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Coordinate transformation   |   Gravitational waves  |   Differential Equation

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Whose speed was faster than light,
She set out one day,
in a relative way and
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# general theory of relativity equation

Before explaining general theory of relativity some ideas and concepts of special theory of relativity are needed to be clarified. Special thoery of relativity unified space and time into a single framework known as spacetime. But spacetime only applies to non-accelerated motions in a stright line. Einstein showed that space and time are not absolute. They are dynamic and relative to the motion of the observers. The consequence was the unification of space and time.

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

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.

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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 :

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).

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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:

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]).

Metric tensor transforms like a general rank two tensor, which can be derived using flat metric and coordinate transformation as shown in first equation as given above. 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.

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.
The relation between flat and curved space can be represented with a diagram more easily:

Flat space on the left is transformed into curved space on the right. Coordinate lines (ζ) are now curved on the right. Two coordinate curves define a surface where another cooridate is constant. Here three dimensional case is depicted. Similar things will happen in four or higer dimensions.
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 ).

"The first rule of fight club is you do not talk about fight club , the second rule of fight club is you do not talk about fight club and the third and final rule of fight club is that if it is your first night at fight club you have to fight"

## Christoffel Symbol

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.

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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:

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.

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

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 :

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.

## 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

Parallelism on the sphere S2. Choose p at the north pole, with tangent vector y pointing along the Greenwich meridian. Which tangent vectors, at other points of S2, are we to regard to being ‘parallel’ to y? (a) The direct Euclidean notion of ‘parallel’, from the embedding of S2 in E3, does not work because (except along the meridian perpendicular to the Greenwich meridian) the parallel ys do not remain tangent to S2. (b) Remedy this, moving y parallel along a given curve γ, by continually projecting back to tangency with the sphere. (Think of g as made up of large number of tiny segments p0 p1, p1 p2, p2 p3 , . . . , projecting back at each stage. Then take the limit as the segments are made smaller and smaller.) This notion of parallel transport is indicated for the Greenwich meridian, but also for a general curve γ.

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 .

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:

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 20 which can represent all the components 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.

The trace is taken over two indices of one contravariant 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.
There are two types of curvatures that constitutes Riemann curvature. These are Ricci and Weyl curvature. Ricci curvature has 10 components and Weyl curvature has 10 components. So Riemann curvature has a total of 20 components.

When we see a distance star , two types of effects arise relating to space time curvature. One is magnification of the object we see and the other is distortion. Distortion is due to sell curvature and magnification happens due to ricci curvature.
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.

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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.

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:

Where δ is the kronecker delta matrix. Geodesic equation can be interpreted in another way which is equivalent to above formulation .

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.
You can view this video for explanation : ( In English)

In bengali

## 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.

# Derivation

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

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 :

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)

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:

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

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:

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:

Lots of terms, but remember the mu <-> nu exchange comprises for half of them.

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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.

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:

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:

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:

Finished. But not finished. 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.

### "space-time interval"

In relativity theory, distant spacetime points has only relations between them , which can be obtained from integration of the relation of distances between neigbouring points. As the distance is always finite , what we call relation is not really a relation but a kind of limit like velocity. Only the language of calculus can tell what this really means. What is apparent, like velocity, "the notion of interval" only tells what is tending to happen each moment. We do not know exactly what is going to happen because because any assigned point is reached something might cause a diversion. This is exactly the case with velocity. We can not tell , given the velocity of a body at a given instant, where the body will be at another instant. We need to know the velocity throughout the interval of time to correctly infer the path of the body. Similarly the interval formula characterizes each separate point of the space-time. To find interval between one point to another, however near we must specify a route and integrate along that route. However, we find that, the routes which are natural are called geodesics.

## Relativistic Lagrangian mechanics

Relativistic Lagrangian mechanics is the the Lagrangian mechanics applied in context with special and general theory of relativity. For a charged particle in electromagnetic field the Lagrangian is :

In general relativity there need some corrections to be made to the usual Newtonian motion :

Now some original manuscripts of Einstein will be discussed. Einstein published many paper prior to discovering special and general theory of relativity. Here are some of his handwritten manuscripts.

This is the manuscript where he calculated curl and pondermotive force.

## Mathematical relations rules the world of physics....

Here is the field equation written by Einstein himself.

Here is the manuscript of calculating Christoffel connection

Do share my page if you like it.

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### 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.

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.

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.

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.

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.

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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.

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.

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.

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 have 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.

