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Coordinate transformation   |   Gravitational waves  |   Differential Equation
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"There was a young lady named Bright,
Whose speed was faster than light,
She set out one day,
in a relative way and
returned on the previous night"

 equation of light

theory of general relativity and theory of relativity equation

General theory of relativity is the generalization of special theory of relativity. It states that laws of physics should be same for all observers whether they are moving at constant speed or accelerating. In special relativity the transformation is linear in coordinates but in General relativity the coordinates transformation is non-linear. The distance between two points in spacetime becomes what is called Riemannian metric. Riemannian metric is the generalization of Gauss's first fundamental form to higher dimensional space- 4 dimensional spacetime of relativity for our purpose. Riemannian metric is defined using metric tensor. Consequently everything becomes generalized in general relativity: straight line becomes geodesics, derivative becomes covariant derivative and many other entities as are required to formulate general relativity.

General theory of relativity has a larger sweep than any other theories that have been developed so far. A certain kind of admixture of geometry and physics had been done in general theory. Geometry which was assumed a constant feature of the world is no more constant according to this new theory. Gravity is not a force according to Einstein but curvature of spacetime. To gain further insight about the theory certain abstruse(very complicated) mathematics need to be pursued and understood. There are few fundamental concepts which can be elucidated(explain) in apparently simple way:

general theory of relativity, this page is all about general relativity

Metric tensor and curved space

metric tensor

Take a flat piece of rubber sheet. The sheet is flat in the sense that the distance between two points corresponds to phythagoras's law. So you lay down the coordinates for each point and you can calculate distance exactly according to Pythagoras's law. Now stretch and twist the rubber sheet without tearing or glueing it with other. Now the coordinates will move away from each other with the amount of distance which must differ from point to point. Equal stretch will not give equal distance. The usual law of Pythagoras will no longer hold. You now have a curved surface. This is Einstein's general theory of relativity!!!

In general theory of relativity it is assumed that events have four dimensional order or an event is assigned four real numbers to be uniquely represented. This amounts to say that the number of events is the number of Cantorian continumm. It is the number 2 ^ℵ(0) where ℵ is the number of all integers. the number 2 ^ℵ(0) is the second transfinite cardinal. Every class of 2 ^ℵ(0) terms in the field of various multiple relations among them arrange the class in a n-dimensional continuum or four-dimensional continumm for our purpose. But we require a little more than this. Among all the ways of arranging the points with multiple relations, there is only one that serve our purpose. All others are just convenient for pure mathematical purpose.

Metric tensor allows one to measure distance between two points on a manifold or Euclidean space. Metric tensor is rank two tensor. So it is a matrix and transforms like a general tensor. More precisely, it takes two tangent vectors and returns a scalar. Tangent vectors are like instantaneous velocity that we find by differentiating position of a particle. It has a magnitude as well as direction. So it is a function of two vectors , which returns a single number. In special relativity it has special form, which contains 1, 1, 1, -1 in its diagonal elements. In general relativity it is called Riemannian metric which components are generally functions of coordinates. These coordinates are called gaussian coordinates. Gaussian coordinates can be assigned only on a curved space , which can be transformed to Gallilian coordinates in suffuciently small region. Metric tensor in First fundamnetal form has 2X2 = 4 components. First fundamental form relates the infintesimal distance or line element on a gaussian surface with the coordinate differential. It has the form :

metric tensor

du and dv are coordinates differentials. E, F and G are functions of coordinates u and v , which is calculated from two tangent vectors r(u) and r(v). Diffential geometry is the branch of mathmeatics, which deals with such kind of curves surface and spaces. It actually describes how geometry changes or varies from point to point. All the concepts of general relativity relies on differential geometry. A long ang lengthy study is need to fully understand differential geometry. The basic ingredient is the metric tensor which characterize a particular geometry. Riemann generalized the notion of Gaussian surface to higher dimensions. Line element (ds) in more than two dimensions can be computed directly from it. Metric tensor used in general relativity is a symmetric 4X4 matrix. So it has a total of sixteen components. Due to symmetry sixteen components reduce to only ten components. In addition , metric tensor generalizes usual dot product of vectors in three dimensional space. A short mathematical illustration of metric tensor and its properties are given below:
general theory of relativity

