# General Relativity

## Notation

• IFs is inertial frames

## Newtonian spacetime

### Stuff

Reformulation of Poisson laws:

in terms of the Newtonian spacetime curvature as

where is the 00-component of the Ricci tensor.

Then Newtonian spacetime is tuple where

### Absolute space

Absolute space at time is defined

and so (Newtonian) spacetime can be written as the disjoint union of the absolute spaces:

One way to visualize absolute time is as follows:

If absolute time didn't "flow uniformly", then we would allow something like the following, where we can have time standing completely still:

## Motivation

### Newton's laws

• Motion: and where
• is the inertial mass
• Gravity: where , which satisfies the (gravitational) Poisson equation

where

• is Newton's gravitational constant
• is density
• is the mass of the "source"
• Experiment: inertial mass is equivalent to gravitational mass, i.e.

which is called the weak equivalence principle, thus

Therefore we get the fact that locally, gravity is uniform acceleration

• Non-uniform gives rise to tidal forces. Follows from the deviation equation

### Special relativity

• Spacetime where points are events, with inertial frames

and we use the Minkowski metric

(a quadratic form), which is invariant in all IFs. Notice

• Motion is described by wordlines
• Free motion follow straight lines
• IFs related by Poincare group (which leaves invariant:
• Rotations
• Translations
• Lorentz boosts (rotation between space and time)
• Maxwell's eqns have Poincaré symmetry, but NOT Galilean symmetry (which is what Newton's gravity follows)

### General relativity

1. Laws of physics take same form in any coordinate system, i.e. invariant under general coord changes
• Will obtain by formuating laws with tensor fields
2. There exists lcoal IF with no gravity, where laws are as in special relativity (called the /strong equivalence principle)
• Spacetime is (differentiable) manifold with Lorentzian metric

## Postulates

Spacetime is a four-dimensional Lorentzian manifold .

More specifically, (relativistic) spacetime is defined by the tuple where

Let be a Lorentzian manifold.

Then a time-orientation is given by a smooth vector field that

1. do not vanish anywhere, i.e. for all

Here denotes an oriented atlas.

• Comparing with Newtonian spacetime we have gone from a "time" to a metric and time-orientation .
• Reason for introducing a metric now is that we want some way of enforcing the fact that a particle cannot move faster than the speed of light
• If we instead considered the metric then we observe that

so if , we are basically saying that the flow along has a "speed" which less than the speed of light (as we want).

• Observe then that (with our convention), that is "illegal"
• The above produces the well-known light-cone, where if we make a choice of time-orientation to enforce the fact that a particle cannot move into the past

## Connections and curvature

### Notation

• denotes a basis of a vector fields on
• denotes the dual basis of covector fields on \$M4
• In GR, the term connection will often be used to refer to the tensor ; this is because of how it relates to the connection 1-form
• Covariant derivative as a (1, 1)-tensor field

with components

• Components of connection

• Symmetric summation and anti-symmetric
• Torsion is denoted or in components and is given by

and in components

### Covariant derivative

Let be a basis of vector fields on . The components of the connection in a basis is defined by

Letting

the covariant derivative can be written

and so

where is the dual basis of . In a particular coordinate basis

Under a basis change and , we have

whre the basis change coefficients .

In particular, under a change of coordinate basis, this becomes

We use to denote fixed indices, and for contractible ones.

If we then let and we obtain our proof (up to commutation between functions, so we're fine).

Moreover, in a specific chart and changed basis we simply have

Substituting back into the above, we obtain our result.

Let be a connection on a smooth manifold .

A vector field is said to be parallely transported along a curve with tangent vector field if

The parallel transport condition in components reads

Along an integral curve we have , and so we can write

A geodesic is an integral curve of a vector field that satisfies

In other words, a geodesic vector field is parallely transported wrt. itself.

In a coordinate basis, the integral curves satisfy , and so the geodesic eqation is

Consider a geodesic with tangent field .

Suppose we reparametrize the curve so that with . The tangent field of the curve is then given by where . Thus,

Thus, the equation describes the same geodesic curve in a different parametrization.

