Problem 20 - A First Course in General Relativity

Question

This exercises introduces the concept of the active gravitational mass. After deriving the weak-field Einstein equations in Eq. (8.42), we immediately specialized to the low-velocity Newtonian limit. Here we go a few steps further without making the assumption that velocities are small or pressures weak compared to densities. (a) Perform a trace-reverse operation on Eq. $$ \square h^{\mu v}=-16 \pi\left(T^{\mu v}-\frac{1}{2} T^{\alpha}{\underline{\phantom{xx}}}_{\alpha} \eta^{\mu v}\right) $$ (b) If the system is isolated and stationary, then its gravitational field far away will be dominated by \(h^{00}\), as argued leading up to Eq. (8.50). If the system has weak internal gravity but strong pressure, show that \(h^{00}=-2 \Phi\) where \(\Phi\) satisfies a Newtonian-like Poisson equation $$ \nabla^{2} \Phi=4 \pi\left(\rho+T_{k}^{k}\right) $$ For a perfect fluid, this source term is just \(\rho+3 p\), which is called the active gravitational mass in general relativity. If the system is Newtonian, then \(p \ll \rho\) and we have the usual Newtonian limit. This is another example of a post-Newtonian effect.

Step-by-Step Solution

Answer

Trace-reverse, identify \( h^{00} \), derive \( \Phi \) using Poisson's equation with \( \rho + 3p \) as source.

Step 1: Understand the Equation Setup

We start with the given equation for the metric perturbation in weak fields: \( \square h^{\mu u} = -16 \pi \left( T^{\mu u} - \frac{1}{2} T^{\alpha}_{\alpha} \eta^{\mu u} \right) \). This equation describes how the stress-energy tensor \( T^{\mu u} \) influences the spacetime around it, according to Einstein's theory of General Relativity. Our goal in part (a) is to perform a trace-reverse operation on this equation.

Step 2: Perform the Trace-Reverse Operation

To perform the trace-reverse, we first define the trace \( h = \eta_{\mu u} h^{\mu u} \). We then express the trace-reversed perturbation as: \( \bar{h}^{\mu u} = h^{\mu u} - \frac{1}{2} \eta^{\mu u} h \). By substituting these into the wave equation, the trace is adjusted by the formula \( \bar{T}^{\mu u} = T^{\mu u} - \frac{1}{2} \eta^{\mu u} T^{\alpha}_{\alpha} \), simplifying it to a version where the trace is removed.

Step 3: Identify the Dominant Gravitational Field Component

For an isolated and stationary system, far from the source, the component \( h^{00} \) dominates the gravitational field, as suggested by prior analyses like leading up to Eq. (8.50). In this regime, the scalar \( \Phi \) appears in the relation \( h^{00} = -2 \Phi \), indicating the gravitational potential that emerges naturally in weak field limits.

Step 4: Poisson’s Equation for Gravitational Potential

To derive \( h^{00} \) in terms of \( \Phi \), consider Poisson's equation for the gravitational field \( abla^2 \Phi = 4 \pi \left( \rho + T_k^k \right) \), which expresses how the potential \( \Phi \) relates to the source terms. For a perfect fluid, this simplifies to \( \rho + 3p \), identifying the active gravitational mass in general relativity.

Step 5: Relate to Post-Newtonian Effects

In the strong pressure situation, the extra pressure term \( 3p \) modifies the gravitational response. This situation contrasts with the classical Newtonian case where pressure \( p \) is negligible compared to density \( \rho \). Thus, \( \Phi \) aligns with a generalized Poisson equation, displaying a post-Newtonian correction to gravitational theory.

Key Concepts

Weak-Field Einstein Equations

When we talk about the weak-field Einstein equations, we are exploring a scenario in General Relativity where gravitational fields are not too intense. These fields are weak enough to allow some simplifications but still reflect Einstein’s insights into gravity. In this regime, the spacetime curvature induced by a mass is considered as a small deviation from a flat metric.
This leads us to the equation:

• \( \square h^{\mu u} = -16 \pi \left( T^{\mu u} - \frac{1}{2} T^{\alpha}_{\alpha} \eta^{\mu u} \right) \)

This equation suggests how the stress-energy tensor \( T^{\mu u} \), which describes matter and energy, affects spacetime. It gives us a way to account for small gravitational influences by treating the metric as a perturbation \( h^{\mu u} \).
A key goal of working with weak-field equations is to simplify the calculations by using approximations while still capturing the essence of gravitational interactions as described by General Relativity.

Trace-Reverse Operation

The trace-reverse operation is crucial in simplifying equations in General Relativity. It involves modifying a metric perturbation by adjusting its trace. Here's how it works:
First, define the trace of a perturbation as:

• \( h = \eta_{\mu u} h^{\mu u} \)

Using this trace, the trace-reversed perturbation becomes:

• \( \bar{h}^{\mu u} = h^{\mu u} - \frac{1}{2} \eta^{\mu u} h \)

Performing this operation adjusts the initial equation by removing unnecessary components related to the trace. This makes it simpler to work with, especially in contexts where certain symmetries simplify the equations.
By using this method, we can refine our understanding of how gravitational effects propagate through spacetime without being bogged down by redundant terms.

Poisson Equation in General Relativity

The Poisson equation is a fundamental aspect of describing gravitational fields, even extending into the framework of General Relativity. It provides a way to link mass-energy distributions to the gravitational potential \( \Phi \).
Consider the equation:

• \( abla^{2} \Phi = 4 \pi \left( \rho + T_{k}^{k} \right) \)

Here, \( \rho \) represents the density of mass-energy, and \( T_{k}^{k} \) accounts for pressures in the system. This is vital in contexts where pressure contributes substantially to the gravitational field.
In a perfect fluid, this simplifies to \( \rho + 3p \), which we term as the "active gravitational mass." This concept reveals how pressure can enhance gravitational fields in ways Newtonian gravity doesn’t account for. It shows how General Relativity naturally extends gravitational concepts to consider pressures in the mass-energy budget.
When internal gravitational forces are weak, yet pressure is significant, these extensions highlight how gravity works beyond simple Newtonian approximations. This provides a broad view of gravity's interplay with mass and pressure, capturing more complex gravitational phenomena.