# Einstein field equations

Coordinate transformation   |   Gravitational waves  |   Differential Equation

### Introduction

Albert Einstein transformed the way we all see the world. Just over one hundred years ago, his theory of Relativity stunned scientists, but today it is integral to modern thought as the most important scientific discovery of twentieth century.
Field equations are a set of equations in the formulation of general relativity, which gave many fruitful and interesting results about gravity. Time was shown to be the fourth dimension earlier in special theory of relativity. Space and time were combined to form spacetime. To come about this unification Einstein simply modified famous phytagoras law to include time in a certain manner. The spacetime interval was proved to be fundamental to his theory. All measurement is relative. Temporal separation and space separation are different for different observer but spacetime interval is the same for all. There is no basic difference between space and time from relativistic viewpoint. Mass , length and time all became relative. There is just one absolute structure and that is four dimensional spacetime. These were the facts that the theory of special relativity revealed about our world. But Einstein later took bold task to include gravity in framework of theory of relativity. He thought what if gravity is bending and warping of spacetime itself. He successfully explained that gravity was in fact so and finally proposed his field equations.
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. Time dimension is excluded for simplicity. This is also known as rubber-sheet analogy of curved spacetime. This is Einstein's general theory of relativity!!!

# Einstein Field Equationa explained

Field equation of Einstein's General theory of relativity is perhaps one of the greatest theories ever discovered by any 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 equations 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. Solutions of field equations gave many interesting fact about the universe like black holes, worm hole and time travel. Wor m hole is a hypothetical gateway that connect s two distance regions of space. it can be used as a time machine also. In the movie interstellar the crew of the ship travel through wormhole , which was shown to be a sphere in three dimensions.

## Principle of equivalence

Principle of equivalence is the most important idea behind developing general theory of relativity. It simply states that acceleration and gravity are equivalent to each other. Einstein always devised thought experiments to invent his theories. He thought about an elevator in deep space.

Suppose you are inside an elevator in deep space where there is no gravity. Now suddenly someone starts pulling the elevator with constant acceleration. You will feel a backward force inside the lift. Can you tell that you are in deep space without looking outside the elevator ? You will not see any difference between gravity and acceleration. You will feel as if you were sitting on the surface of the earth. That was Einstein's insight. He said acceleration is the same thing as gravity. Gravity can be transformed away by going into different coordinate system , at least locally in a small region.

### Derivation

Force in Newtonian framework is the gradient of a gravitational field (φ). The field in Newtonian framework has one quantity and a direction at each point in space. This is equivalent to acceleration that an object experiences due to gravity.

The Laplacian (∇) of the field is proportional to mass density.
Now we consider similar equation of the form given below , which contains Laplacian of metric tensor g(uv).

The "MCRF" are here having the meaning of momentarily co-moving reference frame.

In general relativity field is the metric tensor g(uv). Metric tensor transforms like a second rank tensor. As a result gravity is solely the result of coordinates tranformation. Metric tensor allows one to calculate distance between two points in spacetime. Suppose you want to calculate distance on the surface of a cylinder. The distances in various direction s are different as the curvature changes along each direction. Metric tensor takes account of all these individual distances to calculate total distance between any two points. On the surface of cylinder there is usually two directions. In space time there are many directions and components of metric tensor track all these directions. The idea of metric tensor dates back to Gauss's remarkable theorem egregium . The theorem states that if any surface is developed on to another surface , the curvature at each point will remain the same. This curvature is known as Gaussian curvature. He also stated that the measure of curvature can be determined from the Measurement of length , angle and rate of change of these. The curvature does not depend on the way the surface is embedded in space. The theorem is remarkable because definition of curvature makes direct use of position of the surface in space . So it is quite surprising that the result does not depend on its embedding in spite of all bending and twisting deformations undergone.
Now calculating the divergence of Einsteing tensor G(ab) and energy momentum tensor T(ab ) and setting them equal to zero we can find the value of constant (μ). Putting back everything in equation (11) we get field equations.

Field equations describe basic interplay between space, time, matter and energy. It reflects a great mathematical as well as philosophical insight. It is a second order partial non linear differential equation. The constants of the field equations are Cosmological constant Λ, gravitational constant(G), speed of light(c). The cosmological constant (Λ) was proposed by Einstein. His field equations did not allow for a static universe. He wanted to stop the universe from collapsing in on itself because gravity would contract it. Later he considered it as the biggest blunder of his life, when he knew that universe is indeed expanding. Other parameters of the left hand side are metric tensor, ricci tensor. Divergence of the left hand side is zero and so is the right hand side of energy momentum tensor. Divergence in relativity is defined through the covariant derivative, which is the 4-dimensional analog of that in Euclidean space.
Euclidean space is a mathematical construct. Euclid first created an axiomatic system which helped Euclidean space to be constructed. Euclidean geometry is the geometry of Euclidean space. There are few axioms and propositions of Euclidean system. The axioms are :
1)To draw a straight line from any point to any point.
2)To produce [extend] a finite straight line continuously in a straight line.
3)To describe a circle with any centre and distance [radius].
4)That all right angles are equal to one another.
5)[The parallel postulate]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
One of the axiom was the parallel postulates. Parallel lines are the lines which do not intersect themselves. Given a line in a plane there two class of lines , which cuts the given line and which do not. Paralel lines to this given line is defined to limit of these two classes.
It was refusal to admit the parallel postulates without proof that began the development of meta-geometry or non-euclidean geometry. There are other cases where parallel postulate does not hold. For example on the surface of a sphere (elliptic geometry) where parallel lines do meet. Such elliptic space is positively curved.
Our space is similarly curved due to presence of mass and energy content of the universe . Space in our solar system is curved due to the mass of sun and other planet. But we can not directly percieve such curved space because of our restricted sens e. Think about a bug living on the surface of a sphere. The bug does not know that he is actually living in higher dimension other than its surface. But Gauss had shown that we need not reference higher dimension in order to infer the curvature of any space. The metric of any space can reveal the intrinsic curvature. So we can easily find out any properties of our space without thinking about how our three dimensional space is embedded in higher dimensions.

We can decompose the field equation into individual components like this : .

In empty space the field equations reduce to simpler form : .

So what are the field equations all about? It suggests that the material world is intimately connected with the curvature of spacetime. Geometry is the same as physics. The field equation is a system of ten simultaneous separate equations which correspond to the ten independent components of metric tensor. In the right hand side , the constant quantity is very small. So a large amount of mass is needed to create a little spacetime curvature. In theoretical consideration, even an electron can create some spacetime curvature as it has some mass. As the mass of electron is too small, The curvature created by it is too small.
That the space is curved refers to the fact that the ratio of circumference to the radius is not the constant 2π rather it is greater or less than 2π. When the ratio is greater than 2π , it is characterized by hyperbolic geometry and when it is less than 2π , it is characterized by elliptic geometry. In Euclidean geometry it is exactly 2π.

