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The Weak Field Limit

Using Cartesian space coordinates, the metric determined from the radar method is,
The partial derivatives of the metric are
Using , the index raising operators are
The geodesic equation is
where the Christoffel symbols are
For the Christoffel symbols to be non-zero, two indices must be the same, and, in a constant field, the other must not be zero. For non-relativistic velocities, terms in the order of velocity squared can be ignored, and we have . Then, 3-acceleration is given by, for a ≠ 0,
Writing k = 1 + V, where V = z is small, we have that acceleration is minus the gradient of V,
So, gravitational redshift can be identified with the scalar potential in Newtonian gravity. The time component of the metric is
This is called the weak field limit. mV has units of kinetic energy, ½mv². To convert to conventional units, we must divide by c².
Thus,
and it is seen that very small redshifts are associated with normal accelerations in a gravitational field.

Conservation of Mass-Energy

Consider a particle of matter, using primed coordinates in which the particle is stationary. A fundamental observation is that its mass remains constant in time. We say that mass is conserved. This is expressed in the simple equation,
This is not a covariant equation, as it depends on coordinates in which the particle is stationary. General covariance means this should be replaced by a vector or tensor equation, saying exactly the same thing, but in any coordinates. Replace m by a vector, p^i' =; (m, 0, 0, 0), which expresses the fact that the particle is stationary, and replace d/dt' by the covariant derivate. Then the equation ,
is a restatement of conservation of mass, and is expressed in terms of vectors so that, in any coordinate system, it has the same form. More generally, a particle at a point is replaced by a probability density function, reflecting the fact that we don’t know exact position. p is then a vector field, describing momentum density. We can then sum the densities for all particles to find the distribution of matter and net momentum within neighbourhood. By linearity, the equation of conservation of mass still holds.

In classical physics, this equation appears in a number of contexts, e.g, using, for i = 1, 2, 3,
the mass continuity equation in fluid dynamics,
where ρ is mass density and v is 3-velocity, or the continuity equation in electrodynamics
where ρ is charge density and j is current density.

The Stress-Energy Tensor

We know from observation that matter is the source of the gravitational field. Einstein sought a law to express this. Quantitative science requires that laws are given in the form of equations. A tensor equation is required by general covariance, so as to ensure the general principle of relativity, that local laws of physics are the same for all observers. We need a tensor equation relating curvature to mass density. A four index equation for the Riemann curvature tensor would be too restrictive — tidal forces show that curvature is not zero in empty space. Einstein started to look for an equation involving the Ricci tensor. The Ricci tensor is a rank 2 tensor. To form an equation we need a rank 2 tensor describing the density of matter.

It is trivial to write down a rank 2 tensor describing a stationary particle of mass, m. In primed, locally Minkowski, coordinates, this is
The same tensor is described generally using coordinate transformation. Coordinate transformations for tensors act on both indices. For Minkowski coordinates, coordinate tranformations are Lorentz transformations, and represent a velocity boost such that proper velocity (1, 0, 0, 0) becomes v^a =  (v⁰, v¹, v², v³). To transform the tensor m^i'j', we must apply the transformation to both indices separately. The result is
More generally, mass and velocity of a point particle are replaced by density functions. We then sum over all the particles of matter within a region. The resulting symmetrical tensor, T^ab, is the stress energy tensor. The stress-energy tensor is defined at each point of a spacetime coordinate system (i.e. it is a field). While the momentum vector, p, for a body, has a simple intuitive meaning as the sum of the momenta of its component subparticles, it is less easy to see the meaning of the sum of products, v^av^b, in the stress-energy tensor. In statistics, similar sums of products appear in the study of correlation. The stress-energy tensor may be regarded as a measure of not just of total motion, but also of the correlation between the motions of component particles, as described in the term “stress”.

Observe that
by conservation of mass-energy and by the geodesic equation, which follows from Newton’s first law. The law of local energy-momentum conservation follows by linearity.


