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:

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.

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:

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:

- According to SR, no signal can travel faster than light, so how can gravitational attraction be

instantaneous? - 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). - 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). - The theory can not explain what happens to the gravitational field around a black hole.

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

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**.

energy generate a gravitational field

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

The mathematical description of how energy curves spacetime requires the use of mathematical constructs

or tools called

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,

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:

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

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!).

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:

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:

signs. Notice how the metric changes when a different coordinate system, such as spherical polars, is

used! However, the beauty of

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.

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:

two points is given by a curve called the

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.

accurately! Many experiments and observations have verified these predictions. A few of which are listed

below:

. Planets do not orbit the Sun exactly in ellipses, rather they__The perihelion precession of Mercury__

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!. When a planet is passing, in its orbit, behind the disc of the Sun,__Deflection of light by 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!. Radar beams have been used to measure accurately the distance to the planets__Radar echo delays__

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.. Distant galaxies and other objects with strong gravitational fields can severely__Gravitational lenses__

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.? One solution to Einstein's equation is the wave equation. This predicts the__Gravitational waves__

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!).. When light is radiated from an object with a strong gravitational field, such as__Gravitational redshift__

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.

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

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

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

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.

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.

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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