An affine parameter is one for which . We deduce that this means that for and .

Let open for which is a bijection.

Normal coordinates at , of a point , are given by where are the components of

in a basis .

Consider the geodesic through and .

In normal coords. the geodesic takes the form

Inserting this into the geodesic equation and evaluating at , we deduce that

Hence

Since this is true for all in an open set , we have our proof.

### Torsion and curvature

#### Torsion

Let be a smooth manifold with a connection .

The torsion is a tensor defined by

where are smooth vector fields.

One can easily check that this is indeed a tensor.

Componentwise,

Geodesics are determined by the symmetric part of the connection components , thus the torsion does not affect the geodesics!

Let be a manifold with a connection .

For any function ,

We prove this by working in a basis .

Since , the covariant derivative of the covector is

Therefore, antisymmetrising, we get

Since this is a relation between tensors, it follows that it must hold in any arbitrary basis.

#### Curvature

The Riemann curvature of a connection is a defined by

where are smooth vector fields.

One can verify that this is indeed a tensor by checking that it's linear in and .

The Riemann tensor is the (1, 3) tensor defined by , where the ordering of the arguments is standard notation. The components are then defined

The component-wise expression for the Riemann tensor can be seen as follows:

where we first observe that

from which the Riemann tensor immediately follows.

Let be a torsionless connection.

For any vector field ,

This is called the Ricci identity.

First, observe that for a torsionless connection , we have

Now

Since this holds for all , we conclude our proof.

Let be a torsionless connection.

For any covector field

Ricci identity (for vectors) says that for some vector field we have

We can write , which gives us

We have

so

Locally

where we have dropped the resulting terms since they commute in the lower indices. Therefore

Locally the coefficients of a covector field can be expressed as for some vector field , so we simply let

and map indices and , giving us

Riemann tensor has an important geometrical interpretation.

It can be shown that is the change in upon parallel transport around a small quadrilateral whose opposite sides are integral curves of vector fields and . Hence, if parallel transport is locally path-independent.

The torsion and curvature of a connection vanish if and only if for any , there exists a chart such that and

In short,

The Ricci tensor is the tensor defined by contraction of the Riemann tensor

In components,

Suppose is a torsionless connection. Then we have

We will be working in normal coordinates.

Recall that in normal coordinates at , we have .

Hence, for a torsionless connection, this reduces to .

Therefore, the Riemann tensor in normal coordinates at is simply

1. Due to anti-symmetry in the last two indices,

Substituting in the above identity (since we are in normal coordinates),

since .

2. We have

So we need the last term to equal . To see this we write the expressions out

as we wanted. Substituting into the expression above,

as claimed.

Suppose is a torsionless connection. Then we have

This is called the Bianchi identity.

which follows directly from

Then observe that

since in normal coordinates we have

because since . Finally, substituting this into the above expression:

since partials commute.

### Levi-Civita connection

Let be a psuedo-Riemannian manifold. There exist a unique connection with vanishing torsion and satisfying .

This choice of connection is often called the Levi-Civita connection.

### Curvature of Levi-Civita connection

Let be the Levi-Civita connection. We define the (0, 4)-tensor

We work in normal coordinates at , so

and so

(at ) differentiate wrt. and evaluate at :

at .

AND MORE.

for Levi-Civita connection.

using Proposition proposition:4.61-lectures.

The Ricci scalar of is

The Einstein tensor

From the Bianchi identity

We contract by

Contract with , noting that by

Riemann tensor of vanishes if and only if for every , there exists a chart containing such that

Recall if torsion vanishes, then Riemann tensor vanishes if and only if there exists charts where . Then

in such charts.

This implies that is constant in chart, if we choose basis at to be orthonormal, which implies

Theorem thm:Riemann-tensor-vanish-iff-Euclidean-or-Minkowski-metric shows us that the Riemann tensor sort of measure the deviation from Euclidean or Minkowski metric, locally.