The development of non-Euclean geometry took a long time. When Einstein used it , it became clear that our space is indeed curved and Euclidean axioms and propositions only hold in a limiting case of that. The curvature can be explained in terms of amount of deviation of total angle of a triange in curved space from that in Euclidean space as the area of the triangle shrinks to zero.

Two dimensional surface is characterized by geodesics. Plane in Euclidean geometry is an special case of two dimensional manifolds. At every point on a two dimensional manifold there are infinite number of geodesics that pass through it. These geodesic are arcs which have measures of curvature. Among them there are two of which the measure of curvature is either least or greatest. These two measures of curvature determine the curvature of surface at the particular point i.e the product of the two. If the measure of curvature is the same at every point then the surface is said to be of constant cur vature. Sphere is an example of surface of constant curvature. Such curved surface deviates away from a plane which is tangent to it at every point. So the amount by which the surface pulls away from the plane is quadratic in the coordinates measured from the point of tangency. We expect similar thing to happen in four dimensions. In four dimensions there are a number of surfaces that pass through a point. Each surface has a curvature defined at the point so the total curvature of the space will be a collection of numbers.
"The life on earth may be a expensive but it certainly includes a free trip around the sun once a year"
Field equation is the theory of gravity which Einstein struggled for ten years to find out. Mach's idea influenced Einstein to develop his theory. Mach's idea was that there is no absolute rotation. Newton thought that motion and acceleration was absolute. He demonstrated a famous experiment known as "Newton's bucket" to prove this. But later Mach refuted his claim and told that the said acceleration is relative to all the stars and nebulae in universe. The water in the Newton's bucket is really rotating with respect to the entire mass content of the universe. Field equation is the best description of our physical world in macroscopic perspective. Field equations say not only mass energy creates gravitational field but also pressure and stress contribute to the gravity.
Where there is no matter the equation takes a simple form. This is the so called vacuum condition: R(ik) = 0 . Ricci tensor vanishes but the Riemann curvature tensor does not necessary vanish. Some components of the Riemann curvature tensor can exist. Empty space can be curved too. What it means is that average scalar curvature is zero. Einstein's field equation is very hard too solve. German physicist Karl Swardchild first found the exact solution of the field equation. This solution is used to describe the spacetime around a star when it turns into a black hole. The solution is relatively hard to derive from the field equation. But we can surely give a try.

## Spherically Symmetric Collapse

First we consider an isolated object of mass M. Surrounding spacetime around the object will be curved by its mass. So we need to find the metric that can describe the spacetime distortion around the object. We use spherical polar coordinate system which has coordinates r, θ and φ. we need an additional coordinate t. So a set (r, θ,φ, t) will describe the non-euclidean spacetime around the object.

We apply symmetry under each coordinate change from positive to negative. Using the properties of metric tensor's transformation we see that all the components vanish except the diagonal components. The combination of these diagonal components will be our final form of metric.
Next we let U= g(11) and V= g(44) and compute the components of Christoffel symbols. We can use the formula where it contains the derivative of metric tensors. After tedious computation we can get four independent components of christoffel symbols which can be used in the vacuum equations. These symbols become simplified under certain conditions.

The covariant metric tensor is the negative one (-1) multiplied the inverse of contravariant one. Each individual component of Christofel symbol is calculated using this inverse metric and derivative of metric tensor's components. Most of the Off-diagonal components seems to vanish.

Now the values of some christoffel symbols are calculated.

Similarly some others are calculated for individual indices. Christoffel symbol's component are symmetric in lower indices. Now the task is to compute ricci tensor components. The diagonal components R(00) and R(11) are computed for brevity.

Now the Einstein tensor G(00) and G(11) is determined where Ricci scalar is used.

In the weak field limit when r tends to infinity the value of U=A=g(44) and V=B(r)= must take the value of that of Minkoswki metric. Thus we finally find the value of U in terms of mass (m) of the object and radius r.

Finally putting the value of U and V in the line element we get the metric of the spacetime around the object of mass m. This is so called exact solution of Einstein field equation. It has predicted the existence of black hole. So far large number of such black holes have been discovered. Our milky way contains a black holes that is a million times heavier than the sun. Heavier star collapses under the pull of gravitational field of it after it has run out of its fuel.
The nonlinearity of the EFE makes finding exact solutions hard. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, gravitational field is very feeble and the spacetime approximates to that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term denoting the deviation of the true metric from the Minkowski metric, with terms that are quadratic in or higher powers of the deviation being disregarded. This linearization procedure can be used to investigate the phenomena of gravitational radiation.
The distorted picture of spacetime around a massive object will look like something like this:

### "nature abhors a nacked singularity!"

The most attractive part of the universe is curvy. Space tells matter how to move and matter tells space how to curve. This is the most beautiful theory in physics.

### Geodesics

The importance of geodesics came from the idea that freely moving bodies experiencing no other forces except gravity move in geodesics. In field equation the concept of geodesic is implicit. What is geodesic really?
A lonely pedestrian in Alps might want to go from one valley to another. He surely want to find shortest route from one valley to another. Shotest route is to mean the path between the two places while reamining on the earth everytime. He can not find it by drawing a straight line on the map. For the average gradient of some path can be greater that others which slopes gradually from one pass and down again on the next. What he is seeking is geodesics. The general definition of geodesic is not shortest route. Geodesic is the path such that distance from one point to another on it is stationary. Silght different paths are either longer or slightly differnt paths are shorter. First order change of path length is zero. This is the problem of maximizing and minizing a function. Geodesic path on the surface of the earth is the great circle. For many complicated surface the geodesic can be very complicated curve. In theory of relativity geodesic is the path of longest time. This is so because the distance in relativity is time-like. When it is possible for a material particle to travel between two events , the interval in spacetime is what seems like the lapse of time between those events. It is unlike the usual distance in Euclidean geometry.
There is one matter of great importance , which is not clear in the usual accounts of relativity theory. How are to know that given two events in spacetime, they are happening to the same piece of matter ? An electron and a proton are supposed to preserve its identity thoughout the time considered. But out continumm is a continuum of events. So one must suppose that one piece of matter is a series of events or series of sets of events. There is no specific criterion to determine whether a piece of matter belong to any such series. It is not supposed that events that overlap at the same place and time belongs to one piece of matter belong to no other. It is also supposed that events that have spacelike interval or zero interval does not belong to same piece of matter. But in case of timelike interval there is no obvious criterion. Two events which have time-like interval can always be connected by a geodesic and as the laws of dynamics suggest events on it will belong to the same piece of matter. Still we sometimes think they do and sometimes they don't.

"one must see the hell before he has any right to speak of heaven.."