## CTC and time travel

According to the principles of general relativity closed time like curve can exist in the universe. CTC means closed timelike curve. So any time-like path that is closed is called a CTC. So anyone who travels through this CTC will come back to its starting point. This may sound bizzare but theoretically it is possible. If our orbit on which the earth travels around the sun were a closed time-like curve we would come back to our past. In principle travelling to the past in possibe through time-like curve which is closed.
Godel's found a solution to Einstein's field equation , which permits this kind of weird time travelling.

This rotating universe is known Godel's universe. In this universe the light cone bends as the distance from the centre increases. At some critical distance the light cones tip over. Any observer can start from a point and follow a curve to come back from where he started. The solution was explicitly worked out by Godel

## Consequences of general theory of relativity

One of the major consequences of general theory of relativity is the precession of perihelion of mercury around the sun even for non-rotation massive body. The other effects are frame-dragging and geodetic precession. These three different kinds of effects modifies Newton's law of motion radically.

## Consequences of general theory of relativity

One of the major consequences of general theory of relativity is the precession of perihelion of mercury around the sun even for non-rotation massive body. The other effects are frame-dragging and geodetic precession. These three different kinds of effects modifies Newton's law of motion radically.

The precession of perihelion of mercury is not entirely due to frame-dragging. On the other geodetic precession and frame-dragging effects are analogous. The total change of gyroscope's angular momentum can be decomposed into two components: one is the frame dragging which pulls the gyroscope towards the massive body like earth and the other is the geodetic precession. A picture can make it clear.

Clocks at the satellite run faster than the clocks at earth by a tiny amount. This can be calculated precised using equations of general relativity.

Steller dynamics is explained by general relativity. The pressure and density of gas in the star contributes to self-gravity. On the other hand , the radiation from the star creates an outward pressure , which try to expand the star. When the outward pressure and inward pressure becomes equal the star reaches the state of equilibrium.

## Gravitational doppler effect

Motional doppler effect is

On the other hand gravitational doppler effect is

## Intersteller movie formula of general relativity

The theme of "Intersteller " movie has been based on general theory of relativity. Even many equations related to general relativity and brane cosmology have been explicitly shown inside it. The first equation to be mentioned is the first law of thermodynamics and its relationship with general relativity.

The second equation to be mentioned is evolution equation of branes

Lagrangian and action in brane's cosmology

Quantum gravity equations are also seen

Branes are generalization of point object to higher dimensions. In string theory there are higher dimensional object. For example a point may be called a zero dimensional object and strings are one-dimensional objects. P-branes are p-dimensional objects. Potential function looks like

More equations in brane cosmology

Gravity and wormhole equations

An observer on a brane will find specific set of physical effects and constants

The bulk metric on Brane's cosmos is defined as

dynamics of strings and branes is

Kerr black hole is a rotating black hole with the metric defined as

When a black hole rotates it drags spacetime around it. This is called frame dragging effect. r, φ t are the coordinates .

## Equation of geodesic

To do that we will try to determine a path between two points for which

This absolute track is of fundamental importance in dynamics, but at the moment we are concerned with it only as an aid in the development of the tensor calculus.
Keeping the beginning and end of the path fixed, we give every intermediate point an arbitrary infinitesimal displacement Δx so as to deform the path. Since

The stationary condition is

which becomes by previous equation

Changing the dummy suffixes in the last two terms

Applying the usual method of partial integration, and rejecting the integrated part since Δxa vanishes at both limits,

This must hold for all values of the arbitrary displacements Δxa at all points, hence the coefficient in the integrand must vanish at all points on the path. Thus

Now

Also in the last two terms we replace the dummy suffixes u and v by ε. The equation then becomes

We can get rid of the factor g_&elslon;σ by multiplying through by g(σα) so as to form the substitution operator g_ε(α). Thus

Or

Or
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## Fibre bunde

In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a distinct topological structure. Specifically, the similarity between a space E and a product space { B X F} is defined using a continuous surjective map :
π : E -> B

that in small regions of E behaves just like a projection from corresponding regions of B × F to B. The map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber.
In the trivial case, E is just B × F, and the map π is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles are the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; this is the sense in which the product topology is "natural".

