form of law in all frames of reference,all coordinate systems)–tensor array eq.
    ([R_ij]-1/2.R[g_ij( )]=8PiG/c^4)[T_ij( )]
    The 16PDE are defined by corresponding ij terms
    Francis

  40. Dear Mr. Gurudev
    I have looked over all your answers and only partially digested them,but feel satisfied. The only outstanding question I still have is the one about quantum theory and its influence on spacetimes continuity. However I think I would do well to read over your article more closely.
    In the specific case of the Black Hole I really dont understand using relativistic ideas why the Schwartzchile radius is R=2GM/c^2,doesn’t it have something to do with the “curvature” AT this radius which is such as NOT to allow light to propagate outwards(when it travels along the geodesic of the curved spacetime AT that radius? But If this is so why does the radius depend only on the mass M inside the radius and NOT on all the mass-energy in the cosmos as the curvature at a point usually does in GR??
    Francis

    PS–I guess I should stop for a while and go back and try to put things in GR together for myself.(your article above looks like a good place to start)

  41. Dear Mr. Gurudev
    As you say the physical nature of “spacetime” probably has more to do with Quantum Theory than with relativity. Your statement was, “IF we were able to resolve HOW the spacetime continuum is “curved”(bent) by the mass-energy”, of the momenergy continuum,”that probably would be the unifying principle of GR and QM”(this would explain HOW they go together into one unified theory.
    Now I heard that J. Wheeler came to the conclusion that quantum fluctuations(of uncertainty) destroyed the continuity of the spacetime structure and resulted in it being a foamlike structure and NOT a continuum anymore. Didn’t Wheeler know why the continuity destruction occured because of the quantum fluctuations? I would think that Prof. Wheeler had some reason for feeling that these fluctuations destroyed the continuity of spacetime and maybe THIS would be the answer to WHY mass-energy curves spacetimeby its “grip”?(I really don’t know)
    yours truly
    Francis

  42. Dear Mr. Gurudev
    You mentioned that a body in free fall IS accelerating yet we do NOT feel this acceleration as a force because gravity is NOT a force but the acceleration of the body which occurs simpli because the body follows the curvature-tracks(the geodesic) in the local region. So thay in the local region the free fall frame IS an inertial frame. Now my problem is that in one sense I see this,since the spacetime in the local region is effectively flat,but in another way I do not understand hoew this can be. namel HOW can a body move in a st.line at constant speed IF the frame this motion is referred to is in free fall(accelerating),when it originally is moving in a st line at some constant speed relative to the original inertial frame k ,namely the inertial frame the free fall frame k’ is falling(accelerating)through?
    Francis

    PS I know these are a lot of questions and probably not even framed in a totaly meaningful way so dont feel rushed, Have a happy and well new year.
    Francis

  43. Dear Mr. Gurudev
    In your answere you say that spacetime and mass-energy are dependent on each other and one without the other is meaningless. Now J.Wheeler used the term “momenergy” continuum and I have alway wondered if the momenergy continuum is formed from momentum and energy in the same(or maybe similar) way that the spacetime continuum is formed from the coordinates of space and time,that is from the coordinatization of and event by momentum and energy instead of spatial and temporal coordinates? Could anh interval be defined tor this continuum “”between”” events nwhich would be DEFINED in terms of p and E coordinates the same way ds^2 is given by space and time coordinates as ds^2=c^2dt^2-dx^2. Where [d(icm)]^2-P^2-(E/c)^2=
    SUM SUM{T_ij( )dX_i dX_j} Are these T_ij functions the functions that occur in the Einstein Field Eq(on the RHS). I really don’t have any understanding of these, to me just, symbols but their are some similarities do they mean anything?
    Francis

  44. Dear Mr.Gurudev
    Why should “mass-energy” cause or determine(tell) spacetime to curve into a non-Euclidean structure?
    Wheeler called this a “grip” but I think that was very metaphorical to the concept of “bending” a 2 D surface
    by the “grip” of your hands. I don;t think gravity or accelerated motion can be looked at that literally, but have to be described more by mathematical descriptions. BUT IS THAT ALL PHYSICS IS JUST MATHEMATICAL DESCRIPTIONS? WHAT ABOUT PHYSICAL QUANTITIES AND IDEAS??
    Well in any case I dont see any connection between curvature(non-Euclidean structure) or acceleration and the concept of “mass-energy” of the momenergy continuum? yours truly Francis

    • To see why mass-energy tells space-time HOW to curve, we will analyze a few points.

      First the existence of space-time and mass-energy is dependent of each other. There can be no space-time without mass-energy and vice versa in an evolving universe. By an evolving universe I mean, the universe that came into existence after Big Bang. Prior to big bang, the universe existed only in the form of a singularity with infinite density and there was no space-time. Big bang created space-time and the mass-energy started living inside it in a distributed manner as galaxies, clusters, radiation etc.

      Newton formulated gravitation, but didn’t know HOW sun ATTRACTED earth or HOW apple KNEW earth was down there and NOT up above? His formula for gravitation derived based on observed calculations and his thought experiments worked though. But that could not explain for example, the precession of perihelion of mercury.

      But there had to be some means by which Earth FELT the existence of Sun a few million miles away? Later Einstein concluded through his thought experiments that instant action at a distance as suggested by Newton was NOT possible and that gravity BENDS the space-time around it which causes all masses to move the way they do. This answered Newton’s unanswered question as to HOW apple KNEW that the earth was down there. Apple doesn’t knows anything about the earth below. It only follows the local space-time curvature and that leads down to earth, just like the way a train simply follows its tracks. The train does not know where the tracks lead to or about the existence of any station nearby, it simply follows the local tracks and on a larger scale it looks like the train is moving towards the nearest station. The twist in GR though is that the tracks itself are CAUSED by the mere EXISTENCE of these stations . The presence of EARTH lays down tracks for nearby apples to fall towards it. The presence of SUN lays down tracks for nearby bodies like planets and comets to revolve around it and so on.

      Newton thought gravity was a force which meant that the body being attracted should FEEL the force. Einstein concluded that Gravity is NOT a force in that sense. This is where acceleration comes into picture. In Newton’s world, you HAVE TO feel the force while in acceleration. Not so in Einstein’s world, a body in free fall IS accelerating, but WON’T feel the force, and that is because gravity is not a force, but only local space-time tracks which bodies follow.

      So we see that Newton answered WHAT happens and solved problems like apple falling or earth moving around Sun.
      Einstein clarified WHY it happens and further solved problems like precession of mercury’s orbit, etc.
      Now the next logical question is what you asked which is HOW it happens. How does mass-energy bend space-time? The normal answer we receive is to compare it with a rubber sheet with a heavy ball in the middle of it which causes the sheet to bend at the center. Think of the rubber sheet at the space-time and see how mass-energy represented by the heavy ball bends it. So is space-time like the rubber sheet? Well, it’s much more than that. Some of the GR solutions tell us that space-time can be twisted, cut and pasted, about all those worm holes, time travel etc. All Star Treck stuff.

      Coming back to the HOW question, if we observe the evolution of GR we find that Einstein set out with the need to do away with the requirement for the existence of ether when he started with special relativity and succeeded in explaining the constancy of the speed of light. But finally when GR was completed we see that ether is nothing but the space-time itself! There is nothing like pure vacuum because space-time exists everywhere. And it cannot be curved unless and until it is physical is what is my interpretation. But the physical nature of the space-time is extremely flexible and has more to do with quantum mechanics than relativity is what I guess. If we are able to resolve how the space-time structure can be bent by mass-energy, I guess that would be the unifying principle for GR and QM.

      As for the answer to your question of HOW mass-energy CURVES space-time, nobody knows for sure yet. That is for the next unification theory to explain. Einstein himself only postulated it thinking “If we suppose that the presence of mass bends 4-D space-time, let’s see what this leads to” and like Newton arrived at his set of equations. Newton postulated about mass attracting mass, and Einstein created a better version of it. GR is about geometry of space-time and its relations to mass-energy. HOW that happens in the physical universe is for our future research :)

      Yes as you said, we must try not to get carried away too much by mere numbers and formulae. Mathematics gives us specific solutions to specific problems where we have a set of input numbers and a set of output numbers, but understanding HOW this manifestation takes place in nature is very important, else it won’t be possible for us to ask more questions and seek the knowledge beyond what we know today.

  45. Dear Mr.Gurudev
    In your answer to me of Dec 3,2010 at 4.58 PM, I did not understand what you meant when you said “the curvature is related to the mass-energy because the curvature itself, in the first place, is a product of mass energy”. Did you mean that curvature,in the first place, is given by mass multiplied by energy? Isn’t this “action” ; OR am I all wrong in interpreting your word product as the mathematical operation of multiplication, and that you were using it in a more general sense the way Wheeler used it ” mass energy tells spacetime how to curve”. This is the way I think I should look at it. BUT if so I still don’t see WHY mass-energy tells spacetime to curve and take on a non-Euclidean metric structure?
    I remember that Einstein came to the idea of curvature of spacetime in GR through the thought experiment of a rotating(accelerating) disk. And that he saw curvature as due to acceleration. But I am still baffled because I still dont see what acceleration has to do with mass energy? yours truly Francis

    • When I said “product of mass-energy”, I meant the “resultant curvature” in the general sense, not a mathematical multiplication. Please see my other answer for your question related to WHY mass-energy tells space-time how to curve.

  46. The article seems to say that each ij component of curvature in spacetime is caused by the corresponding component ij of the mass-energy density,so that each point in spacetime is caused to have curvature by the mass-energy AT that point in question. Now my question is isn’t this curvature also cause by all the other mass-energies in the cosmos(even those NOT at this point)?
    I would want to say yes and this is why GR has a Machian flavor to it. Is there any truth in what I have said or am I all wet?

    • Your interpretation is right. The article doesn’t imply the actual mass-energy PRESENT at that point because in that case the curvature outside a giant star just above its surface would then become almost zero which is NOT the case. It is the resultant curvature AT the point in question due to ALL surrounding mass-energy distribution which in theory spreads out all around the cosmos. Yes as you said this gives GR the universal composition of just not in space but also in time, because if we consider the influence of say a distant quasar on the curvature at Earth here TODAY, then it would be due to the mass-energy component of that quasar a few billion years AGO and NOT today. As we know space-time unification is at the heart of GR, for its not just what WE SEE moves back in time as we see further away in space, but it is what the SPACE-TIME around us also FEELS :)

  47. The array that occurs on the RHS of the tensor eq. is said to be the mass-energy density array that causes the spacetime to curve and become non-Euclidean in its geometric structure. In the book by Walter Issacson he says it is the mass -energy of the mass that moves along the geodesic of the curved spacetime. now I tassume these are the same thing BUT why shoulkd we think that the curvature is related in any way to the stress energy array.

    • Yes they are the same – and the curvature is related to the mass-energy because the curvature itself in the first place is a product of mass energy. The way Wheeler put it in his famous quote “Mass tells space-time how to curve, and space-time tells mass how to move”.

  48. I’m a 9th grade homeschool student doing a research report on Einsteins contributions to the Industrial Age one of which is Relativity/Field Equations and most websites were difficult for me to understand until I came across yours. Thanks for posting such a colorful easier explanation!

  49. Gurudev:
    Thanks for a very nice overall explanation of the Field Equations, and how to solve them. It is nice to have a guide when working though all the mathematics!
    Dave Ebert

  50. yes indeed! well written! Gurudev!
    The general Field equations are relatively simple in terms of mathematical complexity.. Einstein’s general field equations are so brilliant as if the nature was trying to adjust to fit his equations, rather equations do the opposite.
    John wheeler said “” The equations tell you how bodies curve space-time and curved space-time tell bodies how to move. The Max Born said the equations show Einstein’s Philosophical penetration ,Physical intuition and Mathematical skill is an unparallel in the
    history of science.
    These equations are pure brilliance of Einstien!!

  51. A friend of mine forwarded this link to me knowing very well that I am crazy about general relativity, and I must admit that I have not yet read any single scientist/physicist explaining the overly complicated math of general relativity in such a simple way, so that it could be very easily understood by an average physics student like me. You have done it.

    Great work Mr Junior Scientist! Thanks
    Pam

  52. This is a real master piece! A true master designer has created your brain. I still cannot believe that such a complicated general relativity can be written down with such a beautiful simplicity. If this was written in any book, that book would have been an instant hit. This article deserves to be published in all physics textbooks. Had you been there when Einstein published relativity, you would have been one of the very few people who understood relativity then.
    I was fortunate that a googling on general relativity got me here. Will read all your other articles. Keep up the great work.

  53. I have been thrashing around for several weeks trying to get a grasp on GR. This has been a great help.

    Thanks very much for a useful article.

    Regards, Nils

  54. Thanks a lot Paul for your comments..

    In fact what prompted me to write this article was that I really never found any book or article that explained general relativity to a common man or to a basic science student, in terms of the general relativistic equations. There were many books that explained GR in terms of its basic fundamentals and thought, but not any in terms of its mathematical equations…

    Yes, sure, will definitely write on the attempt to unify GR and Q Mech..

    Cheers
    Gurudev

  55. I’ve been trying to develop an “”intelligent non-specialists”” understanding of GR for about 5 years, by reading monograph after monograph by a lot of famous scientists. If I had read your work in beginning, it would have saved me a LOT of trouble. You write and explain excellently. So…

    What I would really, really look forward to, from you, would be a piece on the inherent difficulties or challenges ahead for scientist trying to unite GR and Quantum mechanics…to yield the long sought-after ‘quantum gravity theory’.

    Paul Chernabrow, York University, Toronto Canada

  56. HI itzguru,
    Nice article.
    The wave structure of matter in space provides a simple explanation. I wonder as to your thoughts on this.
    Thanks for a nice page – will add a link to it.
    Geoff Haselhurst

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