The spacetime interval is written in terms of metric tensor in quadratic form of coordinate differential. The components of metric tensor as shown, depend on coordinate but the value of interval ds(squared) is an invariant under general coordinate transformation. The matrix notation of metric tensor is also depicted. In flat geometry the metric reduces to Minkowsky metric(eta[n]).

general theory of relativity, this page is all about general relativity

general theory of relativity

Metric tensor transforms like a general rank two tensor, which can be derived using flat metric and coordinate transformation as shown is first equation is second figure. Covariant metric tensor transforms like convariant tensors whereas contravariant metric tensor transforms like contra-variant tensors. The spacetime interval defined in terms of metric tensor is so fundamental that everything in general relativity can be derived using it. Metric tensor is called the fundamental tensor. It can be used to raise or lower index of any tensor.

metric tensor identity

metric tensor identity

If we multiply two metric tensor of type covaraint and contravariant , we get a unity matrix called Kronecker delta. Kronecker delta is also a matrix but its diagonal elements are all 1.
So far all have been plain sailing. It could not be much simpler. Metric tensor involve coordinates and Its derivatives can tell whether the space is flat or curved. If all the derivatives up to second order vanish then the space-time is flat. Otherwise the spacetime is curved. We can always choose a coordinate transformation such that the first derivative of metric tensor vanishes at some point. In other words the metric tensor becomes stationary there. Sufficiently small region surrounding that point will be flat and theory of special relativity can be applied. How small will the region be will depend on the distribution of matter and energy in space. From the definition of manifold it is apparent that such will be the case. But a formal proof can be given. But in general we can not make a coordinate transformation such that first and second derivative of metric tensor can all vanish throughout the whole spacetime unless the Riemann curvature tensor is zero everywhere. This is the mathematical underpinning of Einstein happiest thought which led him discover that gravity is the same thing as acceleration. His happiest thought was that if you jump from the roof top of your building, you will feel weightless as if no gravity were present. Gravity can be transformed away ( at least in a small region ).

Christoffel Symbol

General theory of relativity

Christoffel symbol is 3X3X3 array of numbers. That means we can arrange the numbers in three dimensional grid , indexed by three indices. Christoffel symbol is a combination of first derivatives of metric tensor. It is the parameter that gives rise to connection. Connection is the relation of nearby tangent vectors that live in the spacetime manifold. All the tangents at a point in spacetime form tangent space. Riemann curvature tensor contains Christofel symbols and their first derivative. The connection defined by Christoffel symbol is the covariant derivative operator. In curved spacetime it is harder to distinguish vectors in different points because the basis vectors always changes from point to point. We simply can not compare vectors at different points because of this. So connection is defined to relate vectors between two points. Connection enables us measure how a vector changes from place to place in spacetime manifold. Covariant derivative is defined as follows:

covariant derivative

The nabla(∇ ) is the derivative operator. The (∇ U) here is actually directional derivative, which is evaluated at the direction of vector U. When it is applied to basis vector the changes are only the basis vector multiplied by Christoffel symbol(with indices placed correctly). A operator maps one space of vectors to another space of vectors. In quantum mechanics operator performs same mapping from one space of quantum states to another space of quantum states. When nabla(∇) is applied to a vector other than the basis vector the situation becomes somewhat different. It decomposes the vector into two components. When we take out basis vector from the expression , we get the covariant derivative of the original vector( shown inside first bracket). It can be further simplified and we can get covariant derivative of a vector as a sum of an ordinary derivative component and Christoffel symbol coupled component. It is the last equation of the above figure. The essence of covariant derivative is that it transforms as tensor while the ordinary derivative does not. Now we have necessary tools to explain Riemann curvature tensor.

