I was watching a OCW from UC Irvine on GR. It starts with some discussions about special relativistic. This one about perfect fluid caught me because is not obvious how the Euler equation is derived. So I did it here.

It is an interesting topic in astrophysics since most of the objects we study can be considered (general) relativistic perfect fluid. As a first step, perfect fluid in special relativity is studied.

Starting by write down the stress-energy tensor at a point . First create the reference frame at that point, and the tensor looks like

Then generalize it using Lorentz transformation:

where consider only the boost
where is the Lorentz factor, then the tensor is

Combine the equations:

To get the equation of motion, we can use the energy-momemtum conservation law:

For ,
For ,
Define

This is called relativistic Euler Equaltion for Fluid Dynamics.

In Newtonian transformion, frame of references are related by Galilean transformation:

Thus we have the notion of "absolute" time flow. This does not satisfy . Lorentz transformation multiplies the time coordinate by a factor , which depends on :
Pluggin it into :

In general, the Lorentz transformation relates two coordinates in different reference frames by

has a important property:

where . It is similar to rotation in Euclidean space. It is usually seen as rotation in Minkovsky space.

From now on, we set .

Define proper time that

Also,

, so the proper time in another frame is

General form of boost-only Lorentz transformations:

If we define , we then have a unified form:

We can check if this obeys the identity:

Some 4-vectors

4-force

Define 4-force in that

If the particle is momentarily at rest, then , then the 4-force is

Then apply Lorentz boost

4-velocity

4-momentum

It can be defined as

then we get:

Another way to generalize momentum:
the momentum at rest is , applying Lorentz boost:

When , :
This should become the clasicall limit where , therefore
.

Scalars, Vectors, and Tensors

Contravariant vectors transform by Lorentz transformation; covariant vectors transform by inverse Lorentz transformation:

where
To prove this is indeed inverse Lorentz transformation:

Derivative with respect to a contravariant vector is covariant.

A tensor is a physical value that transform in certain ways.

How to construct tensors

Linear combination of tensors of the same kind

Multiply tensors with different indices (direct product). This gives symmetric tensors;

Contraction. This reduces the total number of indices by 1.

Differentiation/Division. Similar to direct product.

Special tensors

(4-invariant)

Levi-Civita symbol:

(Wikipedia) ... Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems, however it can be interpreted as a tensor density.

Special topics

Current Densities and Electromagnetism

If we have a simple particle that follows .

Defining , This generalizes to

This does not manifestly look like a 4-vector. For: 1. the Dirac delta is (3) multiple, and 2. explicit use of . Now we can turn it in to a manifestly covariant form by add a delta term and a integral

which also means that is a scalar. Then

is indeed a 4-vector.

is conserved

Observe that ,
This can also be written as .

Electromagnetic tensor

From Maxwell's equation:

Source equations:

Non-source equations:

We can write this down using 4-vectors/tensors. First introduce electromagnetic tensor defined as follows:

The source equations can be written as

The non-source equation can be written as

Equation of motion for EM

Energy-Momentum Tensor

The conservation of charse arises from the fact that current density is a 4-vector. In order for momentum to conserve, we need a order-2 tensor to correspond to the field.

The tensor of a particle looks like this:

which is manifestly a Lorentz tensor.

Electromagnetic field itself also contains energy. This energy can be described in terms of energy-momentum tensor:

Then we can guess the relation between the tensors:

The energy-momentum tensor of a point particle is give as:

There will be 4 conserved quantities:

Plugging it in, we get:

In EM, we have only one conserved quantity , which comes from:

Since GR is about the energy momentum tensor, in order for it to be covariant, we are essentially looking for a equation that looks like this:

Einstein didn't realize that this must be an equation of a 10-component tensor until 1912. This is the gravitational field. (In contrary, in Newtonian gravity, you can define everything with a one-component gravitational potential ).

Angular momentum and spin

Starting from energy-momentum conservation:

we define an object (angular momentum density):

Then we take the partial derivative of it:

This implies a conserved quantity (angular momentum) . The spatial components of the angular momentum tensor gives the classical angular momentum:

The time components of it gives mass moment:

A special note is that this quantity contains both orbital part and spin part of angular momentum, the latter does not depend on the origin. (I should make another note on this issue). This is called the Pauliâ€“Lubanski pseudovector.

(This is essentially talking about the Lie algebra of the Lorentz Group as we discussed before in QFT notes.)

In QM, we discussed rotation group in 3 dimensional Euclidian space:

where is a 3x3 matrix.

it corresponds to the transformation of an indexed wave function:

and is the rotation matrix on the state vectors. This mean while we need to change the coordinate to index into a field under rotation, we also need to change the components of the field at the same time given the field has more than one components.

The rotation matrix has a simple form:

where is the angular momentum matrices. It satisfies certain algebra (in terms of commutation relations):
one of the outcome of this algebra is that only certain eigenvalues of the spin is allowed:

Considering special relativity, the true transformation of the world is not only rotation, but also boost.
In SR, , therefore, we need to redo the exercise to make sure we can still only get
spin but not and such.

We write

which is the transformation associated with . They form a group:

To find , consider an infinitesimal Lorentz transformation looks like:

The product of two Lorentz transformation is:

So the product of two infinitesimal Lorentz transformation is the infinitesimal Lorentz transformation with the displacement equals to the
sum of the two transformations.

where is infinitesimal. From its being rotation:

Therefore:

Therefore is anti-symmetric.

For , we will pick an anti-symmetric to make a group:

These s, which we pick to be anti-symmetric., are called the "generators of the Lorentz group".

Lorentz group has 6 different parameters, therefore we need 6 of these 4x4 matrices, denoted as .

If is a 4-vector, then the , and then:

If all components of are independently arbitrary, then we can equalize the coefficients of the them (thus removing from the equation). Unfortunately is anti-symmetric. So we need to expand the summation:

Now we can cancel and get:

(Last time, I derived this from . This time its much simpler.)

The next step is to figure out the commutation relations. The result is:

Set(Pauli)

So this breaks up into two rotation groups. So the eigenvalues of and should be:
Therefore all the representations of the Lorentz group should be classified by the a pair of numbers .

(The relationship between dimension of the field/particle and the pair requires some further explanation).

An easier way to describe this is through potential:

where is the gravitational potential, which only depends on the position. For point mass:
for mass distribution:
It means that we can introduce a gravitational field:

Gravitational force is special:

It is the oldest force.

It couples to everything.

Gravity cannot be screened.

It is long range. (Mediated by massless particle if there is any.)

It is the weakest force.

History of development of Einstein's GR

~1907

Gravitational fields have relative existence. (Gravitational field can be removed by coordinate transformation).

There's a need to extend SR to accelerating frames.