More generally, local energy-momentum conservation still holds when particles interact, since momentum is conserved in interaction. This will be shown in Quantum Electrodynamics. For now, it is treated as a fundamental law with application in specific equations of physics. For example, in the non-relativistic approximation,
reduces to, for a = 0, v⁰ = 1, to conservation of mass in fluid dynamics, and, for a ≠ 0 it is the Navier-Stokes equations for a perfect fluid in the absence of pressure and external forces. T^ab_;b = 0 can also be shown, from Maxwell’s equations, to express conservation of energy-momentum for a classical electromagnetic field, in which energy is given by
and momentum is given by

Einstein’s Law of Gravitation

In his early researches Einstein tried to relate the Ricci tensor to the stress energy tensor, but found that this did not work. If curvature is the consequence of stress-energy, then it must reflect the law of local energy-momentum conservation as an identity. The Einstein tensor, G^ab, is unique as a rank 2 tensor formed by contraction of the the Riemann curvature tensor, which is symmetrical, and which satisfies an identity,
the contracted Bianchi identity. Einstein therefore wrote down the law of gravitation known as Einstein’s field equation
where κ is a constant of proportionality, to be determined by comparison with the Newtonian approximation. This law summarises the two principle tennets of Einstein’s theory of gravity, that matter is the cause of curvature, and, through the Bianchi identity, that energy-momentum is conserved and hence that inertial objects follow geodesics, thereby combining Newton’s three laws of motion and Newton’s law of universal gravitation into a single tensorial law.

Stationary Matter

In the case of a static body of uniform density, ρ, T^ab is,
Einstein’s field equation, written in terms of the Ricci tensor is
Contract the indices a and b, noting from the summation convention that ,
So, the total scalar curvature in the presence of matter is given by
Hence

The Newtonian Approximation

The Christoffel symbols are
The Ricci tensor is
In the Newtonian approximation, the metric, g, is slowly varying in space and constant in time. We may neglect terms of second order in derivatives of the metric, and set time derivatives to zero. Then
In Cartesian, x-, y-, z-coordinates
where is the Laplacian operator. Thus,
Poisson’s equation for a Newtonian gravitational potential, k, due to a mass distribution of density ρ is
where G is Newton’s universal gravitational constant. Thus, κ = 8πG, and Einstein’s field equation is

The Schwarzschild Solution

The first solution to Einstein’s field equation was found, within a few months of the publication of general relativity, by Karl Schwarzschild, who was on active service in Russia. Schwarzschild had been known as a prodigy and the calculation is quite involved. It shows that the metric outside of an isolated, spherically symmetric, non-rotating, gravitating body of mass M is
This is the Schwarzschild metric. In the weak field limit, the redshift factor
is equivalent to a potential,
in accordance with Newton’s inverse square law of gravity.

Black Holes

The Schwarzshild metric becomes singular at the Schwarzschild radius. For a normal star or planet, this is not important because Schwarzchild geometry applies only in empty space outside of the star. The radius of a normal star is greater than its Schwarzschild radius, but there is a theoretical possibility that a body could exist for which its mass is contained within its Schwarzschild radius. Such a body is called a black hole. In practice, we know that when a star burns out it can become a white dwarf. Provided its mass is less than the Chandrasekhar limit (about 1.4 solar masses) electron degeneracy pressure due to the Pauli Exclusion Principle prevents further collapse against the force of gravity. For a burned out star greater than this mass, gravity overcomes electron degeneracy pressure and the star collapses. Energy released in collapse creates a supernova explosion, electrons and protons combine to form neutrons, and a neutron star may form. If the neutron star has a mass greater than about 5 solar masses, gravity overcomes neutron degeneracy pressure, and collapse continues indefinitely. In practice, many galaxies, including our own, appear to have supermassive black holes at their centres. Observations show that Sgr A*, at the centre of the Milky way, is almost certainly a supermassive black hole.

From the pespective of an external observer, the redshift factor, k = (g₀₀)^−½, becomes zero at the Schwarzschild radius, showing that time, and all physical processes, slow down relative to time measured by a distant observer as matter approaches the black hole. The external observer would calculate that matter does not actually fall through the Schwarzschild radius, but stops at it. This is true also of light. The Schwarzschild radius is an event horizon. Processes inside an event horizon are hidden to the observer. An issue was raised as to whether a singularity could actually form according to the equations of relativity. Oppenheimer & Snyder published the first calculation of gravitational collapse in 1939, showing that one can.

In fact the singularity at the event horizon is only a singularity in a particular coordinate system. If instead of using coordinates defined by an external observer, stationary with respect to the hole, we use coordinates determined by an observer falling into it, it can be shown that no singularity arises at the Schwarzschild radius. According to general relativity, the observer simply falls through empty space at the Schwarzschild radius, into a region from which he can no longer communicate with the external observer. There is no singularity in coordinates defined by the falling observer, and physical processes proceed as normal until he meets the true singularity at r = 0. That is the solution according to Einstein’s field equation, but the question still arises as to whether it is a real physical solution. The meaning of the singularity at r = 0 is that known laws of physics break down. We cannot say, from classical general relativity, precisely at what point the laws of physics break down in the vicinity of the singularity. Relational quantum gravity will reexamine this issue in the light of a unification with quantum theory.

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