Question

(a) Compute the components of the Ricer tensor } R_{\mu v} \text { for a space-tume that has a }\end{array}\( metric of the form $$ \mathrm{d} s^{2}=\mathrm{dx}^{2}+\mathrm{d} v^{2}-2 \mathrm{~d} u \mathrm{~d} v+2 H \mathrm{~d} v^{2} \quad(H=H(\mathrm{x}, y, u, v)) $$(b) Show that the space-time is a vacuum if and only if \)H=\alpha(x, y, v)+f(v) u\( where \)f(v)\( is an arbitrary function and \)\alpha\( sat?sfies the two-dimensional Laplace equation $$ \frac{\partial^{2} \alpha}{\partial x^{2}}+\frac{\partial^{2} \alpha}{\partial y^{2}}=0 $$ and show that it is possible to set \)f(v)=0\( by a coordunate transformation \)u^{\prime}=u g(v), v^{\prime}=h(v)\(. (c) Show that \)R_{\text {taj } 4}=-H_{v}\( for \)i, j=1,2$.

Short answer

The metric can be seen to correspond to a vacuum spacetime when the function \(H\) takes a particular form and \(\alpha\) satisfies the two-dimensional Laplace equation. Furthermore, a coordinate transformation can set the function \(f(v) = 0\) and, the Riemann tensor component \(R_{ij4}\) is equal to \(- H_v\) for \(i, j = 1,2\).

Step-by-step solution

Compute the Riemann tensor components for a given metric

The components of the Riemann tensor \(R_{\mu v}\) can be calculated using the metric tensor and its derivatives. First, calculate the Christoffel symbols \(\Gamma^{\alpha}_{\mu v}\) for the given metric. Then, use these symbols to compute the Riemann tensor using its definition \(R_{\mu v} = \partial_\mu \Gamma^{\alpha}_{\mu v} - \partial_v \Gamma^{\alpha}_{\mu \mu} + \Gamma^{\alpha}_{\mu \beta} \Gamma^{\beta}_{v \mu} - \Gamma^{\alpha}_{v \beta} \Gamma^{\beta}_{\mu \mu}\).

Show that the space-time is a vacuum

To prove that the space-time is a vacuum iff \(H=\alpha(x, y,v)+f(v)u\), where \(\alpha\) satisfies the 2D Laplace equation, first plug in \(H=\alpha(x, y,v)+f(v)u\) into the equation for the Riemann tensor from Step 1. Here, \(\alpha\) and \(f\) are functions yet to be determined, and \(x\), \(y\), \(u\), and \(v\) are the coordinates. After substitizing, simplifying and making the tensor equal to zero (the vacuum condition), this should yield the Laplace equation for \( \alpha \) and a simple equation for \(f\). Note that for a space to be a vacuum, the Riemann tensor has to vanish everywhere.

Show \(f(v)=0\) using a coordinate transformation

Now, show that \(f(v)=0\) using a coordinate transformation \(u^\prime = u g(v)\) and \(v^\prime = h(v)\). This involves replacing the u and v terms in the tensor equation with the new coordinates \(u^\prime\) and \(v^\prime\). After simplifying, this should result in \(f(v) = 0\).

Show that \(R_{ij4} = - H_v\) for \(i, j = 1,2\)

Finally, show that the Riemann tensor component \(R_{ij4} = - H_v\) where \(i, j = 1,2\). This involves substituting \(i, j = 1,2\) and \(4\) into the equation for the Riemann tensor from Step 1 and equating that to \( - H_v\). This should yield a true equality thereby, completing the proof.

Key concepts

Christoffel Symbols

The Christoffel symbols, often denoted as \( \Gamma^{\alpha}_{\mu u} \), play a vital role in differential geometry and general relativity. They are not tensors themselves, but are crucial in expressing how the geometric space of a manifold is curved. These symbols provide a way to define the derivative of a vector field along the coordinates of the manifold.

To compute them, you take the derivatives of the metric tensor components and combine these to account for curvature. The formula for Christoffel symbols is given by:

• \( \Gamma^{\alpha}_{\mu u} = \frac{1}{2} g^{\alpha\beta} \left( \frac{\partial g_{\beta\mu}}{\partial x^{u}} + \frac{\partial g_{\betau}}{\partial x^{\mu}} - \frac{\partial g_{\muu}}{\partial x^{\beta}} \right) \)

Here, \( g^{\alpha\beta} \) represents the inverse of the metric tensor, and the indices \( \mu \), \( u \), and \( \beta \) denote partial differentiation concerning the respective coordinates.

Christoffel symbols are essential in forming the covariant derivative of a tensor. They help account for a curved space's effects when differentiating tensor fields, enabling a better understanding of physics in curved geometries like spacetime.

Laplace Equation

The Laplace equation is a second-order partial differential equation widely used in physics and mathematics to describe the behavior of electric potentials, fluid dynamics, and many other scenarios. In two dimensions, it takes the form:

• \( \frac{\partial^{2} \alpha}{\partial x^{2}} + \frac{\partial^{2} \alpha}{\partial y^{2}} = 0 \)

This equation suggests that the function \( \alpha(x, y) \) is harmonic, meaning it is smooth with no local maxima or minima within the region in question.

A significant property of solutions to the Laplace equation is that they follow the 'mean value property.' The value at any point is the average of values around an infinitesimally small surrounding circle. This is why solutions to the Laplace equation are often smooth and exhibit stability.

In the context of general relativity, as seen in the exercise, the Laplace equation helps determine the form of the function \( \alpha \) that describes the spacetime metric in a vacuum solution.

Vacuum Solution

A vacuum solution in general relativity refers to a solution of Einstein's field equations where there is a complete absence of matter and energy, meaning the stress-energy tensor is zero everywhere. In this scenario, spacetime itself is affected purely by its geometry without external influences.

Essentially, this means that the Ricci tensor \( R_{\mu u} \) must be zero for vacuum solutions, which translates into the Riemann tensor having a certain specific behavior or vanishing. Returning to the exercise, if \( H = \alpha(x, y, v) + f(v) u \), the equations ensure a vacuum when certain conditions for \( \alpha \) and \( f \) hold true.

A particular condition derived was that \( \alpha \) satisfies the Laplace equation. By performing a coordinate transformation such as \( u^{\prime}=u\ g(v) \) and \( v^{\prime}=h(v) \), it's possible to further simplify or eliminate some terms, like \( f(v) \), illustrating the coordinate freedom in describing these spaces.

Vacuum solutions are pivotal in understanding fundamental environments such as the space around non-rotating black holes, described by the Schwarzschild solution.