To sum up the above lengthy discussion about field equation, spacetime has a four dimensional order of events. At each event of spacetime, the field equations are satisfied by the value of the coordinates of the event. In particular , at each event metric tensor has ten components which corresponds to ten non linear differential equations. There are twenty one numbers at each event, corresponding to various curvature components of Riemann curvature tensor. The events satisfying field equations have very complicated non linear realtions among them.

## Newtonian Limit

In the weak and static field approximation Einstein's field equation reduces to Newtonian limit. That is to say, if we take metric tensor which components are slowly varying and space-varying part is much higer than time-varying part, then the field equation of Einstein reduces to Newtonian law of gravitation. Let us prove it mathematically.:
First equation relates geodesic equation with the time-time component of metric tensor:

Now we let g(00) = φ and h(00) will be 1+2φ. And off course the time-time component of T(ab) , T(00) is &rou; . This will together imply the following poisson's equation of Newtonian gravity. It is not hard to prove.
We need to show that the potential φ function correspond to poisson's equation for Newtonian gravity. The poisson's equation can be stated again :

## Stephen Hawking manuscript on cosomology

This is actually Robertson-Walker solution of Einstein's field equation

## Gravitational field energy

Gravitational field energy is the total energy required to move a body from infinity to distance R (for example earth's redius R). In classical physics it can be easily calculated by intergration method:

So due to negative value of the energy the system's energy increases as the distance between the two bodies increases. Two bodies have more energy when they are close together than they are far apart.

Non-locality of gravitational potential energy. Imagine two planets (which for simplicity we may suppose to be instantaneously relatively at rest). If (a) they are far apart, then the (Newtonian) negative potential energy contribution is not so great as (b) when they are close together. Thus the total energy (and hence the total mass of the whole system) is larger in case (a) than in case (b) despite the total energy densities, as measured by the energy–momentum tensors T(ab), being virtually the same in the two cases.
Now let us consider that the bodies are in motion, in orbit about one another. It is a consequence of Einstein's field equation that gravitational waves—ripples in the fabric of spacetime—will emanate from the system and carry (positive) energy away from it. In normal circumstances, this energy loss will be very small. For example, the largest such effect in our own solar system arises from the Jupiter–Sun system, and the rate of energy loss is only about that emitted by a 40-watt light bulb! But for more massive and violent systems, such as the Wnal coalescence of two black holes that have been spiralling into each other, it is expected that the energy loss would be so large that detectors presently being constructed here on Earth might be able to register the presence of such gravitational waves at a distance of 15 megaparsecs or about 4:6 X 10^23 metres

The Hulse–Taylor double neutron star system PSR 1913 + 16. One member is a pulsar which sends out precisely timed electromagnetic signals that are received at Earth, enabling the orbits to be determined with extraordinary accuracy. It is observed that the system loses energy in exact accord with Einstein’s prediction of energy-carrying gravitational waves emitted by such a system. These waves are ripples in the spacetime vacuum, where the energy–momentum tensor vanishes. (Not to scale.)

## Wave function of universe

The wave function of the entire universe in Wheeler -Dewitt can be derived from Einstein's fied equations. The derivation is avoided for the moment:

Where a is the scale factor of the universe , which is a function of time t.

## Drake equation

Drake developed an equation to calculate the probability that intelligent life form exists outside our galaxy or in the universe.

## Hubble's constant

Hubble's constant was discovered by astronomer and scientist Hubble. Our universe is expanding in itself. The real cause of this expansion is a mystery. Scientists explained it by using inflationary cosmology. Right after the big bang there was a pull or force which made everything in the hot primordial universe expand at a very fast rate. Since then our universe is expanding. Galaxies in it are moving away from one another at a very fast rate. This has been experimentally verified by the measurement of relative redshift of light coming from distance galaxies. Hubble derived a law which tells us exactly how fast the galaxies should be receding. Before explaining this law let us explore Friedman's equation of general relativity. Friedman also postulated that universe is expanding. His equation implies expansion of space.
The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, i.e. the cosmological principle; empirically, this is justified on scales larger than ~100 Mpc. The cosmological principle implies that the metric of the universe must be of the form

where ds3^2 is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature. The parameter k discussed below takes the value 0, 1, −1, or the Gaussian curvature, in these three cases respectively. It is this fact that allows us to reasonably speak of a "scale factor", a(t).
Einstein's field equation now relates the evolution of scale factor a(t) with the pressure and energy of the matter in the universe.

which is derived from the 00 component of Einstein's field equations. The second is:

which is derived from the first together with the trace of Einstein's field equations. a is the scale factor, H = a``(t)/a(t) is the Hubble parameter. H is a function of time t and so is scale factor a(t). This is the brief history of the Hubble's parameter.
Then Georges Lemaître, in a 1927 article, independently derived that the universe might be expanding, observed the proportionality between recessional velocity of and distance to distant bodies, and suggested an estimated value of the proportionality constant, which when corrected by Hubble became known as the Hubble constant. Though the Hubble constant H(0) is roughly constant in the velocity-distance space at any given moment in time, the Hubble parameter H, which the Hubble constant is the current value of, varies with time, so the term 'constant' is sometimes thought of as somewhat of a misnomer.
Hubble's law is then mathematically V = H(0) D where v is the velocity and D is the distance.

### Historical background

You told us how an almost churchlike atmosphere is pervading your desolate house now. And justifiably so, for unusual divine powers are at work in there. Besso to Einstein, 30 Oct 1915
The basis of Einstein's general theory of relativity is the bold idea that not only do the metrical relations of spacetime deviate from perfect Euclidean flatness, but that the metric itself is a dynamical object. In every other field theory the equations describe the behavior of a physical field, such as the electric or magnetic field, within a constant and immutable arena of space and time, but the field equations of general relativity describe the behavior of space and time themselves. The spacetime metric is the field. This fact is so familiar that we may be inclined to simply accept it without reflecting on how ambitious it is, and how miraculous it is that such a theory is even possible, not to mention comprehensible. Spacetime plays a dual role in this theory, because it constitutes both the dynamical object and the context within which the dynamics are defined. This self-referential aspect gives general relativity certain characteristics different from any other field theory. For example, in other theories we formulate a Cauchy initial value problem by specifying the condition of the field everywhere at a given instant, and then use the field equations to determine the future evolution of the field. In contrast, because of the inherent self-referential quality of the metrical field, we are not free to specify arbitrary initial conditions, but only conditions that already satisfy certain self-consistency requirements (a system of differential relations called the Bianchi identities) imposed by the field equations themselves. The self-referential quality of the metric field equations also manifests itself in their non-linearity. Under the laws of general relativity, every form of stress-energy creates gravitation, including gravitation itself. Furthermore, after eschewing the comfortable but trusting principle of inertia as a suitable foundation for physics, Einstein concluded that "in the general theory of relativity, space and time cannot be defined in such a way that differences of the spatial coordinates can be directly measured by the unit measuring rod, or differences in the time coordinate by a standard clock...this requirement ... takes away from space and time the last vestige of physical objectivity".
It seems that we're completely at sea, unable to even begin to hypothesize a definite solution, and lacking any definite system of reference for defining even the most elementary quantities. It's not obvious how a viable physical theory could emerge from such an austere level of abstraction. These difficulties no doubt explain why Einstein's path to the field equations in the years 1907 to 1915 was so twisted, with so much confusion and backtracking. One of the principles that heuristically guided his search was what he called the principle of general covariance. This was understood to mean that the laws of physics should be expressible in the form of tensor equations, because such equations automatically hold with respect to any system of curvilinear coordinates (within a given diffeomorphism class). He abandoned this principle at some stage, holding that he and Grossmann had proven it could not be made coherent with the Poisson equation of Newtonian gravitation, but he subsequently realized the invalidity of their arguments, and re-embraced general covariance as a fundamental principle of formulation of GR.