## Surfaces

Since, by considering functions of more than one variable, we are now beginning to venture into higher-dimensional spaces, some remarks are needed here concerning 'calculus' on such spaces. As we shall be seeing explicitly in the chapter following the next one, spaces—referred to as manifolds—can be of any dimension n, where n is a positive integer. (An n-dimensional manifold is often referred to simply as an n-manifold.) Einstein's general relativity uses a 4-manifold to describe spacetime, and many modern theories employ manifolds of higher dimension still. We shall explore general n-manifolds later, but for simplicity, in the present chapter, we just consider the situation of a real 2-manifold (or surface) S. Then local (real) coordinates x and y can be used to label the different points of S (in some local region of S). In fact, the discussion is very representative of the general n-dimensional case.
A 2-dimensional surface could, for example, be an ordinary plane or an ordinary sphere. But the surface is not to be thought of as a 'complex plane' or a 'Riemann sphere', because we shall not be concerned with assigning a structure to it as a complex space (i.e. with the attendant notion of 'holomorphic function' defined on the surface). Its only structure needs to be that of a smooth manifold. Geometrically, this means that we do not need to keep track of anything like a local conformal structure, as we did for our Riemann surfaces, but we do need to be able to tell when a function defined on the space (i.e. a function whose domain is the space) is to be considered as 'smooth'.
In order to understand what this means a little more precisely, let us introduce a system of coordinates on our surface S. These coordinates need apply only locally, and we can imagine 'gluing' S together out of local pieces—coordinate patches—in a similar manner to our procedure for Riemann surfaces (For the sphere, for example, we do need more than one patch.) Within one patch, smooth coordinates label the different points; see Figure below. Our coordinates are to take real-number values, and let us call them x and y (without any suggestion intended that they ought to be combined together in the form of a complex number). Suppose, now

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that we have some smooth function φ defined on S. In the modern mathematical terminology, φ is a smooth map from S to the space of real numbers R (or complex numbers C, in case φ is to be a complex-valued function on S) because φ assigns to each point of S a real (or complex) number i.e. φ maps S to the real (or complex) numbers. Such a function is sometimes called a scalar field on S. On a particular coordinate patch, the quantity φ can be represented as a function of the two coordinates, let us say
φ = f (x, y),
where the smoothness of the quantity φ is expressed as the differentiability of the function f(x, y).

φ = F(X, Y);
for the values of F on the new coordinate patch. On an overlap region between the two patches, we shall therefore have F(X, Y) = f (x, y),
But, as indicated above, the particular expression that F represents, in terms of the quantities X and Y, will generally be quite different from the expression that f represents in terms of x and y. Indeed, X might be some complicated function of x and y on the overlap region and so might Y, and these functions would have to be incorporated in the passage from f to F. Such functions, representing the coordinates of one system in terms of the coordinates of the other, X = X(x, y) and Y = Y(x, y)
and their inverses x = x(X, Y) and y = y(X, Y) are called the transition functions that express the cordinate change from one patch to the other. These transition functions are to be smooth—let us, for simplicity, say C(infinity)-smooth—and this has the consequence that the 'smoothness' notion for the quantity F is independent of the choice of coordinates that are used in some patch overlap. By "local path" , it is to mean any part of the whole manifold. Like small patches of coordinate chart that can be used to cover the whole of the globe.

## Philosophical consequences

There is a lot of philosophical consequences of general relativity. Theory of relativity as a whole has lot of consequences. In Newtonian framework of gravity the sun was like a despotic government who emits decrees from the metropolis. He is the supreme authority. But in Einstein's universe the solar system is like the societ of Kropotkin's dreams, in which everybody does what he prefers at every moment. The result is perfect order.
General theory of relativity tries to explain the origin and fate of the universe. According to the principles of general relativity spacetime is smooth. It gives a complete physical world view. It thus supports realism in a sense.
Another consequence is that of the abandonment of the notion of force. Equation of motion in spherical polar coordinates will be

Equation of motion of torsion system can be derived also

Gyroscopic motion also has mathematical expression like other motion

Elastic and inelastic collisions obey different sets of equations.
In elastic collision momentum or kinetic energy is conserved

In non-elastic collisions momentum or kinetic energy is not conserved.

What is thrust ?
Thrust is a force

The equation of simple harmonic motion is given by

Conservation of energy equation can be mentioned for convenient.

This equation has been accounted for both rotational and translational motion.
In a rectangula cavity momentum is conserved as the particles bounch from the opposite walls.