#### Weyl tensor

• Ricci tensor and Ricci scalar contain information about "traces" of the Riemann tensor
• Sometimes useful to consider separately those parts of the Riemann tensor which the Ricci tensor does not tell us about

The Weyl tensor is defined (in dimensions)

The Weyl tensor is basically the Riemann tensor with all of its contractions removed. The above formula is designed so that all possible contractions of vanish, while it retains the symmetries of the Riemann tensor:

One of the most important properties of the Weyl tensor is that it's invariant under

## Special relativity

Spacetime is just a Lorentizan manifold (Minkowski spacetime),

where

and are the inertial coordinates, i.e. .

Free motion: are timelike geodesics, ligthrays null geodesics

Physics: described by tensor fields on which obey evolution equations.

### Examples

#### Scalar field

Let be a scalar field, then

which is just the wave-equation:

#### Energy-momentum tensor

satisfies

which is conservation of energy / momentum.

#### Maxwell's theory of E. M.

• Electromagnetic field strength which obeys the Maxwell's equations:

• The corresponding energy-momentum tensor

Any matter distribution is described by energy-momentum tensor

#### Fluids

• Described by vector field , and typically normalized such that
• Perfect fluids:

where

• is the energy density
• is the pressure
• Then is the relativitic eqns. of fluid dynamics

## General relativity

Main idea: there exist local freely flowing frames with no gravity.

This is achieved by the following postulates.

Postulates:

1. Space-time is a 4D Lorentzian manifold
2. Free particles follow timelike / null-geodesics wrt. Levi-Civita of
3. Energy-momentum distribution of matter fields described by symmetric tensor field which is conserved

4. The curvature of is related to energy-momentum tensor of matter by the Einstein's equations:

where is the Newton gravitational constant.

• Note that might also depend on , so we cannot simply fix and solve

Laws of physics governed by:

1. General covariance: laws indep. of basis / coord system
2. Equivalence principle: in any local inertial frame (normal coordinate system) laws reduce to the laws in Minkowski spacetime (Minkowski space)

Do not fix laws uniquely! But suggests simple rule (called minimal coupling):

• given any equation in (Minkowski space), we replace

to get laws on curved spacetime

• Rules ensure general covariance as they output tensor equations.
• Local intertial frames are normal coords at , i.e.

### Examples

#### Wave equation

Applying these rules to wave-equation of S.R. we get

Consider the wave equation

also reduces to the wave-equation on spacetime since .

In local inertial frame, observe that we have

where the RHS is the wave-equation in .

Postulate 3: we have

and by wave-equation

#### Maxwell's equations

Applying these rules to Maxwell's equations in spacetime, we get

Postulate 3: we have

combined with Maxwell's equations we get

#### Fluids

• Described by vector field , and typically normalized such that
• Perfect fluids:

where

• is the energy density
• is the pressure

Then

Motion of fluid given by

Observe that if the pressure vanish, i.e. , then we're left with

which is the equation describing geodesic, fluid moves as "free particles".

### Einstein's equations

• Motivation:
• Newtonian
• Graviational field described by scalar potential
• Equation of motion

where in cartesian coordinates

• Relative acceleration of nearby particles ("tildal force") governed by deviation equation

where is the separation vector of the particles

• GR:
• Relative acceleration of two nearby particles following timelike geodesics is given by the geodesic deviation equation

where is the tangent to the geodesics and is now the deviation vector.

• Comparison suggests

## Spacetimes

### (anti-)de Sitter

de Sitter spacetime in dimensions with radius of curvature is locally isometric to the quadratic

Anti de Sitter spacetime in dimensions with radius of curvature is locally isometric to the quadrics

More precisely, the de Sitter spacetimes are the simply-connected universal covers of these quadrics.

Taking the limit is equivalent to the zero curvature limit in which we recover Minkowski spacetime.

The real affince space with a metric which, when expressed relative to affine coordinates, is given by

where we have introduced the speed of light .

We may take limits:

• non-relativistic limit (on the co-metric): gives us galilean spacetime
• ultra-relativistic limit (on the co-metric): gives us carrollian spacetime

These spacetimes are no longer lorentzian: the (co-)metric becomes degenerate in the limit, leading to a galilean and a carrollian structure, respectively.