Of course, the presence of any particle of matter with a determinate state of motion automatically breaks the symmetry, but a particle of matter is a dynamical object of the theory. The general principle that Einstein had in mind was that only dynamical objects could be permitted to introduce asymmetries. This leads naturally to the conclusion that the coefficients of the spacetime metric itself must be dynamical elements of the theory, i.e., must be acted upon. In this way, Einstein believed he had pointed out what he regarded as the strongest of Mach's criticisms of Newtonian spacetime, namely, the fact that Newton's space acted on objects but was never acted upon by objects.

By convention we define the Ricci tensor R(bc) as the contraction g(ad)R(abcd). In seeking favorable conditions for the metric field in empty space, Einstein observes that …there is only a minimum arbitrariness in the choice... for besides R(mn) there is no tensor of rank two which is formed from the g(mn) and it derivatives, contains no derivatives higher than the second, and is linear in these derivatives. This prompts us to require for the matter-free gravitational field that the symmetrical tensor R(mn) ... shall vanish. Thus, guided by the belief that the laws of physics should be the simplest possible tensor equations (to ensure general covariance), he proposes that the field equations for the gravitational field in empty space should be
Noting that R(mn) takes on a particularly simple form on the condition that we choose coordinates such that = 1, Einstein originally expressed this in terms of the Christoffel symbols as

(In his 1916 paper Einstein had a different sign because he defined the symbol Γabc as the negative of the Christoffel symbol of the second kind.) He then concludes the section with words that obviously gave him great pleasure, since he repeated essentially the same comments at the conclusion of the paper: These equations, which proceed, by the method of pure mathematics, from the requirement of the general theory of relativity, give us, in combination with the [geodesic] equations of motion, to a first approximation Newton's law of gravity, and to a second approximation the explanation of the precession of the perihelion of the planet Mercury discovered by Leverrier. These facts must, in my opinion, be taken as a convincing proof of the validity of the theory. To his friend Paul Ehrenfest in January 1916 he wrote that "for a few days I was beside myself with joyous excitement", and to Fokker he said that seeing the abnormalities in Mercury's orbit emerge naturally from his purely geometrical field equations "had given him palpitations of the heart". (These recollections are remarkably similar to the presumably spurious story of Newton's trembling hand when he learned, in 1675, of Picard's revised estimates of the Earth's size, and was thereby able to reconcile his previous calculations of the Moon's orbit based on the assumption of an inverse-square law of gravitation.)
The identity R(mn) = 0 gives ten distinct equations in the ten unknown metric components gmn at each point in empty spacetime (where the term "empty" signifies the absence of matter or electromagnetic energy, but obviously not the absence of the metric/gravitational field.) Since these equations are generally covariant tensor equations, it implies that given any single solution we can construct infinitely many others simply by applying arbitrary (continuous) coordinate transformations. So each individual physical solution has four full degrees of freedom which allow it to be expressed in different ways. In order to uniquely fix a particular solution we must impose four coordinate conditions on the g(mn), but this gives us a total of fourteen equations in just ten unknowns, which could not be expected to have any non-trivial solutions at all if the fourteen equations were fully independent and arbitrary. Our only hope is if the ten formal conditions represented by our basic field equations automatically satisfy four identities for any values of the metric tensor's components, so that they really only impose six independent conditions, which then would uniquely fix a solution when augmented by a set of four arbitrary coordinate conditions. It isn't hard to guess that the four "automatic" conditions to be satisfied by our field equations must be the vanishing of the covariant derivatives, since this will guarantee local conservation of any energy-momentum source term that we may place on the right side of the equation, analogous to the mass density on the right side of Poisson's equation

where we’ve chosen units so that Newton’s gravitational constant equals 1. In tensor calculus the divergence generalizes to the covariant derivative, so we expect that the covariant derivatives of the metrical field equations must identically vanish too. The Ricci tensor R(mn) itself does not satisfy this requirement, but we can form a tensor that does satisfy the requirements (using the Bianchi identity as explained below) with just a slight modification of the Ricci tensor, and without disturbing the relation R(mn ) = 0 for empty space. Subtracting half the metric tensor times the invariant R = g(mn)R(mn) gives what is now called the Einstein Tensor
Obviously the condition R(mn) = 0 implies G(mn) = 0. Conversely, if G(mn) = 0 we can see from the mixed form
that R must be zero, because otherwise Rmn would need to be diagonal, with the components R/2, which doesn't contract to the scalar R (except in two dimensions). Consequently, the condition Gmn = 0 is equivalent to Rmn = 0 for empty space, but for coupling with a non-zero source term we must use G(mn) to represent the metrical field. To represent the "source term" we will use the covariant energy-momentum tensor T(mn), and regard it as the "cause" of the metric curvature (although one might also conceive of the metric curvature as, in some temporally symmetrical sense, "causing" the energy-momentum). The components of this symmetrical tensor are the fluxes of the four components of momentum in each of the four directions. Thus the time-time component T(00) is the mass-energy density, T(0j) are the translational momenta, and the remaining T(jk) are momentum fluxes signifying pressures and shear stresses. (For this reason, Tμν is sometimes called the stress-energy tensor.) Einstein acknowledged that the introduction of this tensor is not vindicated by the relativity principle alone, but it has the virtues of being closely related by analogy with the Poisson equation from Newton's theory, it gives local conservation of energy and momentum, and finally it implies gravitational energy gravitates just as does every other form of energy. On this basis we surmise that the field equations coupled to the source term can be written in the form G(mn) = kT(mn) where k is a constant which must equal −8π (remembering that Newton's gravitational constant is 1 in our units) in order for the field equations to reduce to Newton's law in the weak field limit. Thus we have the complete expression of Einstein's metrical law of general relativity
------(2)

The minus sign of the right hand side is due to our choice of g(ad)R(abcd) for the definition of the Ricci tensor. As noted above, this is the negative of g(ac)R(abcd), which we could just as well have chosen as the definition of the Ricci tensor, in which case the sign of the right side of (2) would be positive. The choice is purely conventional. It's worth noting that although the left side of the field equations is quite pure and almost uniquely determined by mathematical requirements, the right side is a hodge-podge of miscellaneous "stuff". As Einstein wrote,

