Question

Consider Einstein's equation \(\mathbf{R}-\frac{1}{2} R \mathbf{g}=\kappa \mathbf{T}\). (a) Take the trace of both sides of the equation to obtain \(R=-\kappa T_{\mu}^{\mu} \equiv\) \(-\kappa T .\) (b) Use (a) to obtain \(R_{00}=\frac{1}{2} \kappa\left(T_{00}+T_{j}^{j}\right.\) ). (c) Now use the fact that in Newtonian limit \(T_{i j} \ll T_{00} \approx \rho\) to conclude that agreement of Einstein's and Newton's gravity demands that \(\kappa=8 \pi\) in units in which the universal gravitational constant is unity.

Short answer

The result of solving the given steps in the exercise led to the coupling constant \(\kappa = 8 \pi\) in Einstein's field equations, in units where the universal gravitational constant \(G\) is unity.

Step-by-step solution

Taking the trace

To take the trace of the Einstein's field equation, multiply both sides of the equation by the metric tensor \(g^{\mu \nu}\). This yields the equation: \(g^{\mu \nu} (\mathbf{R}_{\mu \nu}-\frac{1}{2} R g_{\mu \nu}) = \kappa g^{\mu \nu}\mathbf{T}_{\mu \nu}\). This simplifies into \(R=-\kappa T\), since the trace of the Ricci tensor \(R = g^{\mu \nu} R_{\mu \nu}\) and the trace of the stress-energy tensor \(T = g^{\mu \nu} T_{\mu \nu}\).

Transforming the field equation

Now, substitute \(R\) from Step 1 into the original field equation. This gives the equation: \(\mathbf{R}_{00} - \frac{1}{2}(-\kappa T)g_{00} = \kappa \mathbf{T}_{00}\) which simplifies to \(R_{00}=\frac{1}{2} \kappa\left(T_{00}+T_{j}^{j}\right)\).

Applying Newtonian limit

In the Newtonian limit, \(T_{ij} \ll T_{00} \approx \rho\). Substitute these approximations into the equation from Step 2 to yield \(R_{00} \approx \frac{1}{2} \kappa(2T_{00})\). Simplification gives \(R_{00} \approx \kappa T_{00}\). In Newtonian gravity, \(R_{00}\) corresponds to the Laplacian of the gravitational potential, which, as per Poisson’s equation, is \(4 \pi \rho\). Substituting this in gives \(4 \pi = \kappa\), however, in units where the universal gravitational constant \(G\) is unity.

Concluding

By applying the units where the universal gravitational constant \(G\) is unity, one finds \(4 \pi = \kappa\). In the original formula, however, \(G\) appears as \(G/c^4\), where \(c\) is the speed of light. Thus, in the actual units, \(\kappa = 8 \pi\), because the factor 2 appears from \(c^4\).

Key concepts

General Relativity

General Relativity is a revolutionary theory of gravitation that Albert Einstein developed, which has transformed our understanding of physics and the universe. Unlike the Newtonian view where gravity is a force between masses, General Relativity describes gravity as the curvature of spacetime caused by mass and energy. According to this theory, objects move along paths called geodesics, which are the 'straightest' possible paths in a curved spacetime.

This curvature is expressed mathematically by Einstein's field equations, which represent the core of General Relativity. These equations are complex and consist of ten interrelated differential equations that relate the geometry of space (described by the metric tensor) to the distribution of mass and energy in that space (represented by the stress-energy tensor). These equations are highly nonlinear and typically require advanced mathematical techniques to solve, often relying on numerical analysis and approximation methods.

Stress-Energy Tensor

The stress-energy tensor is a mathematical construct in the field of General Relativity that encapsulates the distribution of energy, momentum, and stress within spacetime. Simply put, it's like a ledger that records how much energy and momentum are present in each tiny segment of space and time. The components of this tensor include energy density, momentum density, and stress (like pressure and shear stress).

Importance in Einstein's Field Equations

In Einstein's field equations, the stress-energy tensor acts as the source of spacetime curvature. It's a pivotal term that tells spacetime how to bend in the presence of matter and energy, and this bending is what we perceive as gravity. The equations essentially state that the more energy and momentum present in a region of space, the stronger the curvature of spacetime will be in that region. Interestingly, light, which carries energy and momentum, can also influence the curvature of spacetime despite being massless—a radical departure from Newtonian theory.

Newtonian Limit

The Newtonian limit signifies the conditions under which General Relativity simplifies down and aligns with the well-established Newtonian mechanics. This is a realm where velocities are much lower than the speed of light, and gravitational fields are weak, which is mostly what we experience in our daily life.

Connecting General and Newtonian Gravity

Under the Newtonian limit, certain terms in the stress-energy tensor become negligibly small, and Einstein's field equations can be approximated in a way that they yield results consistent with Newton's law of universal gravitation. This limit is essential as it provides an essential check for the correctness of General Relativity—any new theory of gravity must replicate the successful predictions of Newton's theory in the appropriate limit. These include the motions of planets in the solar system and the behavior of objects near Earth's surface, because these phenomena were already well-explained by Newtonian mechanics before Einstein crafted his theory of General Relativity.

Gravitational Constant

The Gravitational Constant, often symbolized as 'G', is a fundamental physical constant that appears in both Newton's law of universal gravitation and Einstein's field equations of General Relativity. In Newtonian physics, 'G' quantifies the strength of the gravitational attraction between two masses. It is a proportionality factor that allows us to calculate the force of gravity acting between bodies.

Role in Einstein's Field Equations

In the context of General Relativity, the Gravitational Constant is related to the constant \(\kappa\) which appears in Einstein's field equations when considering a specific system of units where the speed of light \(c\) is set to unity. In these units, if we want to recover the predictions of Newtonian gravity in the appropriate limit, 'G' must effectively take on a value that ensures the equations 'bridge the gap' between the two theories, leading to the specific conclusion that \(\kappa = 8 \pi G/c^4\) in standard units.