general theory of relativity, this page is all about general relativity

Geodesic equation in General Relativity is partial differential equation . It is actually a system of several separate equations. Physical significance of the equation is
Differential geodesic equation

that freely falling particle follows a path which is called geodesic. The whole relativity theory is concerned with how geodesics are shaped in the vicinity of matter. Geodesic is the Newton's equation of motion in curved spacetime. The symbol "Γ" with three indices involves the partial derivative of metric tensor. Thus geodesic equation is a partial differential equation. The derivation of the geodesic equation can be done in several ways. In four dimension of spacetime, the geodesic equation is a system of four differential equations.The geodesic equation is valid in any reference frame under general coordinate transformation. But The symbol "Γ" called Christoffel-symbols is not a tensor itself. If the metric tensor has the properties of a flat spacetime, the symbol "Γ" becomes zero. Thus the above equation reduces to Newton's equation of motion. The spacetime around sun is curved by the present of mass of the sun. All the planets follow geodesic in their course around the orbits. Earth's relativistic path satisfies the geodesic equation too. Relativistic path is all the points in spacetime that the earth traces out when she orbits the sun. That is if we solve the above equation we will get four equations of coordinate variable x, y, x and t, which will be function of some parameter. The parameterized equation will define the path of the Earth in the same way we can parameterize a circle in two dimensions with parameter angle(θ). In relativity theory the parameter is the proper time (τ). Proper time is invariant quantity which is the same for all observers. It may be defined as the time order of events in the neighborhood of a body. It is a temporal relationship whereas the interval in spacetime is a spatio-temporal relation between events. The proper time , expressed mathematically, is :

General theory of relativity
proper time
proper time

In the very first equation as in the figure the proper time (τ) multiplied by c is assumed to be equal to spacetime interval (ds) as the proper time is time-like interval. Time-like interval is the real valued space-time interval(ds>0) when an observer can travel between two events. Corresponding proper time in Swardchild coordinate is defined. Thus proper time becomes total space-time interval ( the integration of right hand side of the first equation, divided by light speed c). After doing some algebraic manipulation we arrive at the equation which says proper time is integral of coordinate time multiplied by inverse of Lorenzt factor. If we differentiate this equation with respect to t, we get proper time as the coordinate time divided by Lorenzt factor. Coordinate time is the proper time multiplied by Lorenzt factor, which is the same thing, mathematically. Whichever coordinate frame is chosen the value of coordinate time multiplied by Lorenzt factor remains unchanged. Proper time decreases as the speed of the observer increases. As he travels at the speed of light his time comes to a halt. This is so called time dilatation of special relativity. Proper time has a similar expression in general relativity perspective. Gravity slows down time rate of clock. Gravity, as we will see , is , nothing but curvature of space and time. Although the mathematics is too esoteric(hard), general relativity is considered the most beautiful theory in physics. It has many applications as well. There may be many branch of physics which lies outside the scope of general relativity but there is no part of physics which, to some degree, is not related to general relativity.
Some equations are derived using variational principle. The variational principle is very analogous to least action principle which can be defined as the action quantity that should be minimized. In case of space-time distance the action is the metric distance itself. So when we take variation of metric ds^2, it must be set zero.
General theory of relativity

Parallel transport and Riemann curvature tensor

The parallel transport is the displacement of a vector such that its covariant derivative vanishes. So what can be physical interpretation when we want it to apply in 4-dimesional spacetime manifold? When we displace a vector along a path such that the vector does not change it direction , the notion of such displacement is called parallel transport. When we transport a vector in such way in Euclidean plane around a closed circuit , the resultant vector comes to initial point unchanged. That means the direction between them remain the same. But when we do that in curved spactime the resultant vector does not maintain its original direction. Here is a such a portrait on the surface on a sphere

parallel transport

A vector at the point C is first transported along CA to point A so that it stays parallel to itself along the way. Then from point A to B it is again transported parallely to point B. Finally the vector in transported parallely from point B to point C to its initial position. But the direction is now changed. It is now pointing all the way to the right along the equator making an angle 90 degree to the initial vector. This change of direction is caused by the curvature of the sphere, which is not Euclidean. So we can come to conclusion that any such manifold should have a curvature at each point. In spacetime this curvature is captured by Riemann curvature tensor which we now going to derive .