The energy tensor can be considered only as a provisional means of representing matter. In reality, matter consists of electrically charged particles... It is only the circumstance that we have no sufficient knowledge of the electromagnetic field of concentrated charges that forces us, provisionally, to leave undetermined in presenting the theory, the true form of this tensor... The right hand side [of (2)] is a formal condensation of all things whose comprehension in the sense of a field theory is still problematic. Not for a moment... did I doubt that this formulation was merely a makeshift in order to give the general principle of relativity a preliminary closed-form expression. For it was essentially no more than a theory of the gravitational field, which was isolated somewhat artificially from a total field of as yet unknown structure. Alas, neither Einstein nor anyone since has been able to make further progress in determining the true form of the right hand side of (2), although it is at the heart of current efforts to reconcile quantum field theory with general relativity. At present we must be content to let T(mn) represent, in a vague sort of way, the energy density of the electromagnetic field and matter. A different (but equivalent) form of the field equations can be found by contracting (2) with g(mn) to give R - 2R = -R = -8πT, and then substituting for R in (2) to give

## Exact solution of Einstein field equations

First there is a solution for rotating spherical massive body. This kind of solution is known as Kerr exterior solution. The external metric for a rotating mass is

Where m is the mass and a is the angular momentum associated with it.
Tholman found a solution for neutron star. In this case the solution takes the form as :

There is another solution of Lemaitre-Tolman(-Bondi) . It is for spherically symmetric non homogeneous models

## Unpacking field equation

Field equation is a complicated tensor equation, which is generally covariant. The very simple looking equation has a lot of other equations as already discussed. The unpacking of the field equation will results some basic fundamental ingredients of general relativity.

Brian Greene was a host in tv show where he explained exactly the above unwraping of field equation.

## Loop quantum gravity

As stated earlier, gravity breaks down at the instance of big bang and inside the balck hole. So we need a quantum theory of gravity to explain these special occurences in nature. Somewhat more accurate, they regard Ricci flatness as being an implication of only the first term in an infinite power series in the string constant α`, the higher-order terms providing us with ‘quantum corrections’ to Ricci flatness. The string lagrangian is defined as 1/2α` multiplied by the area of the world sheet which is 1-surface history.

The mathematics of string theory uses 'string histories' that are Riemann surfaces—having Riemannian (positive definite) metrics. But, physically, the string histories are Lorentzian. Passing from one to the other involves a kind of 'Wick rotation'.
The Calabi-Yau spaces in string theory are complex manifolds.

There are tens of thousands of distinct classes of such spaces. In fact, within a one particular class there are infinitely many different ones , distinguished by one parameter called moduli, describing its shape just like Riemann surfaces. The higher dimensions in string theory are very small. They are believed to exist as a curled up tiny dimension so we can not see with our bare eyes.
From string theory the idea of branes or membranes has sprung.

Membranes (or p-branes, or just branes), have p spatial dimensions and 1 time dimension, the worldsheet being (1 + p)-dimensional. These structures are involved, together with ordinary strings (1-branes), as part of the undefined M-theory.
We now recite Hawking-Bekenstein formula for a black hole's entropy in a slightly modified form:

Where m is the mass and e is the charge content of the hole.
A D-brane is classical entity (though possessing supersymmetry properties), representing a solution of 11-dimensional supergravity theory (a type of 'BPS state').

The two ends of an open string are supposed to reside on a timelike (q + 1)-dimensional subspace of spacetime called a D-brane, or D-q brane.

## Graviational lensing

Gravitational lensing is a distribution of matter and energy in space. This distribution make distance stars and galxies visible to the observer that look beyond into the sky. The light bends due to warped spacetime around this distribution of masses like the lens which alters the path of light rays. Thus distance galxies which are actually stright behind the mass distribution can be seen with naked eyes.

## Dark Matter and Dark energy

Dark matter is a form of matter that constitutes almost 85% of the matter content of the universe and about a quarter of its total energy density.

Dark energy is a hypothetical form of energy that is assumed to fill all of the space. It is believed to accelerate the expansion of the universe. Einstein considered his cosmological constant in the field equation to be a fudge factor which does not actually exist. But later it turned out that Cosmological constant is the cause of the apparent expansion of the universe. This acts as an anti-gravity. There is a relationship between dark energy and curvature my by Hubble's constant H.

## Perturbation method in General Relativity

The motion of the precession of perihelion of mercury is calculuted using perturbation method. It is a method to find an approximate solution by using a known solution. In other words, it is a solution whose terms form tylor's series polynomian in some functions of the unkown. The anomaly of the motion of the planet Mercury arises because of spacetime curvature. In Newtonian terms it is just a small correction to the original equation of motion of planets. We can , thus, find an approximate solution to this perturbed condition :

G(mn) is a rank two tensor. So it R(mn) and g(ab). Field equation is a tensor equation. Without the knowledge of tensor theory of general relativity is hard to understand. Tensor formalism has the beauty of simplicity and elegance. For interested reader s this tensor analysis page can be very helpful.
We know that general theory of relativity connects geometry with physics. But what is geometry ?
Geometry is a scientific knowledge and branch of Mathematics. It deals with spatial magnitudes. Any kind of geometry is possible if four axioms are satified :
a) continuity and dimensions : In a space of n dimension , a point is uniquely determined by n continuously changing variables.

b) lines and points: two points uniquely determine a straight line. Between 2n coordinates of a rigid body there exist a equation, which is the same for all congruent point pairs of the body. By considering sufficient number of point-pairs, we get more equations than unknown quantities: this gives us a method for detrmining the form of these equations , so that all can be satisfied.
b) axiom of free mobility: eEvery point can pass freely from one position to other. From (2) and (3) it follows that if two systems A, B can be brought into congruence in any position, it can be done in any position.
c) Axiom of rotation of rigid body(Monodromy): if (n-1) points of a rigid body remain fixed so that every other point describes a certain curve then the curve is closed

### Slipsism

Solipsism is the idea that only the mind exist. Nothing outside the mind exists. There is certain reason to believe in solipsism. This material world is the projection of our mind. What we see is just The figment of our imagination. What if this world is just a scientific construct. The people who believe in solipsism is called solipsists. solipsism is philosophically the same thing as idealism. Idealism states that only ideas and concepts create our reality. All these evidences certainly support in favour of solipsism.

