Matter

There are to approach to deal with matter in the context of general relativity. Below we simply provide the actions for massive and massless particles, and for field matter; however, we will see later that this actions actually arise as a consequence of Einstein’s field equations (incorporate to this notes from Schuller GR when I get to that part). Their equation of motions can be obtained from an action by the corresponding Euler-Lagrange equations.

1. Point matter

One approach is to consider matter as massive or massless particles (see postulates 1 and 2 in worldline).

Free particles

For massive particles:

S_{massive} [γ] := m \int d λ \sqrt{g_{γ (λ)} (v_{γ, γ (λ)}, v_{γ, γ (λ)})},

For massless particles:

S_{massless} [γ, μ] := \int d λ μ g_{γ (λ)} (v_{γ, γ (λ)}, v_{γ, γ (λ)}),

where $μ$ is a Lagrange multiplier, which is introduced so that when you vary with respect to it, you get the condition:

g_{γ (λ)} (v_{γ, γ (λ)}, v_{γ, γ (λ)}) = 0,

which corresponds to condition (i) in Postulate 2. Of course, we also impose the condition:

g_{γ (λ)} (T, v_{γ, γ (λ)}) > 0

on our actions.

Why are we starting from actions instead of just starting from the Euler-Lagrange equations? The answer is simply that we can easily add different actions together and then find the corresponding equations of motion (e.o.m.) for that composite system. That is, composite systems have an action given by the sum of the constituent actions, possibly including interaction terms, and we then vary this composite action to obtain the complete e.o.m.

External forces

Roughly speaking, in special relativity, the reaction of a particle to a force is not instantaneous, but has some time delay. This time delay is explained by the fact that forces are mediated by fields, and if the particle is to react to the field, it must be coupled. So what we really mean by ‘presence of external forces’ is the ‘presence of fields to which the particles couple.’

The prime example of a particle coupling to an external field is that of a massive charged particle coupling to the electromagnetic field. The action can be written as:

S [g; A] := \int d λ [m \sqrt{g_{γ (λ)} (v_{γ, γ (λ)}, v_{γ, γ (λ)})} + q A (v_{γ, γ (λ)})],

where $A \in Γ^{*} T M$ is the electromagnetic potential on $M$ and $q \in R$ is the charge of the particle.

2. Field matter

Definition (Classical Field Matter). Classical field matter is any tensor field on spacetime whose equations of motion derive from an action. $◼$
The definition is broad and not very informative on its own. To understand what classical field matter is, we look at a specific example: Maxwell’s action. This action describes the behavior of the electromagnetic field, which is a $(0, 1)$ -tensor field $A$ .
Suppose, for simplicity, that the whole $M$ is covered by one chart $(M, x)$ . The action $S_{M a x w e l l} [A; g]$ for the electromagnetic field is given by:

S_{M a x w e l l} [A; g] := \frac{1}{4} \int_{M} d x^{4} \sqrt{- g} F_{a b} F_{c d} g^{a c} g^{b d}

where:

$F_{a b}$ : This is the electromagnetic field strength tensor or Faraday tensor (see electromagnetic field). Recall that it is, by definition, $F_{a b} := 2 \partial_{[a} A_{b]} = 2 \nabla_{[a} A_{b]}$ .
$g^{a c}$ and $g^{b d}$ : These are the components of the inverse metric tensor, which raises and lowers indices of tensors. In this expression, the metric $g_{a b}$ is assumed to be fixed. This means that we’re considering a situation where the curvature of spacetime (due to gravity) doesn’t change. This simplifies the analysis to focus solely on the electromagnetic field.

We can particularize this example even more. If we take our spacetime to be Minkowski spacetime and use the chart $(R^{4}, 1_{R^{4}})$ , we have $g = - 1$ , $g^{a b} = η^{a b}$ , and so the Maxwell action just becomes

S_{Maxwell}^{Mink} [A; g] = \frac{1}{4} \int_{R^{4}} d x^{4} F_{a b} F^{a b},

which is in the derivation given in the note electromagnetic field. Note, however, that it only takes this form in this chart. If we chose to use polar coordinates, we would not have $g = - 1$ nor $g^{a b} = η^{a b}$ .

Other examples

The action for a free matter field depends on the type of field (scalar, spinor, or vector). Below are examples for three common types of fields in relativistic field theory, formulated in a Lorentzian manifold:

Free Scalar Field (Klein-Gordon Field)
For a real or complex scalar field $ϕ (x)$ , the action is:

S = \int d^{4} x \sqrt{- g} (\frac{1}{2} g^{μ ν} \nabla_{μ} ϕ \nabla_{ν} ϕ - \frac{1}{2} m^{2} ϕ^{2}) .

Free Spinor Field (Dirac Field)
For a free spin-1/2 field $ψ (x)$ , the action is:

S = \int d^{4} x \sqrt{- g} (i \bar{ψ} γ^{μ} \nabla_{μ} ψ - m \bar{ψ} ψ)

where:

$ψ$ is the spinor field, and $\bar{ψ} = ψ^{†} γ^{0}$ is the Dirac conjugate,
$γ^{μ}$ are the gamma matrices in curved spacetime, satisfying the Clifford algebra ${γ^{μ}, γ^{ν}} = 2 g^{μ ν}$ ,
$\nabla_{μ}$ is the spinor covariant derivative, which includes both the usual connection and the spin connection.