 theory of relativity general

First we take double covariant derivative of vector v in the order from 2 to 1. A simple use of the formula of covariant derivative can be applied twice and we should get somewhat complicated expression involving derivative of the vector and Christoffel symbol. Now we reverse the order and compute the new double covariant derivative. Something similar is now apparent. Finally we take the difference to calculate the change of vector along the route on which the derivative was taken. The reason for taking double covariant derivative is that we need to find difference of difference of the vector around the loop. When the vector completes the loop , vector changes along two paths which are also separated by a path of the same parallelogram creating the loop. Why we need parallelogram to define curvature? Eddington used it when deriving the Riemann curvature tensor. He used it as an assertion of the fact that parallelogram is the absolute structure to compare dispacement. End result is the components that are in a parenthesis , multiplied by the vector and tensor product of three basis vectors. Do not worry about the tensor product now. It is the definition of tensor product (⊗) which is used here. We are dealing with three basis vector : one for index v, one for index 1 and one for index 2. So we can expand the components of the tensor using these three basis vectors. The final result can be put in this way:
parallel transport

The combination of the terms in the parenthesis can be grouped by a general tensor quantity of rank 4. This tensor is called Riemann curvature tensor which has 256 components in total if we replace indices 1 and 2 by alpha (α) and beta(β), each of which can have 4 values as other two indices ( u, sigma). Due to symmetry and other conditions 256 components reduces to only 21 which can represent all the component of spacetime curvature at a single event. We can contract the Riemann tensor by taking trace of it. Taking trace of a tensor over two indices reduces its rank by 2. For example if we take trace of a matrix we will get only a single value as the summation of its diagonal elements. So if we take the trace of Riemann tensor over its two indices we get a tensor of rank two. This is called Ricci tensor.

ricci tensor

The trace is taken over two indices of one contranvariant index and one covariant index. The resultant ricci tensor still contains derivative of Christoffel symbol and its products. Ricci tensor measures the extent to which the volume of a geodesic ball in curved spacetime deviates from that in a Euclidean space. Ricci tensor is used in the Einstein field equation. If we take trace of ricci tensor again we get ricci scalar which assigns a single number at every event in spacetime. We have almost everything to cover Einstein field equations. One last entity is needed before we go to explain field equation. That is the Energy-momentum tensor. It is the piece that is needed inevitably to make field equation significant. Let us now review it.
Energy momentum tensor encodes energy density , momentum density and mechanical stress-energy components. As we see it is a rank two tensor. It is symmetric in components above and below the diagonal elements. The equation that is used to calculate those elements of the matrix is T(a,b) = p.(du/ds)(du/ds). When non-interacting particles are considered the coordinate density of the particles is p = p(rest)(ds/dt)(sqaured). The reason for the squaring the quantity ds/dt is that one accounts for increase of mass and one accounts for decrease of volume.

energy momentum tensor

The energy flux is the flow of energy (t-component) through the surface of constant x(β). We have labelled the components of the matrix by t(0) and (β = 1,2,3) for space elements. The momentum density is represented by last three elements of the first column of the matrix, which is flow of momentum per unit volume. The mechanical components of the energy momentum tensor is the stress tensor which have both shear and normal stress components. Normal stress is the pressure and shear stress is the momentum density. If we look at the equation that generates the matrix the pressure(p = F/A) is the same as density multiplied by velocity squared. On the other hand the shear stress is the flow of some alpha(α) component of momentum through a surface of some constant beta(β). The first component is the energy density. It is the time-time (t-t) component. It is the mass density which implies energy in spacetime physics.
Then we have the energy momentum tensor for perfect fluid. The decomposed components are only pressure and density caused by the fluid. The equation is shown below.

 theory of relativity general
That is the final entity that we needed to establish Einstein filed equation. As it needs some more explanation , it is done in a separate page.

Geodesic Deviation

Geodesic equation describes how two nearby geodesics get separated as the curvature changes from point to point. Two test bodies falling freely under the influence of gravity will be moving away or towards each other as their coordinates change. This is analogous to tidal force that the earth experiences due to the attraction of moon. The deviation vector (ζ) will be related to the four velocity V of the particle in the following way:
geodesic deviation
Where δ is the kronecker delta matrix. Geodesic equation can be interpreted in another way which is equivalent to above formulation .
general theory of relativity
V(a) and V(b) are tangent vector or coordinate vector fields but ζ (a) is a separation vector.