DEFINITION OF A DYNAMICAL WORLD

Before roceeding to the laws of motion, which introduce new complications of which some are difficult to express in terms of pure mathematics, I wish briefly to define in logical language the dynamical world as it results from previous chapters. Let t be a one-dimensional continuous series, s a three-dimensional continuous series, which we will not assume to be Euclidean as yet. If R be a many-one relation whose domain is t and whose converse domain is contained in s, then R defines a motion of a material particle. The indestructibility and ingenerability of matter are expressed in the fact that R has the whole of t for its field. Let us assume further that R defines a continuous function in s. In order to define the motions of a material system, it is only necessary to consider a class of relations having the properties assigned above to R, and such that the logical product of any two of them is null. This last condition expresses impenetrability. For it asserts that no two of our relations relate the same moment to the same point, i.e. no two particles can be at the same place at the same time. A set of relations fulfilling these conditions will be called a class of kinematical motions. With these conditions, we have all that kinematics requires for the definition of matter; and if the descriptive school were wholly in the right, our definition would not add the new condition which takes us from kinematics to kinetics. Nevertheless this condition is essential to inference from events at one time to events at another, without which Dynamics would lose its distinctive feature.

### Russell's philosophy of theory of gravity

Although physics has worked with differential equations ever since the invention of calculus , geometry was supposed to be able to start with laws applying to finite spaces. If we adopt the viewpoint of Einstein , there can be no separation between geometry and physics; every propositionns of geometry will be to some extent causal. Take first the special theory. Relatively to axes (x,y,z, t) we can obtain propositions of geometry by keeping t constant; but relatively to other axes these propositions will refer to events at different times. It is true that these events (simultaneous events), in any system of co-ordinates , will have space-like interval, and will have no causal relations with each other, but they will have indirect causal relations derived from a common ancestry. Let us take for example , say : The sum of the angles of a triangle is two right angles. Our triangles may be composed of rods or of light rays.In either case , it must preserve a certain constancy while we measure it. Both rods and light rays are complicated physical structures, and the physical laws of their behaviour are involved in taking them as approximations to ideal straight lines. Neverthless, so far as the special theory is concerned, all this might be allowed , and yet we might maintain a certain distinction between geometry and physics, the former being a set of laws supposed exact, and approximately verified , for the relations of the x, y, z coordinates in any Galilean frame when t is kept constant.
But in general theory the intermixture of geometry and physics is more intimate. We can not accurately reduce ds(2) to the form :
dx(2)+dy(2)+dz(2) - dt(2)
and therefore can not accurately distinguish one coordinate as representing time. We can not therefore obtain a timeless geometry by putting t=constant. With this goes a change in our axioms. We no loger have, as in Euclid, in Lobatchevsky and Bolyai , and in projective geometry, axioms dealing with straight lines and finite lengths. We now have as our initial apparatus , infinitesimal from which large-scale results must be found by method of integration. From this point of view , Weyl's extension of Einstein seems natural. The proposition is that the statement that the distances AB and CD are equal is the assertion of the relation between the four points A, B, C and D. If all the relations that constitute our initial apparatus are confined to infinitesimal , so must this relation; if so points A, B, C and D must be close together and Weyl's geometry results.

## Tidal Effect

Imagine an astronaut Albert, whom we shall refer to as ‘A’, falling freely in space, a little away above the Earth’s atmosphere. It is helpful to think of A as being just at the moment of dropping towards the Earth’s surface, but it does not really matter what Albert’s velocity is; it is his acceleration, and the acceleration of neighbouring particles, that we are concerned with. A could be safely in orbit, and need not be falling towards the ground. Imagine that there is a sphere of particles surrounding A, and initially at rest with respect to A. Now, in ordinary Newtonian terms, the various particles in this sphere will be accelerating towards the centre E of the Earth in various slightly different directions (because the direction to E will differ, slightly, for the different particles) and the magnitude of this acceleration will also vary (because the distance to E will vary). We shall be concerned with the relative accelerations, as compared with the acceleration of the astronaut A, since we are interested in what an inertial observer (in the Einsteinian sense)—in this case A—will observe to be happening to nearby inertial particles. Those particles that are displaced horizontally from A will accelerate towards E in directions that are slightly inward relative to A’s acceleration, because of the finite distance to the Earth’s centre, whereas those particles that are displaced vertically from A will accelerate slightly outward relative to A because the gravitational force falls of with increasing distance from E. Accordingly, the sphere of particles will become distorted. In fact, this distortion, for nearby particles, will take the sphere into an ellipsoid of revolution, a (prolate) ellipsoid, having its major axis (the symmetry axis) in the direction of the line AE. Moreover, the initial distortion of the sphere will be into an ellipsoid whose volume is equal to that of the sphere.

The astronaut A (Albert) surrounded by a sphere of nearby particles initially at rest with respect to A. In Newtonian terms, they have an acceleration towards the Earth’s centre E, varying slightly in direction and magnitude (single-shafted arrows). By subtracting A’s acceleration from each, we obtain the accelerations relative to A (double-shafted arrows); this relative acceleration is slightly inward for those particles displaced horizontally from A, but slightly outward for those displaced vertically from A. Accordingly, the sphere becomes distorted into a (prolate) ellipsoid of revolution, with symmetry axis in the direction AE. The initial distortion preserves volume. (b) Now move A to the Earth’s centre E and the sphere of particles to surround E just above the atmosphere. The acceleration (relative to A = E) is inward all around the sphere, with an initial volume reduction acceleration 4pGM, where M is the total mass surrounded.
This last property is a characteristic property of the inverse square law of Newtonian gravity, a remarkable fact that will have significance for us when we come to Einstein’s general relativity proper. It should be noted that this volume-preserving effect only applies initially, when the particles start at rest relative to A; nevertheless, with this proviso, it is a general feature of Newtonian gravitational Welds, when A is in a vacuum region. (The rotational symmetry of the ellipsoid, on the other hand, is an accident of the symmetry of the particular geometry considered here.) Now, how are we to think of all this in terms of our spacetime picture?

I have had to discard a spatial dimension, because it is hard to depict a genuinely 4-dimensional geometry! Fortunately, two space dimensions are adequate here for conveying the essential idea.) Note that the distortion of the sphere of particles (depicted here as a circle of particles) arises because of the geodesic deviation of the geodesics that are neighbouring to the geodesic world line of A. I indicated why this geodesic deviation is in fact a measure of the curvature R of the connection ∇.
In Newtonian physical terms, the distortion effect that I have just described is what is called the tidal effect of gravity. The reason for this terminology is made evident if we let E swap roles with A, so we now think of A as being the Earth’s centre, but with the Moon (or perhaps the Sun) located at E. Think of the sphere of particles as being the surface of the Earth’s oceans, so we see that there is a distortion effect due to the Moon’s (or Sun’s) non-uniform gravitational field. This distortion is the cause of the ocean tides, so the terminology ‘tidal effect’, for this direct physical manifestation of spacetime curvature, is indeed apposite. In fact, in the situation just considered, the eVect of the Moon (or Sun) on the relative accelerations of particles at the Earth’s surface is only a small correction to the major gravitational eVect on those particles, namely the gravitational pull of the Earth itself. Of course, this is inwards, namely in the direction of the Earth’s centre (now the point A, in our spatial description; as measured from each particle's individual location. If the sphere of particles is now taken to surround the Earth, just above the Earth’s atmosphere (so that we can ignore air resistance), then there will be free fall (Einsteinian inertial motion) inwards all around the sphere. Rather than distortion of the spherical shape into that of an ellipse of initially equal volume, we now have a volume reduction. In general, there could be both eVects present. In empty space, there is only distortion and no initial volume reduction; when the sphere surrounds matter, there is an initial volume reduction that is proportional to the total mass surrounded. If this mass is M, then the initial 'rate' (as a measure of inward acceleration) of volume reduction is in fact
4pGM
where G is Newton’s gravitational constant
Tidal force is proportional to the mr^(-3) . That is , it is proportional to mass multiplied by cube of distance from the gravitating body.