## Parallel force law

If two parallel force acts on a body in the same diraction then the resultant force divides the line between them in proportion to the forces . If F1 and F2 is the magnitude of the two forces then the following condition will hold.
F1a = f2b where a and b are the distaces of the resultant force from the two forces F2 and F1.
classical electrodynamics is governed by maxwell's equations and Coulomb's law :

The force between two current loops can be determined using principles of classical electrodynamics.
In general theory of relativity , metrical geometry plays an important role. By metrical geometry , it is to be understood that between two points there is always a notion of distance which is the function of the coordinates of those. More analysis can be made with pure mathematics.
The subject of the present chapter is elementary Geometry, as treated by Euclid or by any other author prior to the nineteenth century. This subject includes the usual analytical Geometry, whether Euclidean or non-Euclidean; it is distinguished from projective and descriptive Geometry, not by any opposition corresponding to that of Euclid and non-Euclid, but by its method and its indefinables. The question whether its indefinables can, or cannot, be defined in terms of those of projective and descriptive Geometry, is a very difficult one, which I postpone to the following chapter. For the present, I shall develop the subject straightforwardly, in a manner as similar to Euclid’s as is consistent with the requisite generality and with the avoidance of fallacies. Metrical Geometry is logically subsequent to the two kinds which we have examined, for it necessarily assumes one or other of these two kinds, to which it merely adds further specifications. I shall, as a rule, assume descriptive Geometry, mentioning projective Geometry only in connection with points in which it shows important metrical differences from descriptive Geometry. In the former case, all the first twenty-six propositions of Euclid will hold. In the latter, the first, seventh, sixteenth and seventeenth require modification; for these propositions assume, in one form or another, that the straight line is not a closed series. Propositions after the twenty-sixth—or, with a suitable definition of parallels, after the twenty-eighth—depend, with few exceptions, upon the postulate of parallels, and are therefore not to be assumed generally.

## General theory of relativity equation revisited

From Newton's equation we get

First some equations of special relativity must be recognized. The relation between two reference frames is

The Lorentz-invariant is

A partcle worldline is

Infinitesimal interval is

The final result is

## Tensors

All the general theory of relativity equations are basically tensor equations. Without the understanding of
tensor it is hard to understant general theory of relativity completely. Tensor product of two contravariant tensors T and S is

Tensor quotient law is

The quotient of two tensor is a tensor but it is not like the usual quotient law of algebra.
Quotient law of tensors states that any quantity which on inner multiplication by any covatiant (and alternatively by any contra-variant) vector is always a tensor. The only fundamental tensors which do not contain derivatives of g^ beyond the second order are functions of g_uv and B(ε_bac).
R(n) denotes a succession of real numbers (x1, x2, x3,....xn). A one-to-one mapping f from space M to space N is a rule that assigns to each point x in M, anothe point f(x) in N. It is also demanded that different point is mapped to different point. This concept is illustrated as

A manifold is a structure satisfying these general properties.
1. there is a family of open neighborhoods U(i) together with f(i): U(i) -> R(n) such that the mapping f is continuous and invertible.
2. The family of neighborhoods covers the whole M as

Such mapping is called coordinate system. U is called the coordinate region of M.

If two regions U and V have non-empty intersection U(intersection ) V != 0 with coordinates x` and x, then we can define an invertible transformation x = x`(x) in that intersection region.

## Special theory of relativity revisited

Einstein alway made thought experiments to find new ideas. Thought experiments motivated him to discover theory of relativity. He thought that what would it be like to ride alongside the beam of light. If you rush toward the beam of light at the exact same vecolity of light then you sould be able to see satationary light. Light should be frozen in front of you. But such scenerio is impossible according to the principle of Maxwell'e equations.
One of those thought experiments is given below:

A thought experiment was done to explain the equivalence principle too.

The clock at the surface of the earth is ticking slower than a clock away from the surface. The ratio of the ticking rate can be calulated easily as

Gravitational time dilation explains the mysterious dark matter

Time dilatation due to rotational velocity can be calculuted using

## Graviational waves emitted by an electron

In principle even a single electron will loose mass due to emission of gravitational waves. Accelerated electron in the atom will emit graviational waves in the same way as an accelerated electric charge emit electromagnetic radiation. Eddington has calculated exactly how much energy an elctron will emit due to its rotational velocity around the nucleus. But the amount is so tiny that it will not affect anything outside the atom. Let us delve into the mathematical explanation.
Let us consider a simple system of oscillating mass below.

This problem differs from that of the gravitational field of a particle in that the electric field spreads through all space, and consequently the energy-tensor is not confined to a point or small sphere at the origin.
For the most general symmetrical field we take as before

Since the electric field is static, we shall have F, G, H= k1, k2,, k3 = 0, which are the components of maxwell
stress-energy tensor.
and ka will be a function of r only. Hence the only surviving components of F_uv are

the accent ` denoting differentiation with respect to r. Then

The condition for no electrical charge and current (except at the singularity at the origin) is

So that

Substituting in

We find

When there is only electro-magnetic stress-energy tensor the field equation takes the form G_uv = -πE_uv
containing no matter.
we have to substitute -8πE^r for zero on the right-hand side of

. The first and fourth equations give as before &lamba;` = - v `; and the second equation now becomes

Hence writing e^v = γ

So that

where 2m is a constant of integration.
Therefore the gravitation field of electron is given by

with

Two particles of the same mass M and charge Q is oscillating vertically. The ratio of gravitational wave energy and electromagnetic energy is equal to

There are three kinds of tensor quantities:

Black and dark matter are characterized by particular equations

Generalized Riemann Christoffel tensor is

The symbol * refers to the in-tensor. The generalized Riemann Tensor is a consequences of gauge-invariance. Gauge invariance is the symmetry arisen due to change of gauge units.
Sometimes an entirely different kind of derivative operator is encountered in the equations of general relativity. This is called hamiltonian derivative and defined as

A tensor of rank (r, s) is defined as

There is another useful definition of co-variant derivative which presuppose connection term

## Perihelion Advance

The track of particles in general theory of relativity is given by geodesic equation

Taking first α = 2 we get after substituting the components of Chrstoffel symbols.
-----1

-----2
Choose coordinates so that the particle moves initially in the plane θ = π/2; Then dθ/ds = and cos 6 = initially, so that d^2 0/ds^2 = 0. The particle therefore continues to move in this plane, and we may simplify the remaining equations by putting θ = π/2 throughout. The equations for α = 1, 3, 4 are found in like manner, viz.
-----2
Where &neu; and λ are functions of r only. . And
-----2
The last two equations can be integrated easily to get
-----22
Instead of troubling to integrate (22) we can use in place of it
-----2
which plays here the part of an integral of energy. It gives
-----2
Eliminating dt and ds by means of (22)
-----2

## This derivation or proof has been taken from Mathematical theory of relativity by Eddington..

whence, multiplying through by γ or (1 — 2m/r),
-----2
Writing u = 1/r ;

-----2
Differentiating with respect to φ, and removing the factor du/dφ ,
-----3
With r^2dφ/ds = h
Writing the equation of planetary orbit in general relativity again
-----3

-----2
we obtain the astronomical results more directly by a method of successive approximation. Neglecting small term 3mu^2 the solution is
---2
as in Newtonian dynamics. The constants of integration e and Ω are the eccentricity and the longitude of the perihelion.
Substitute this first approximation in the small term 3mu^2 , then (1) becomes

Of the additional terms the only one which can produce an effect within the range of observation is the term in cos (φ - Ω) ; this is of the right period to produce a continually increasing effect by resonance. Remembering that the particular integral of

is

This term gives part of u.

which must be added to the complementary integral (2). Thus the second approximation is

The second order term (ΔΩ)^2 is neglected. Whilst the planet moves through 1 revolution, the perihelion Ω advances a fraction of a revolution equal to

using the well-known equation of areas h^2 = ml = ma (1 — e^2).
Another form is obtained by using Kepler's third law.

giving

where T is the period, and the velocity of light c has been reinstated. This advance of the perihelion is appreciable in the case of the planet Mercury, and the predicted value is verified by observation.
For a circular orbit we put dr/ds, d^r/ds^2 = 0; so that equation of planetary motion

becomes

Whence

So the law of Kppler is accurately fulfilled. This result has no observational significance, being merely a property of the particular definition of r here adopted. Slightly different coordinate-systems exist which might with equal right claim to correspond to polar coordinates in flat space-time ; and for these Kepler's third law would no longer be exact. We have to be on our guard against results of this latter kind which would only be of interest if the radius-vector were a directly measured quantity instead of a orthodox coordinate. The advance of perihelion is a phenomenon of a different category. Clearly the number of years required for an eccentric orbit to make a complete revolution returning to its original position is capable of observational test, unaffected by any convention used in defining the exact length of the radius-vector.
For the four inner planets the following table gives the adjustment to the centennial motion of perihelion predicted by Einstein's theory

The measure eΔ&Omaga; is a better measure of the observable effect to be looked for, and the correction is only appreciable in the case of mercury.
After applying these correction to eΩ the following discrepancies between theory and observation remains in the secular change of the elements of inner planets , i and Ω being the inclination and longitude of the node.

The probable errors here given include errors of observation, and also errors in the theory due to uncertainty of the masses of the planets. The positive sign indicates excess of observed motion over theoretical motion.
Einstein's correction to the perihelion of Mercury has removed the principal discordance in the table, which on the Newtonian theory was nearly 30 times the probable error. Of the 15 residuals 8 exceed the probable error, and 3 exceed twice the probable error—as nearly as possible the proper proportion. But whereas we should expect the greatest residual to be about 3 times the probable error, the residual of the node of Venus is rather excessive at 4| times the probable error, and may perhaps be a genuine discordance. Einstein's theory throws no light on the cause of this discordance.

## Deflection of light by gravitational field

For the motion of light ds = 0 and h = &inginity; the equation of orbit then becomes

The track of a light-pulse is also given by a path with ds=0. Accordingly the orbit described by the equation above give the path of the light ray. We integrate by successive approximation. Neglecting 3mu^2 the solution of the approximate equation

is the straight line

Substituting this in the small term 3mu^2 , we have

A particular integral of this equation is

So that the complete second approximation is

Multiplying through by rR

or in rectangular coordinates, x = rcosφ and y = rsinφ

The second term measures the very slight deviation from the straight line x = R. The asymptotes are found by taking y very large compared with as. The equation then becomes

and the small angle between the asymptotes is (in circular measure) 4m/R
For a ray grazing the sun's limb, m = 1*47 km., R = 697,000 km., so that the deflection should be 1""75. The observed values obtained by the British eclipse expeditions in 1919 were
Sobral expedition 1"98 ± 0"12 Principe expedition 1"*61 + 0""30
It has been explained in Space, Time and Gravitation that this deflection is double that which might have been predicted on the Newtonian theory. In this connection the following paradox has been remarked. Since the curvature of the light-track is doubled, the acceleration of the light at each point is double the Newtonian acceleration ; whereas for a slowly moving object the acceleration is practically the same as the Newtonian acceleration. To a man in a lift descending with acceleration m/r* the tracks of ordinary particles will appear to be straight lines ; but it looks as though it would require an acceleration 2m/r- to straighten out the light-tracks. Does not this contradict the principle of equivalence ?
The fallacy lies in a confusion between two meanings of the word " curvature." The coordinate curvature obtained from the equation of the track (41'4) is not the geodesic curvature. The latter is the curvature with which the local observer—the man in the lift—is concerned. Consider the curved light-track traversing the hummock corresponding to the sun's field ; its curvature can be reckoned by projecting it either on the base of the hummock or on the tangent plane at any point. The curvatures of the two projections will generally be different. The projection into Euclidean coordinates (x, y) used in (41 '4) is the projection on the base of the hummock; in applying the principle of equivalence the projection is on the tangent plane, since we consider a region of the curved world so small that it cannot be discriminated from its tangent plane.

## gyroscope physics

To understand general theory of relativity it is imperative that you understand all the physics of Newton and post Newtonian physics. Gyroscopic physics is very useful concept in classical mechanics and relativity. It is all concerned with angular momentum conservation. Let us now review it :
A typical gyroscope has these components

Where: ws is the constant rate of spin of the wheel, in radians/second
wp is the constant rate of precession, in radians/second
L is the length of the rod
r is the radius of the wheel
θ is the angle between the vertical and the rod (a constant)
As the wheel spins at a rate ws, the gyroscope precesses at a rate wp about the pivot at the base (with θ constant).
The question is, why doesn't the gyroscope fall to ground due to gravity?!
The reason is quite simple:
Due to the resultant rotation of ws and wp, the particles in the top half of the spinning wheel experience a component of acceleration a1 normal to the wheel (with distribution as shown in the figure below), and the particles in the bottom half of the wheel experience a component of acceleration a2 normal to the wheel in the opposite direction (with distribution as shown). Due to Newton’s second law, this implies that a net force F1 must act on the particles in the top half of the wheel, and a net force F2 must act on the particles in the bottom half of the wheel. These forces act in opposite directions. Therefore a clockwise torque M is needed to sustain these forces.
The force of gravity pulling down on the gyroscope creates the necessary clockwise torque M.