To sum up the above lengthy discussion is to say that gravity is not a force but it is space time curvature. The planets are in inertial frames which follows straight lines in spacetime. As the space-time is curved , they orbit around the sun. Force can be seen as the deviation of world geometry from Euclidean geometry. And last of all, space-time is a four dimensional manifold which looks like Euclidean space locally.

Field Equation

Field equation of Einstein's General theory of relativity is perhaps one of the greatest feats ever done by human intellect. It reveals certain structure of the physical world, which has great impact on philosophy and science. Gravity can be easily explained using the field equation as a curvature of spacetime. It is a tensor equation and so valid in any reference frame. Although field equation is harder to solve , many exact solutions had been found over a century.


Force in Newtonian framework is the gradient of a gravitational field (φ).
metric tensor

The Laplacian (∇) of the field is proportional to mass density. From the perspective of special relativity, the equation suffers a fatal flaw: if there is a change in the mass density rho, then that must propagate everywhere instantaneously. Let us start with the Hilbert Action :

hilbert action

The square root of the determinant of the metric is a part of the volume element. That is required so the volume element can be in curved spacetime. It plays a vital role in the derivation, so I wish I had a better handle on why that factor in that form is required so that the differential volume element transforms like a tensor.

Now vary with the metric tensor g(uv)
hilbert action

Now Pull back the factor of the square root of the metric and use the product rule on the term with the Ricci scalar R:
hilbert action

Focus on the first term, using the definition of a Ricci scalar as a contraction of the Ricci tensor:

hilbert action

A total derivative does not make a contribution to the variation of the functional, so can be ignored in our quest to find an extremum value. This is Stokes theorem in action.
Show that the variation in the Ricci tensor is a total derivative. Since I don't understand this all in detail, I will try to get you in the neighborhood of getting it. SB1. Start with the Riemann curvature tensor:

rieman curvature tensor

Lots of stuff there, but here is a simplifying viewpoint. One is comparing two paths, that is why there is a subtraction here. The two paths are found by switching the order of the mu and the nu. This is a really complex structure, but that should be obvious :-)
SB2: Vary the Riemann curvature tensor with respect to the metric tensor:

rieman curvature tensor

Lots of terms, but remember the mu <-> nu exchange comprises for half of them.
One cannot take a covariant derivative of a connection since it does not transform like a tensor. Apparently the difference of two connections does transform like a tensor. I say "apparently" because this is an example where I have to rely on authority, I don't appreciate the details.
SB3: Calculate the covariant derivative of the variation of the connection:
Notice that the third terms of these two expressions are identical because the mu and nu are neighbors in the connection. Again, this is a step whose details I don't understand enough to clarify should others have questions.
SB4: Rewrite the variation of the Riemann curvature tensor as the difference of two covariant derivatives of the variation of the connection written in step SB3.

ricci curvature tensor

This now looks to my eye like a total derivative, so will not contribute to the action.
Since that was such a long sidebar, what has been done is the first of three terms in the variation is the Ricci tensor. 5. Focus on evaluating the variation of the second term in the action. Transform the coordinate system to one where the metric is diagonal and use the product rule:

product rule metric tensor

Notice there was a flip of the metric in the variation which required one more sign change. That is the kind of detail I always carry on.
6. Define the stress energy tensor as the third term:

product rule metric tensor

That factor of a minus a half? I don't get it. Bet it comes out of some classical limit. Hopefully I can research that later in the week.
7. The variation of the Hilbert action will be at an extremum when the integrand is equal to zero:
product rule metric tensor

Fini. But not fini. This was a math exercise. Note how little physics was involved. There are a huge number of physics issues one could go into. As an example, these equations bind to particles with integral spin which is good for bosons, but there are quite a few fermions that also participate in gravity. To include those, one can consider the metric and the connection to be independent of each other. That is the Palatini approach.
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experimental test

More than a century after Albert Einstein proposed it, his theory of general relativity has passed another test. With giant telescopes pointed at the centre of our galaxy, a team of European researchers observed a fast-moving star that got close to a monstrous black hole. They saw that the black hole distorted the light waves from the star in a way that agrees with Einstein’s theory. The result was reported overnight in the journal Astronomy & Astrophysics. Einstein’s theory says the fabric of the universe is not simply space, but a more complex entity called space-time, which is warped by the presence of heavy objects. Black holes offer a good opportunity to test that idea. The one that lies at the heart of the Milky Way is 4 million times as massive as our sun.