## Black hole dynamics

Gravitional lensing is the effect of bending of light rays by mass distribution in space. Such a lensing effect is seen around our sun. Astronomer and physicist Aurther Eddington first proved gravational lensing effect experimentally.

The wavefront from a flash of light being focused and dragged back by a strong gravitational field
Sherically symmetric collapse of a black hole can be depicted with a diagram

Symmetric solution of field equation is

If the mass M is less than about 1.5-2M_0, this contraction can be halted by degeneracy pressure of electrons or neutrons resulting in a white dwarf or neutron star respectively. If, on the other hand, M is greater than this limit, contraction cannot be halted. But there is a singularity at 2m , which can be removed by a transformation

the metric takes the Eddington-Finkelstein form

This metric is perfectly regular at r = 2M but still has a singularity of infinite curvature at t = 0 which cannot be removed by coordinate transformation.
As the collapse continues and the density increases, the focusing effect will get bigger until there will be a critical wavefront whose rays emerge from the surface of the star with zero divergence. Outside the star the area of this wavefront will remain constant and it will be the surface r = 2M in the metric (refered above). Wavefronts corresponding to flashes of light emitted after this critical time will be focused so much by the matter that their rays will begin to converge and their area to decrease. They will then form tmpped surfaces. Their area will continue to decrease, reaching zero when they hit the singularity at r = 0.
A trapped surface is one where light is not moving away from the black hole. The boundary of the union of all trapped surfaces around a black hole is called an apparent horizon. Trapped surface is space-like surface, which area tends to increase in the future direction.

A future-trapped set S, together with the aesooiated aohronel sets E = E+(S), B = IyS], H+(B'), H = H+(IE). (For the proof of lemma (2.12)J The figure is drawn aooording to tho oonventions whereby null linen are inolined at 45'. The diagonally shaded portions are excluded from the space-time and some identieoations are made. The symbol co indicates regions 'at infinity' with reapeat to the metric. A future-inextendible timelike oupve 76D+(E)is depioted.
final area of the event horizon after collapse will be

The collapse of a star followed by the collapse of a thin shell of matter. The apparent horizon moves outwards discontinuously but the event horizon moves in a continuous manner

## Conformal Infinity

What can be seen from infinity is determined by the light-cone structure of spacetime. This is unchanged by a conformal transformation of the metric, i.e., g_ab -> Ω^2g_ab where Ω is some suitably smooth positive function of position. It is therefore helpful to make a conformal transformation which squashes everything up near infinity and brings infinity up to a finite distance. To see how this can be done consider Minkowski space

Introduce lagging and advanced time coordinates, w = t - r, v = t + r. The metric then takes the form

Now introduce new coordinates p and q defined by tan p = v, tan q = w, p - q => 0. The metric then becomes

This is of the form ds^2 = Ω^(-2)ds`^2 where dii2 is the metric within the square brackets. In new coordinates

the conformal metric ds`^2 becomes

This is the metric of the Einstein universe, the static spacetime where space sections are 3-spheres. Minkowski space is conformal to the region bounded by the null surface t' - T' = -π/2 [this can be regarded i18 the future light-cone of the point t' = 0, t' = -(π/2)] and the null surface t' + r' = π/2 (the past light-cone of T' = 0, t' = π / 2 )(Fig. below). Following Penrose (1963, 1965b) these null surfaces will be denoted by 2- and Z+ respectively. The point T' = 0, t' = (+-)π/2 will be denoted by t* and the points r' = π/2, 2' = 0 will be denoted by io. (It is a point because sin^2(2r') is zero there.) Penrose originally used capital 1's for these points but this would cause confusion with the symbol for the timelike future which will be introduced in the next section

Minkowski space M conformally imbedded in the Einstein Static Universe. The conformal boundary is formed by the two null surfaces I+, I- and the points i+io and i -

## Tensor field

If associated with each point on a manifold there is a tensor, a tensor field is defined. The figure below shows a simple example, that of a vector field on a sphere in Euclidean space.

## Covariant derivative

A tensor at a manifold point P is not defined on the manifold. It is, instead, defined in a plane tangent to the manifold at P. Hence, in curved spacetime, tensors at different points cannot be subtracted tensorially since they “live” in different tangent planes (see Poisson for more details).

The figure shows the different tangent planes at different points of the manifold
In other words, subtracting tensors at different points does not result in a tensor. The concept of differentiation must, therefore, be redefined. The new type of derivative is called a covariant derivative and it involves the operation of parallel transport.

Let us now put these statements in a mathematical form. First, suppose we have a vector field V defined around a curve on a manifold, which is parameterized by λ. Let u be a vector tangent to the curve (such as in Figure). Now, Take two points with coordinates x and x + dx on the curve. As noted above the following object

is not a tensor since after a coordinate transformation it becomes:

To obtain a proper tensorial derivative, one first needs to parallel transport the vector V, defined at point x, to the point x+dx, and then subtract it from the vector V defined at x+dx. The result of this subtraction must be linear on V and dx which leads to the definition of a (non-tensorial) object known as the connection Γ

The covariant derivative, which will now be a tensorial object, involves the sum of two derivatives with respect to the parameter λ along the curve:

Where the covariant derivative piece is

## Tangent vectors and coordinate function

A curve on a cylinder can be parameterized by &lamda;

We can then write the equations for γ using an arbitrary real parameter λ:

For concreteness consider an example in 3D flat space (instead of spacetime, for simplicity). A curve that spirals along the surface of a cylindrical surface is given by the following parametric equations:

At each point P of the manifold, there is a tangent plane that contains the vectors tangent to the curve γ.

