Problem 36 - A First Course in General Relativity

Question

A four-dimensional manifold has coordinates \((t, x, y, z)\) and line element $$ \mathrm{d} s^{2}=-(1+2 \phi) \mathrm{d} t^{2}+(1-2 \phi)\left(\mathrm{d} x^{2}+\mathrm{d} y^{2}+\mathrm{d} z^{2}\right) $$ where \(|\phi(t, x, y, z)| \ll 1\) everywhere. At any point \(P\) with coordinates \(\left(t_{0}, x_{0}, y_{0}, z_{0}\right)\) find a coordinate transformation to a locally inertial coordinate system, to first order in \(\phi .\) At what rate does such a frame accelerate with respect to the original coordinates, again to first order in \(\phi ?\)

Step-by-Step Solution

Answer

The locally inertial transformation is \(T = t(1-\phi), X^i = x^i(1+\phi)\). The frame accelerates as \(a^t = \nabla^i \phi\).

Step 1: Understand the Problem Statement

We need to find a coordinate transformation to a locally inertial coordinate system at a point \(P\) with coordinates \((t_0, x_0, y_0, z_0)\) in a four-dimensional manifold. The line element is given, and we need the transformation to be accurate to first order in \(\phi\). We also need to determine the frame's acceleration with respect to the original coordinates.

Step 2: Approximate the Metric Tensor

The given line element can be expressed through the metric tensor \(g_{\muu}\). The metric is: \[g_{tt} = -(1+2\phi), \quad g_{xx} = g_{yy} = g_{zz} = 1-2\phi, \quad g_{ij} = 0 \text{ for } i eq j.\] Since \(|\phi| \ll 1\), this represents a small perturbation from the flat Minkowski metric \(\eta_{\muu}\).

Step 3: Set Up the Transformation to Locally Inertial Coordinates

We aim to perform a transformation to new coordinates \((T, X, Y, Z)\) such that the metric becomes Minkowskian \(\eta_{\muu}\) at point \(P\). This is done by choosing transformations of the form \(T = t + \xi^t, X = x + \xi^x, Y = y + \xi^y, Z = z + \xi^z\), where \(\xi^\mu\) are small compared to the coordinates themselves.

Step 4: Apply the Transformation to the Metric

Calculate the transformed metric components to first order in \(\phi\). Using \(g'_{\muu} = \frac{\partial x^\alpha}{\partial x'^\mu} \frac{\partial x^\beta}{\partial x'^u} g_{\alpha\beta}\) and substituting the transformation equations, we find that the metric components change due to the derivatives of \(\xi^\mu\), essentially cancelling the terms involving \(\phi\).

Step 5: Solve for the Correction Terms \(\xi^\mu\)

For metric perturbations to vanish at \(P\), solve \(\frac{\partial \xi^\mu}{\partial x^u} + \frac{\partial \xi^u}{\partial x^\mu} - 2\Gamma^\lambda_{\muu} \xi^\lambda = g_{\muu} - \eta_{\muu}\), where \(\Gamma^\lambda_{\muu}\) are Christoffel symbols of the original metric. Solving yields \(\xi^t = -\phi t\) and \(\xi^i = -\phi x^i\) for the spatial coordinates.

Step 6: Calculate the Frame's Acceleration

The acceleration \(a^\mu\) of the locally inertial frame in the original coordinates can be computed using \(a^\mu = \frac{\mathrm{d}^2 \xi^\mu}{\mathrm{d} \tau^2}\) where \(\tau\) is proper time. For the given transformation, only \(a^t \approx abla^i \phi\) contributes, meaning the acceleration is directed along the gradient of \(\phi\).

Key Concepts

Metric Tensor

In General Relativity, the metric tensor is crucial for describing the geometry of space-time. It serves as a mathematical tool to define distances and angles in a given manifold. In our problem, the line element is expressed using this tensor. The metric tensor for a given line element can be written as:
- The temporal component: \(g_{tt} = -(1+2\phi)\)
- The spatial components: \(g_{xx} = g_{yy} = g_{zz} = 1 - 2\phi\)
- Off-diagonal components: \(g_{ij} = 0\) for \(i eq j\)
This indicates a small deviation from the Minkowski metric by a factor of \(\phi\), which is assumed to be small \((|\phi| \ll 1)\).
The metric tensor not only describes spacetime but also dictates how objects move and light propagates within it.

Coordinate Transformation

Coordinate transformations are used to relate different perspectives or coordinate grids in space-time. In our problem, we are shifting from the original coordinates \((t, x, y, z)\) to a new set \((T, X, Y, Z)\) to simplify calculations and interpretation.
The transformation equations are:

• \(T = t + \xi^t\)
• \(X = x + \xi^x\)
• \(Y = y + \xi^y\)
• \(Z = z + \xi^z\)

Here, \(\xi^\mu\) represents small corrections to the coordinates, specifically crafted to remove the effects of \(\phi\) at the point \(P\). The goal of these transformations is to regain a Minkowski spacetime description locally, making computations more straightforward.

Locally Inertial Frame

A locally inertial frame is a reference frame in which the laws of physics simplify, making them look like those in a non-accelerating, flat space-time for a small region around a point. Essentially, this means gravity appears to vanish or is uniform in this localized area.
Achieving a locally inertial frame requires transforming the metric so its components resemble those of flat Minkowski space (\(\eta_{\muu}\)) at a specific point, typically a difficult task without appropriate transformations.
These frames are particularly useful because they simplify understanding of local physics, eliminating the impact of gravitational curvature and making the equations solvable with simpler techniques.

Christoffel Symbols

Christoffel symbols play a key role in describing gravitational effects in curved spacetime. They act as connection coefficients which reveal how vectors change as they move parallelly along the surface of a manifold.
Mathematically, they are not tensors but instead coordinate-dependent expressions derived from the metric tensor that convey how curvature affects motion. Their formula is given by:\[\Gamma^\lambda_{\muu} = \frac{1}{2} g^{\lambda\alpha} \left(\frac{\partial g_{\alphau}}{\partial x^\mu} + \frac{\partial g_{\mu\alpha}}{\partial x^u} - \frac{\partial g_{\muu}}{\partial x^\alpha}\right)\]In our exercise, we used these to ensure that the local metric, post-transformation, achieves a form that matches the flat metric, thereby defining a locally inertial frame and allowing us to calculate how the frame accelerates with respect to the original coordinates.