In this notes, we are particularly interested in some special model of general relativity. More precisely, here we consider the scalar fields since, for example, a real scalar field more or less provides one of the simplest sources of stress-energy in GR.

**1. Derivation of stress-energy tensor**.

To derive the stress-energy tensor associated to a real scalar field, we make use of the Einstein-Hilbert action. Suppose that the full action of the theory is given by the Einstein–Hilbert term plus a term describing any matter fields appearing in the theory, then we have

The action principle then tells us that the variation of this action with respect to the inverse metric is zero, yielding . Calculating this equation gives

where the right hand side is nothing but the stress-energy tensor .

In modern cosmology, one can introduce on the spacetime a real scalar field with potential as a smooth function of . A particular Einstein field theory is specified by the choice of an action principle with

To find its associated stress-energy tensor, we first look at . Since the term does not depend on the metric, we get

where we have used the fact that partial derivatives also do not depend on the variation of metric. Therefore,

**2. Derivation of the momentum constraint equation.**

Following the notes where we have derived the momentum constrain equation, we know that the momentum constraint equation is nothing but

here we simply assume that . Using components and assuming known as the timelike vector sitting in the frame, we write

It is important to note that is a projection on and on the normal to of the stress-energy tensor .

Therefore, by raising one index and let the other to be zero, we found that the last two terms in the formula for vanish, which yields

where is the induced of on and is also the induced metric on of the spacetime metric .

**3. Derivation of the Hamiltonian constraint equation.**

We now turn to the Hamiltonian constraint equation which can be written as

Clearly, in terms of components and again we assume that , it is nothing but

To calculate we simply get

**4. The constraint equations with the adapted frame**.

First, we take local coordinates adapted to the product structure as follows . For a natural frame on , we choose

The dual coframe is found to be such that

while the -form is nothing but . The last thing we need to find is the vector . To make it correct, we choose

As in this post and thanks to , we find that

Similarly, one can also obtain

Finally, it is easy to see that

for all . By collecting the above calculation, we arrive at

The components of the inverse matrix can be calculated as given below

Therefore, the stress-energy tensor has the following

Thanks to the fact that, in the local frame,

we find that

here is being considered as a vector field, , which yields the following constraint equation

For the momentum constraint equation, we can check that

here is also being considered as a vector field, . Hence, we have

The term is sometime written as with

known as the normalized time derivative.

See also:

- Why do the Einsteins equations describe the propagation of wavelike phenomena?
- Derivation of the Einstein constraint equations: The Hamitonian constraint
- Derivation of the Einstein constraint equations: The momentum constraint
- Construction of spacetimes via solutions of the vacuum Einstein constraint equations and the propagation of the gauge condition
- The evolution equations for the constraint equations
- decomposition of the spacetime metric
- The harmonic slicing

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