One denotes these tangent vectors by:

For concreteness, let us consider the simple example of a circular path in the 2D plane. The (parametric) equations are:

The components of the tangent vectors are, therefore:

After an arbitrary coordinate transformation

such as, for example,

the new components of the tangent vector can be obtained by a direct application of the chain rule

Objects that transform like u are called contravariant vectors. Note here that we are using the so-called summation convention (or Einstein notation) according to which whenever an index is repeated, a sum over all possible values of the index is implicit. For example:

Another important mathematical object in general relativity is the covariant vector, which transforms as (see Poisson):

Since dual vectors are not as well known as vectors, let us consider an example and compute the components of a covariant vector A in spherical coordinates given its components in cartesian coordinates:

To find the components in spherical coordinates we use Eq. 101 with the following transformation equations:

where the following designations were used:

The first component of A in the new coordinate system is:

A quick calculation gives:

The other two components of A are obtained analogously.
(task: check to see if the first component is ok)
One can add several tensors to get a general tensor of arbitrary rank

metric tensor of Minkowski space-time is

Metric of Minkowski spacetime becomes

Consider a region of space where the energy-momentum T=0 and let Λ=0 for simplicity. Its is simple to show that the EFE(enstein field equation) simplifies to:
R_uv = 0;
In any new reference frame, the Ricci tensor remains equal to zero since:

To better understand the geometric meaning of R it is advantageous to understand the concept of local flatness according to which for any point P in spacetime one can always build a coordinate system such that:
The metric g becomes Minkowskian at P

The connection is also zero at P

Now, using Fermi normal coordinates, the spatial components of the metric tensor g can be written as follows:

Similar expressions exist for the other components of g. From this expression, it becomes clear that R measures some kind of departure from spacetime planeness.
The nonlinearity of general relativity implies that the linear combination of two solutions is not necessarily a solution:

Einstein hilbert action is

## Age of flat universe

In flat universe the curvature of space is exactly zero everywhere. Using Friedman's equation the age of this flat universe can be calculated as

critical density of our universe is (according to Friedman equations)

Equations of states are

Equations of standard model of particles

## A simple model for quantum cosmologist

Consider a closed Universe (k=1), “born from nothing” with constant vacuum energy density only. With this choice of parameters, the Friedmann equations describe the so-called inflationary phase. In this phase, the Universe expanded at an exponential rate. The first Friedman equation becomes (setting c=1):
--1
where “vac” stands to vacuum. Now, note that Eq. 136 has the same formal structure as the Hamiltonian with energy E=0. We can, therefore, associate the first term of Eq. 136 with a “kind of” kinetic energy corresponding to the “velocity” (which is the expansion rate of the cosmic scale factor):
--2
and a “potential energy” with the second term (inside the parenthesis). Adhering to the canonical quantization protocol we define:
p_a = -i(d/da)----3 where d/da is the derivative term (where the reduced Planck constant was set to 1). To write Eq. 136 in terms of the momentum operator we must express the conjugate momentum in terms of the expansion rate. This can be done using the principle of least action. The action, in this case, can be shown to be equal to:
--4
where:
--5
From S we obtain:
--6
then the first equation becomes
--7
We are now ready to quantize the Einstein equation for this simple case, using Eq.2. We obtain a simple version of the famous Wheeler-DeWitt equation (WdW):
--8
which has as its solution the “wave function the Universe”

## The Notion of Spacetime Vanishes

Quantization in the present context implies that the notion of spacetime itself becomes ill-defined. To better comprehend this, consider the following analogy:
--9 they imply that the scale factor and its rate of change a and da/dt can not be known simulaneously. ut to fully determine the FLRW metric at any time t, and therefore, characterize the structure of spacetime, we need both quantities. Therefore, the classical notion of spacetime vanishes in quantum mechanics.

## Initial conditions

To find the solution(s) of the WdW equation we need initial or boundary conditions. The best-known cases are quantum tunneling and the Hartle-Hawking (or no-boundary proposal) boundary conditions. They are both illustrated in the figure below.
--10
The solutions corresponding to the Hartle-Hawking (HH) and tunneling (T) boundary conditions (modified version of a figure in source).
There are actually three graphs: one is for potential function V(a) which is a function of scale factor. This potential has given rise to the classically forbidden region which is the left region separated by the vertical dotted line. The other two are the two wave function for two type of boundary conditions. Both the wave functions seem to have tunneled through the potential barrier.
The tunneling of the “particle” originally at the “origin” a=0 corresponds to the evolution from a zero-sized Universe (“nothing”) that pops spontaneously into existence non-singularly (with size a=a₀>0). The probability that this event occured can be calculated using basic quantum mechanics. The result is
--11
his expression can be interpreted using the concept of decoherent histories where probabilities are assigned to possible histories of the Universe.
This approach applies quantum mechanical rules to a single system (our universe) in contrast to standard interpretation in which universe is divided into an ensemble of systems.
In the general case, the spatial part of the metric, or 3-geometry, does not have all symmetries of the FLRW-type universes. As a consequence, the argument of the wave function of the universe Ψ is not a single function — a(t) in the FRW case — but a set of functions that describe the 3-geometry. More particularly, Ψ becomes a functional instead of a function (a functional takes a whole function and returns a single number) defined on superspace which is the configuration space of general relativity, the space of all 3-geometries.
--12

## Parametrized Particle Dynamics

Modified version of principle of least action is
--1
Parametrizing the path in phase space using the label τ and promoting the time variable t to the rank of a dynamical variable one defines the following four-component vectors
--2
where p is the momenta canonically conjugate to x. One can then rewrite the action as follow
--3
Noting from Eq 3 that the conjugate momentum corresponding to t is given by: p_0 = -H
and varying the action pₒ, one obtains, the following constraint:
--4
Following Dirac’s Hamiltonian formalism for constrained systems, to take the constraint Eq. 149 into account, a Lagrange multiplier term is added to the action:
--5

## Where are we actually? we do not have any exit? what are we doing here?...

--6

--7
If sciene can not solve the ultimate secret then why do we still practice science? I am thinking about the universe deeply recently. We are stuck here on the Earth. How can we escape this prison? We are not only imprisoned withing us but also within the earth and the solar system. How can we go to another galaxy? I must remember our late scientist Stephen Hawking. He said that it is not safe to store all the eggs in one planet. We must colonize other planet in solar system or perhaps in other solar system. There is danger that our planet be destroyed anytime however unlikely it may seem.

the differential dx in terms of new coordinates

extrinsic curvature of the hypersurface

eometric representation of the extrinsic curvature .

The Ricci scalar R (which we also previously discussed) can be written in terms of the extrinsic curvature K, its trace K, and the 3-curvature ³R (the 3-dimensional version of the Ricci scalar):

The next step is to write the Lagrangian density by using the same three quantities. This is an elaborate calculation, given in detail, for example, in Poisson. One obtains

with the corresponding action:

From the Lagrangian density, we find that the conjugate momenta corresponding to the lapse function and the shift vector are zero:

Associate momenenta of 3 metric is

The Hamiltonian is built in the usual way:

This Hamiltonian can be re-expressed as follows

where the two objects in the integrand are

From the Poisson brackets of the conjugate momenta with the Hamiltonian, one obtains:

The secondary constraints are then: