General Relativity
Einstein's theory of General Relativity (GR) extends his theory of Special Relativity (SR) by including the
effects of acceleration, such as that due to gravity. More than that, as we shall see, by the Equivalence
Principle, it equates acceleration due to gravity to other forms of acceleration and develops into a theory of
gravity itself.

We shall not, in this article, give a full mathematical description of GR, as this would require a lot of space to
explain such things as curvature, tensor notation, Christoffel symbols and metric connections. A good
account, aimed at undergraduate physics/maths students is given by Kenyon (1995). However, for the
mathematically curious we will include some derivations and results and a basic mathematical description of
the theory. Those who do not wish to follow the maths can simply skip these parts and still potentially gain a
good insight simply bey reading the text.

As a prerequisite it would help to read the pdf on Special Relativity first.

Newton's Theory of Gravity

We begin then with Isaac Newton and his theory of gravity. This theory states that a force acts between
masses and that the strength of this force is proportional to the amount of mass but diminishes with the
square of the distance between the masses (inverse-square law). This force is taken to be instantaneous.
For example, the gravitational force between the Earth and the Sun is large and this keeps the Earth
orbiting the Sun. Likewise, the force attracting an apple to the Earth is large, though the masses involved
are less, the distance between them is much greater. (Note, however, that the apple would not fly off into
space toward the Sun, since it is, along with the Earth, doing so anyway! One has to consider local forces.)
The gravitational attraction between an electron and a proton, however, is tiny and can (almost?) always be
ignored.

A mathematical summary of some of the key features of Newton's theory is given below:
Why do the planets orbit the Sun and the Moon orbit the Earth?

Should not the Earth fly straight toward the Sun? Actually, according to Newtonian mechanics, the centrifugal
force due to the Earth's motion as it orbits the Sun, exactly balances gravity and the Earth is in a stable orbit
(more-or-less) so that the Earth neither flies away nor toward the Sun.

Newton's theory successfully  accurately predicts how a projectile falls back to Earth, how the Moon orbits the
earth or the earth orbits the Sun and how an apple falls to the ground! It is still a useful theory for such
calculations. However, GR is more accurate and has to be used in some cases, such as when describing the
intense gravitational field around a black hole.

Already we can see some problems with Newton's theory:

  1. According to SR, no signal can travel faster than light, so how can gravitational attraction be
    instantaneous?
  2. Newton's theory does not accurately predict the orbit of Mercury around the Sun (Mercury experiences
    a more intense Solar gravitational field than the other planets).
  3. Photons are massless and yet light responds to the force of gravity (light rays are for example, bent or
    curved if they pass close to the Sun, an effect which can be measured).
  4. The theory can not explain what happens to the gravitational field around a black hole.


General Relativity (GR)

According to GR, gravity is actually due to the curvature of space-time in the presence of matter. Space-time
is a four-dimensional (4D) entity, with one dimension of time and three of space. Matter warps or curves this
space-time (as if curving it in another hidden 4th dimension of space, which is not, however, taken as a normal
dimension and can be referred to as a pseudo-dimension). Specifically, it is not mass but
energy itself which
warps the fabric of spacetime. This accounts for the deflection of light in a gravitational field, and the attraction
of photons toward gravitational sources. This is not surprising when Einstein's famous equation relates mass
to an energy equivalent:
So mass can be thought of as a 'form of energy' (not too literally) or as an energy-equivalent. All forms of
energy generate a gravitational field
. All forms of energy warp space-time, resulting in apparent
gravitational force. For example, the Sun warps space-time around itself, so that the planets, which would
otherwise be drifting through space in straight lines travel in elliptical orbits because spacetime is curved! It is
the
energy density that determines the curvature.

The mathematical description of how energy curves spacetime requires the use of mathematical constructs
or tools called
tensors.
Einstein's Equation

In relativity we have to deal with 4-vectors. Often in physics we deal with normal 3-vectors, such as a vector
describing position or momentum with three coordinates for the three spatial dimensions (such as: x, y, z in
Cartesian coordinates). The position 4-vector now comprises time ('position' in time) and the 3D position
vector: (t,
r), where r = (x,y,z). Similarly, we can pair energy with momentum, since energy, like momentum, is
conserved. Energy conservation arises from the homogeneity (uniformity) of time and momentum conservation
from the homogeneity and isotropy (the same in all directions) of space. Thus, we see that energy is the 'time
equivalent' of spatial momentum and it is natural to pair these together in the energy-momentum 4-vector.