general theory of relativity

“I, just like every physicist in the world, would have loved to finally see a fissure in Einstein’s relativity,” said Ohio State University astrophysicist Paul Sutter. “But he’s outsmarted us.” But confirming Einstein’s work — again, “feels like we’re kind of beating a dead horse,” said Sutter, who wasn’t part of the research team led by Reinhard Genzel of the Max Planck Institute for Extraterrestrial Physics in Garching, Germany. Scientists know that the theory still doesn’t reveal everything about the universe. So they keep testing it time and again. So far, nobody has been able to overthrow it. Although the effects of general relativity have been observed before, this was the first detection made by observing the motion of a star near a supermassive black hole. “To me, that’s what makes this so cool,” said Clifford Will, a University of Florida physicist who did not participate in the research. Will wills his colleagues will be able to discover stars even closer to the black hole, where the effects of relativity would be stronger. This finding “is really the opening episode,” he said. “The future, I think, is going to be very exciting.”

Black Hole

Black hole

The most common way to form a black hole in the Universe is to have a massive star reach the end of its life and explode in a catastrophic supernova. However, while the outside portions of the star are blown apart, the innermost core collapses, forming a black hole if the progenitor star is massive enough. But most real stars, including our Sun, are spinning. Therefore — since angular momentum is always conserved — they shouldn't be able to collapse down to a single point. How does this all work? That's what our Patreon supporter Aaron Weiss wants to know, asking: How is angular momentum conserved when stars collapse to black holes? What does it mean for a black hole to spin? What is actually spinning? How can a singularity spin? Is there a "speed limit" to this spin rate and how does the spin affect the size of the event horizon and the area immediately around it? These are all good questions. Let's find out.

Black hole

When Einstein first put forth his theory of gravity, General Relativity, he forged an inseparable link between spacetime, which represents the fabric of our Universe, and all the matter and energy present within it. What we experience as gravity was simply the curvature of space, and the way that matter and energy responded to that curvature as they moved through spacetime. Matter and energy tell spacetime how to curve, and that curved space tells matter and energy how to move. Almost immediately, Einstein recognized that this picture came along with a strange consequence that was difficult to reconcile with the Universe we have: a matter-filled Universe was unstable. If you had, on average, space that was filled with a uniform amount of stationary matter — no matter the shape, size, or amount — it would inevitably collapse to form a perfectly spherical black hole.

Black hole

Once you get matter with a sufficient amount of mass confined to a small enough volume, an event horizon will form at a particular location. A spherical region of space, whose radius is defined by the quantity of mass inside of it, will experience such extreme curvature that anything passing interior to its boundary will be unable to escape. Outside of this event horizon, it will appear as though there is just an extreme region where gravity is very intense, but no light or matter can be emitted from within it. To anything that falls inside, however, it inevitably gets brought towards the very center of this black hole: towards a singularity. While the laws of physics go hay wire at this point — some physicists cheekily refer to singularities as places where "God divided by zero" — no one doubts that all the matter and radiation that passes inside the event horizon heads towards this point-like region of space.

Black hole

I can hear the complain already. After all, there are a legitimate number of ways the actual Universe works differently from this naive picture of gravitational collapse. The gravitational force isn't the only one in the Universe: nuclear forces and electromagnetism play a role when it comes to matter and energy, too. Black holes aren't formed from the collapse of a uniform distribution of matter, but rather by the imploding of a massive star's core when nuclear fusion can no longer sustain. And, perhaps most importantly, all stars we've ever discovered spin, and angular momentum is always conserved, so black holes should be spinning, too. So let's do it: let's go from the realm of a simplistic approximation to a more realistic picture of how black holes truly work.

All stars spin. Our Sun, a relatively slow rotator, completes a full 360° turn on timescales ranging from 25 to 33 days, depending on which particular solar latitude you're monitoring. But our Sun is huge and very low-density, and there are far more extreme objects in the Universe in terms of small physical sizes and large masses. Just as a spinning figure skater speeds up when they bring their arms and legs in, astrophysical masses rotate more quickly if you decrease their radius. If the Sun were a white dwarf — with the same mass but the physical size of Earth — it would rotate once every 4 minutes. If it became a neutron star — with the same mass but a radius of 20 km — it would rotate once every 2.4 milliseconds: consistent with what we observe for the fastest pulsars.