In other words, due to the nature of the kinematics, the particles in the wheel experience acceleration in such a way that the force of gravity is able to maintain the angle θ of the gyroscope as it precesses. This is the simplest explanation behind the gyroscope physics.
Mathematical analysis can be done in a simple manner

Where is g is the graviational acceleraion and G is the center of mass of the wheel. Point P is the pivot position of the wheel at the ground. The global XYZ axes is fixed to ground and has origin at P.
I, J, and K are defined as unit vectors pointing along the positive X, Y, and Z axis respectively. The angular velocity of the wheel, with respect to ground, is

The angular acceleration of the wheel, with respect to ground, is

Looking at the first term:

Therefore

The angular velocity of the rod, with respect to ground, is

The angular acceleration of the rod, with respect to ground, is zero since w_r is constant and does not change direction.
Notice that the unit vectors I, J , K are here subject to change direction as the time passes. This is why the vector derivative of are them is necessary to compute. Note that the terms dJ/dt and dK/dt (given above) are calculated using vector differentiation.

## Large scale structure of space and time

A physical theory like general theory of relativity is just the mathematical model. It is meaningless to ask whether it corresponds to reality. All that one could ask is that its prediction should be in agreement with the experiments.
The crucial structure to be studied to understand the singularities and black holes is the causal structure of spacetime. Define I+(p) to be the set of all points of the spacetime M that can be reached from pby future-directed timelike curves (see figure below). One can think of I+(p) as the set of all events that can be influenced by what happens at p. There are similar definitions in which plus is replaced by minus and future by past. I shall regard such definitions as self-evident.

The boundary of the chronological future cannot be timelike or spacelike. On now consider the boundary I+(S) of the future of a set s. t is fairly easy to see that this boundary cannot b timelike. For in that case, a point qjust outside the boundary would be to the future of a point p just inside. There are geodesics as usually .

Top: The point q lies in the boundary of the future, so there is a null geodesic segment in the boundary which passes through q. Bottom: If there is more than one such segment, the point qwill be their future endpoint.
The condition for causal structure is global hyperbolicy. To show that each generator of the boundary of the future has a past endpoint on the set, one has to impose some global conditio on the causal sbucture. The strongest and physically most important co dition is that of global hyperbolicity. An open set U is said to be globally hyperbolic if 1. For every pair of poin p and q in U e intersection of future of p and e past of q has compact closure. In 0 er words, it is a bounded diamond haped region (fig below). 2. Strong causality holds on U. That· ere are no closed or almost closed timelike curves co tained in U.

The intersection of the past of q and the future of p has compact closure.

A family of Cauchy surfaces for U. The physical significance of global hyperbolicity comes from the fact that it implies that there is a family of Cauchy surfaces I:(t) for U . A Cauchy surface for U is a spacelike or null surface that intersects every timelike curve in Uonce and once only. One can predict what will happen in U from data on the Cauchy surface, and one can formulate a well-behaved quantum field theory on a globally hyperbolic background.

## Most of the topics are taken from Penrose and Hawking's large scale structure of spacetime..

The minimal geodesic is

Left: if there is a conjugate point r between p and qon a geodesic, it is not the geodesic of minimum length. Right: The nonminimal geodesic from p to q has a conjugate point at the south pole.
One can illustrate this by considering two points p and q on the surface of the Earth. Without loss of generality, one can take p to be at the north pole. Because the Earth has a positive definite metric rather than a Lorentzian one, there is a geodesic of minimal length, rather than a geodesic of maximum length. This minimal geodesic will be a line of longitude running from the north pole to the point q. But there will be another geodesic from p to q which runs down the back from the north pole to the south pole and then up to q. This geodesic possesses a point conjugate to p at the south pole where all the geodesics from p intersect. Both geodesics from p to q are stationary points of the length under a small variation. But now in a positive definite metric the second variation of a geodesic containing a conjugate point can give a shorter curve from p to q. Thus, in the example of the Earth, we can deduce that the geodesic that goes down to the south pole and then comes up is not the shortest curve from p to q. This example is very obvious. However, in the case of spacetime one can show that under certain assumptions there should be a globally hyperbolic region in which there should be conjugate points on every geodesic between two points. This establishes a contradiction which shows that the assumption of geodesic completeness, which can be taken as a definition of a nonsingular spacetime, is false.
The reason that one gets a conjugate points is that gravity is an attractive force. It bends spacetime such a way that the neighboring geodesics are bent toward each other rather that way. One can see this from the Raychaudhuri or Newman-Penrose equation, which I will write in a unified form.