As a result, our tensors will have 4 rows and 4 columns, but they will still be rank 2. Einstein's equation in terms
of tensors is summarised below:
The equation relates space-time curvature (described by a rank 2 curvature tensor with 4 rows and 4
columns) to the energy density tensor (also rank 2 with 4 rows and 4 columns). The curvature tensor is a
function of a tensor called the
metric tensor.

The stress-energy tensor contains components for all the sources of gravity, including the static
energy density, energy flows (e.g. heat flow in a dust cloud), the flow of momentum (pressure and viscous
drag or viscosity), momentum density and gravitational energy itself! Even gravity generates a gravitational
field! (However, this does not lead to infinite gravity since the recursion converges to a finite value!).

Metric, Metric Equations and Geodesics

Metric refers to scales of measurement. We are particular concerned with measured the distances between
events in space-time. Considering just 2D space for a moment, we can measure the separation between two
points using Pythagoras's theorem:
Pythagoras' Theorem is an example of a metric equation - an equation which gives the rules for
calculating the distance between points. This metric applies to flat space.

How do the rules or metric equations compare on a flat surface and on the surface of a sphere? If you
have ever peeled an orange and tried to flatten the peel, you will find there is not enough peel and it
splits - it simply can not be made to cover a flat surface perfectly! Similarly one can not wrap a sheet of
paper around an orange without introducing creases into the paper! The metric on the surface of a
sphere is different!

Similarly, we need different metrics to describe different regions of space-time. A region of flat space-time,
or Minkowski space-time, such as results in the absence of matter (though some situations approximate to
flat space-time, in particular the inertial frames considered in special relativity) has the metric shown
below:
Note that, whichever we take to be negative, the time and spatial components must have opposite
signs. Notice how the metric changes when a different coordinate system, such as spherical polars, is
used! However, the beauty of
tensor equations, like Einstein's Equation is that the form of the
equation does not change when we change coordinates - the physics is independent of our arbitrary
coordinate system as one would expect, and all correctly written natural laws should be similarly
invariant.

The Scwarzschild Metric

Scharwschild obtained the metric for curved space-time in the (outer) region of a stationary spherical
object, like a stationary planet or star. This is the Schwarzschild Metric. It can be applied approximately
to spacetime around slowly rotating objects, like the Sun. It also describes space-time around a
stationary black hole. This metric is given below:
This metric can be used to predict the orbits of the planets around the Sun. The shortest distance between
two points is given by a curve called the
geodesic. In flat space this is simply a straight-line. Obviously it will
not always be straight in other metrics - the geodesics on the surface of a sphere are the great circles of
latitude and longitude. However, to a being living on the surface of a sphere who had no idea their world was
spherical, a geodesic would appear to be a straight line. Geodesics can be thought of as the metric's
equivalent to straight lines. From Newton's first law of motion we know that an object will continue either in a
state of rest or moving in a straight line in the absence of an external force (force causes acceleration, which
can be a change in speed or a change in direction). The planets can be thought of as bodies drifting in
space along 'straight lines' or strictly geodesics. They only move in orbits around the Sun because space-
time is curved! To go in a straight-line actually requires a force (rockets to escape the gravitational tug of the
Sun!). According to GR, then, gravity is not a force in the conventional sense, rather it is the curvature of
space and time tending to direct the motion of objects through it.

Evidence for General Relativity

GR is not just a theory, rather it is a well-established scientific theory; meaning it makes testable predictions
accurately! Many experiments and observations have verified these predictions. A few of which are listed
below:

  1. The perihelion precession of Mercury. Planets do not orbit the Sun exactly in ellipses, rather they
    precess as the orientation of the orbital ellipse slowly rotates around the Sun (which is at one focus of
    the ellipse). The perihelion is the point of closest approach of a planet to the Sun and the position of
    perihelion thus gradually moves around the Sun. Mercury is closest to the Sun and so exposed to a
    stronger magnetic field and a more tightly curved space-time. It is not surprising, therefore, that
    Newton's theory does not accurately predict the rate of perihelion precession around the Sun for
    Mercury. This precession is dues in large part to interactions with the gravitational fields of other
    planets. GR does accurately predict the rate of precession!
  2. Deflection of light by the Sun. When a planet is passing, in its orbit, behind the disc of the Sun,
    light reaching us from the planet passes very close to the Sun and so is deflected to a measurable
    degree as it curves through a very curved region of space-time. GR accurately predicts the amount of
    deflection!
  3. Radar echo delays. Radar beams have been used to measure accurately the distance to the planets
    by bouncing the beams off the planets and listening for the returning echo. However, in escaping from
    the curved space-time around each planet, the returning radar echo is delayed by an amount of time
    predicted by GR.
  4. Gravitational lenses. Distant galaxies and other objects with strong gravitational fields can severely
    distort light passing them by. This can act like a gravitational lens, focusing the light. For example,
    when the light from a distant galaxy passes by another galaxy it sometimes gets bent (refracted) so
    much that multiple images of the distant galaxy are seen! This is predicted by GR.
  5. Gravitational waves? One solution to Einstein's equation is the wave equation. This predicts the
    occurrence of gravitational waves, for example from a binary neutron star - two compact and very
    dense stars with very intense gravitational fields orbiting closely to one-another are predicted to emit
    gravitational waves, losing energy and slowly spiralling in towards one-another. These waves are
    ripples in the space-time fabric. Space-time is extremely stiff and these waves are tiny, but contain vast
    amounts of energy! So far experiments have not been able to detect these tiny ripples, but the study
    of binary pulsars has shown that their orbits decay by the amount predicted by GR, as if they are
    radiating gravitational waves. (Not to be confused with gravity waves on water, a very different
    phenomenon!).
  6. Gravitational redshift. When light is radiated from an object with a strong gravitational field, such as
    a star, the light becomes redshifted (due to time dilation) - it's wavelength increases as if its is
    stretched when tugging against gravity. An observer some distance from the star will see the light
    redshifted by an amount that increases with distance from the star, up to a maximum according to the
    strength of the gravitational field. An observer far from a spaceship orbiting a star would see light
    beamed from the spaceship to be redshifted, whilst on observer closer to the star 9and downhill as it
    was) would see the light blueshifted. Observations confirm gravitational redshift.
Shortcomings of GR and Quantum Gravity

However, like newton's theory of gravity, GR is still not the complete picture of gravity, though it is more
accurate and applies to a wider range of phenomena. In particular, GR is thought to fail when describing
very intense gravitational fields and on the particle scale. For example, the Schwarzschild metric predicts
that the matter of a black hole shrinks to a mathematical point of infinite energy density - a
singularity. This
happens when matter is either too dense or two massive such that the gravitational field is so intense that
light can no longer escape from its surface and we have by definition a black hole. In
supermassive black
holes, such as the one in the nucleus of the Milky Way Galaxy, there is so much mass that even if the
density approximates that of water then light can not escape! For a stellar black hole, the collapsing core of
a dying star which has above the critical Chandrasekhar mass then matter can not compete against the
strong gravitational field and nothing can halt the collapse of matter.  Singularities cause all sorts of
conceptual and mathematical problems and nature seems to have ways of avoiding them (they usually tell
us that our mathematical description is incomplete).

It is thought that a theory of quantum gravity may remove the singularity. Attempts to quantise GR have not
been successful. Quantum gravity is needed to explain the very small, like the singularity in our black hole.
Alternative theories that aim to unite GR with quantum mechanics are being developed. These include
String Theory.

If gravity waves do indeed exist, then we would expect them to be quantised (like all physical waves) - that is
we expect gravity to be composed of quanta or particles called
gravitons. What we are really saying is that
the space-time matrix itself is quantised, or made up of particles and so granular and not the smooth
continuum we imagine it to be. (Interestingly, thinking of it in either way introduces conceptual problems).
This means that there is a minimum time interval, called the
Planck Time, and a minimum length, called the
Planck Length. To have any time shorter than the Planck time or any length shorter than the Planck
Length is either impossible or devoid of precise meaning - perhaps length and time lose meaning below this
scale as we enter the realm of 'chaotic' quantum fluctuations.
A Derivation of the Schwartzschild Metric
The presence of matter (energy), like this star warps the space-time matrix, curving space and time. This
causes planets to orbit the Sun instead of drifting through space in straight lines and accurately predicts the
orbit of Mercury where Newton's theory of gravity fails to do so.
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