Black hole

Well, if our star (or any star) collapsed down to a black hole, we'd still have to conserve total angular momentum. When something spins in this Universe, there's no way to just get rid of it, the same way you can't create or destroy energy or momentum. It has to go somewhere. When any collection of matter collapses down to a radius smaller than the radius of an event horizon, that angular momentum is trapped inside there, too.
This is okay! Einstein estableshed his theory of General Relativity in 1915, and it was only a few months later that Karl Schwarzschild found the first exact solution: for a point mass, the same as a spherical black hole. The next step in modeling this problem in a more realistic fashion — to consider what if the black hole also possesses angular momentum, instead of mass alone — wasn't solved until Roy Kerr found the exact solution in 1963.

Black hole

There are some fundamental and important differences between the more naive, simpler Schwarzschild solution and the more realistic, complex Kerr solution. In no particular order, here are some fascinating contrasts: 1.Instead of a single solution for where the event horizon is, a rotating black hole has two mathematical solutions: an inner and and outer event horizon.
2.Outside of even the outer event horizon, there is a place known as the ergosphere, where space itself is dragged around at a rotational speed equal to the speed of light, and particles falling in there experience enormous accelerations.
3.There is a maximum ratio of angular momentum to mass that is allowed; if there is too much angular momentum, the black hole will radiate that energy away (via gravitational radiation) until it's below that limit.
4. And, perhaps most fascinatingly, the singularity at the black hole's center is no longer a point, but rather a 1-dimensional ring, where the radius of the ring is determined by the mass and angular momentum of the black hole.

neutron star

All of this is true for a rotating black hole from the instant you create the event horizon for the first time. A high-mass star can go supernova, where the spinning core implodes and collapses down to a black hole, and all of this will be true. In fact, there is even some hope that if a supernova goes off in our own local group, LIGO might be able to detect the gravitational waves from a rapidly rotating black hole's ringdown. If you form a black hole from a neutron star-neutron star merger or the direct collapse of a star or gas cloud, the same possibilities hold true. But once your black hole forms, its angular momentum can constantly change as new matter or material falls in. The size of the event horizon can increase, and the size of the singularity and ergosphere can grow or shrink depending on the angular momentum of the new material that gets added.

neutron star

This leads to some fascinating behavior that you might not expect. In the case of a non-rotating black hole, a particle of matter outside of it can rorate, escape, or fall inside, but will remain in the same plane. When a black hole rotates, however, it gets dragged around through all three dimensions, where it will fill a torus-like region surrounding the black hole's equator.
There's also an important contrast between a mathematical solution and a physical solution. If I told you I had the (square root of 4) oranges, you would conclude that I had 2 oranges. You could have just as easily decided, mathematically, that I had -2 oranges, because the square root of 4 could just as easily be -2 as it could be +2. But in physics, there's only one meaningful solution. As scientists have long noticed, though:
...you should not physically trust in the inner horizon or the inner ergosurface. Although they are certainly there as mathematical solutions of the exact vacuum Einstein equations, there are good physics reasons to suspect that the region at and inside the inner horizon, which can be shown to be a Cauchy horizon, is grossly unstable — even classically — and unlikely to form in any real astrophysical collapse.

neutron star

Now that we've finally observed a black hole's event horizon for the first time, owing to the incredible success of the Event Horizon Telescope, scientists have been able to compare their observations with theoretical predictions. By running a variety of simulations explaning what the signals of black holes with various masses, spins, orientations, and accreting matter flows would be, they've been able to come up with the best fit for what they saw. Although there are some substantial uncertainties, the black hole at the center of M87 appears to be:
rotating at 94% of its maximum speed, with a 1-dimensional ring singularity with a diameter of ~118 AU (larger than Pluto's orbit), with its rotational axis pointing away from Earth at ~17°, and that all of the observations are consistent with a Kerr (which is favored over a Schwarzschild) black hole.

black hole picture

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