Here v is an affine parameter along a congruence of geodesics with tangent vector fa which is hypersurface orthogonal. The quantity p is the average rate of convergence of the geodesics, while (σ measures the shear. The term R_abl_aL_b gives the direct gravitational effect of the matter on the convergence of the geodesics.

By the Einstein equations, it will be nonnegative for any null vector La if the matter obeys the so-called weak energy condition. This says that the energy density T_oo is nonnegative in any frame. T
Suppose the weak energy condition holds, and that the null geodesics from a point p begin to converge again and that p has the positive value Po. Then the Newman-Penrose equation would imply that the convergence p would diverge to infinity at a point q within an affine parameter distance ~ if the null geodesic can be extended that far.

Infinitesimally neighboring null geodesic from p will intersect at q. The situation with timelike geodesics is similar, except that the strong energy condition that is required to make R_abl_al_b nonnegative for every timelike vector l_a is, as its name suggests, rather stronger.

I will end this discussion on causal structure with the defintion of spacetime singularity .

## Spacetime singularities

IN THE FIRST LECTURE by Stephen Hawking, singularity theorems were discussed. The essential content of these theorems is that under reasonable (global) physical conditions, singularities must be expected. They do not say anything about the nature of the singularities, or where the singularities are to be found. On the other hand, the theorems are very general. A natural question to ask is, therefore, what the geometric nature of a spacetime singularity is. It is usually assumed that the characteristic of a singularity is that the curvature diverges. However, this is not exactly what the singularity theorems by themselves imply. Singularities occur in the big bang, in black holes, and in the big crunch (which might be regarded as a union of black holes). They also might appear as naked singularities. Related to this question is what is called cosmic censorship, namely the hypothesis that these naked singularities do not occur. To explain the idea of cosmic censorship, let me recall a bit the history of the subject. The first explicit example of a solution of Einstein's equations describing a black hole was the collapsing dust cloud of Oppenheimer and Snyder (1939). There is a singularity inside, but it is not visible from outside, as it is surrounded by the event horizon. This horizon is the surface inside of which events cannot send signals out to infinity. It was tempting to believe that this picture is generic, i.e., that it represents the general gravitational collapse. However, the OS model has a special symmetry (namely, spherical symmetry), and it is not obvious that it is really representative. As the Einstein equations are generally hard to solve, one looks instead for global properties that imply the existence of singularities For example, the OS model has a trapped surface, which is a surface whose area will decrease along light rays that are initially orthogonal to it (fig. below). One might try to show that the existence of a trapped surface implies that there is a singularity. (This was the first singularity theorem I was able to establish, on the basis of reasonable causality assumptions but no spherical symmetry being assumed; see Penrose 1965.) One can also derive similar results by assuming the existence of a converging light cone (Hawking and Penrose 1970; this occurs when all the light rays emitted in different directions from a point start to converge toward each other at a later time).
The spacetime picture of the collapse of a star would look like

## Russell's acount on general theory of relativity equation

There is one important aspect of Eddington's theory , which Russell was supposed to have neglected in his analysis of matter. In addition to the fact that the whole theory of general relativity can be deduced from a few simple assumptions, interest attaches to the manner of the deduction and the considerations by which the substantial import of mathematical formulae is made less, or at least other, than would naturally be supposed. A good example is afforded by a paragraph headed " Interpretationtion of Einstein's Law of gravitation". The law is not G_uv = 0 , which is not supposed to be quite accurace where steller distances are concerned. It is modified law:
G_uv = λg_uv,
Where λ must be very small, so small that within the solar system the new law gives the same resuls, within the limits of observation as G_uv = 0. The new law is shown to be equivalent to the assumption that , in empty space, the radius of curvature in every direction is everywhere &root;λ . But this it interpreted as a law about our measuring rods-namely , that they adjust themselves to the radius of curvature at any place and in any direction. It is interpreted as meaning:
"The length of a specified material structure bears a constant ratio to the radius of the curvature of the world at the place and in the direction in which it lies". And the following gloss is added:
"The law no longer appears to be have any reference to the constitution of an empty continuum. It is a law of material structure showing what dimensions a specified collection of molecules must take up in order to adjust itself to equilibrium with the surrounding conditions of the world".
In particular , electrons must make these adjustments ,and it is suggested elsewhere that the symmetry of an electron and its equality with other electrons are not substantial facts, but consequences of the method of mesurements. One cannot complain of an author for not doing everything, but at this point most readers will feel a desire for some discussion of the theory of measurement.