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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.

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