Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 20; Problem 20.10.
The dominant energy condition (DEC) states that if is any four-vector that is causal, that is, it satisfies the conditions
then we require the stress-energy tensor to satisfy the condition that if
then is also a causal four-vector. The causal condition is just a way of saying that a four-vector is either timelike (if ) or lightlike (if ). The DEC is a condition on the stress-energy tensor which amounts to saying that taking the scalar product of one of its rows or columns with a causal vector cannot produce a non-causal (spacelike) vector. Physically, this says that nothing can move faster than light. Note that it’s not a property that is automatically true of any stress-energy tensor; rather it is a condition imposed on the tensor to make it physically realistic.
We can use the DEC to show that the momentum density of a perfect fluid is always causal. The tensor in the fluid’s rest frame is
The momentum density is defined as the first row (or column) of the tensor:
In the rest frame,
so is causal in this frame. In a local orthonormal frame (LOF) the tensor’s components are
where are the orthonormal basis vectors in the LOF. If we plug in the definition 5 we get
where we got the last line by using the fact that is diagonal and . The term in square brackets looks like 3, as long as is a causal vector. However, this vector is just the observer’s four-velocity measured in the fluid’s frame, so
Thus is indeed causal, so we can invoke the DEC to say that if we define a vector by
then must be causal. We then get
With this definition, we can calculate
However, we know that is causal because that’s how we defined it in 15 and since its magnitude is a scalar, it is the same in all coordinate systems, so we must have . As for showing that , we can observe that
Since we see that is the Lorentz transformation of the causal vector and a Lorentz transformation never changes the spacetime nature of a four-vector (that is, timelike remains timelike, etc) so since is causal, and therefore as well. This condition is known as the weak energy condition or WEC.
Another property of the stress-energy tensor that can be derived from the DEC is as follows. In the rest frame of a perfect fluid, 4 holds, so the DEC condition 3 says, for some arbitrary causal vector we get another causal vector :
Since is causal, we must have for all choices of :
Because is causal, we have
and since this amounts to
Finally, we can revisit the case of the stress-energy tensor that gives negative pressure, , where is a positive scalar. In this case, if we apply the DEC to some causal vector we get
Since is a positive scalar multiplied by a causal vector, it too must be causal, so this stress-energy tensor satisfies the DEC.