|
|
|||||
|
|||||
|
Demystifying Einstein’s Field Equations Let’s start with a
bang!! A Small Mistake 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.
One or Many? A great difficulty in solving the field equations is its non-linear nature. In quantum mechanics the Schrödinger'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.
Metric Tensor 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. Related LHS 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 :-) Just like the the metric tensor, the stress energy tensor is just a set of 10 numbers in 4D space-time.
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 mass of radius 5 kilometers and if we take the center of the mass 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 - 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! 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! Cosmological Constant 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. Complete Picture So we have a complete picture now. Field equations tell the way
'Matter tells space-time how to curve' 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' - by Gurudev
|
|||||
| Rate this article | |||||
| Comment on this article: Criticism is most welcomed | |||||
|
Home - Physics - Chemistry - Mathematics - Bio-Sciences - Computers |
|||||
Home
