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Ch. 2 Problems

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Basic

Medium

Advanced

Basic

B-1. Determining the Tensor Rank of Physical Quantities

For each of the following physical quantities, determine the tensor rank (rank 0, rank 1, or rank 2).

(a) Temperature \(T\) (b) Force \(\vec{F}\) (c) Mass \(m\) (d) Velocity \(\vec{v}\) (e) Spacetime interval \(ds^2\) (f) Metric tensor \(g_{\mu\nu}\) (g) Energy-momentum tensor \(T_{\mu\nu}\) (h) Four-velocity \(U^\mu\)

Hint

Focus on the number of indices. If there are 0 indices, it is a scalar (rank 0); if there is 1 index, it is a vector (rank 1); if there are 2 indices, it is a rank-2 tensor.

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B-2. Limitations of Newton's Equation of Motion

Newton's equation of motion \(\vec{F} = m\vec{a}\) is invariant under spatial rotations and translations. However, it does not fully satisfy the requirement demanded by general relativity that "equations take the same form in all coordinate systems." State and explain two points that are lacking.

Hint

(1) Three-dimensional vectors in space alone cannot accommodate "rotations in spacetime" (transformations between inertial frames). (2) As long as gravity is treated as a force, the contradiction with special relativity (instantaneous propagation of gravity) cannot be resolved.

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B-3. The Two Pillars Correspondence between Newton and Einstein

Einstein's model of gravity is constructed from two pillars: the equation that determines particle motion and the equation that determines the shape of spacetime. By drawing correspondences with Newton's model of gravity, state the name and role of each equation.

Newton Einstein
Particle motion ? ?
Field equation ? ?
Hint

Newton's side: \(\vec{F} = m\vec{a}\) and the Poisson equation \(\nabla^2 \Phi = 4\pi G\rho\). Einstein's side: the geodesic equation and the Einstein equation \(G_{\mu\nu} = (8\pi G/c^4) T_{\mu\nu}\).

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Medium

M-1. Understanding the Geodesic Equation

Consider the geodesic equation

\[ \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0 \]

and answer the following questions.

(a) What is the physical meaning of the right-hand side being zero? Describe the correspondence with the right-hand side of Newton's \(\vec{F} = m\vec{a}\).

(b) What determines the connection coefficients \(\Gamma^\mu_{\alpha\beta}\)?

(c) When a charged particle moves in an electromagnetic field, what happens to the right-hand side of this equation? In that case, can the particle's trajectory still be called a geodesic?

Hint

(a) Right-hand side equals zero = no force = a geodesic is "the worldline of an object on which no forces other than gravity are acting." (b) It is determined from the first-order derivatives of the metric tensor \(g_{\mu\nu}\). (c) The four-vector version of the electromagnetic force \(qF^{\mu}{}_{\nu}(dx^\nu/d\tau)\) appears on the right-hand side, and the trajectory is no longer a geodesic.

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M-2. Coefficient of the Einstein Equation

Answer the following questions about the Einstein equation \(G_{\mu\nu} = (8\pi G/c^4) T_{\mu\nu}\).

(a) Explain the difference between \(G_{\mu\nu}\) (the Einstein tensor) on the left-hand side and \(G\) (the gravitational constant) on the right-hand side.

(b) How is the coefficient \(8\pi G/c^4\) determined?

(c) In the weak-gravity, slow-velocity limit, to which Newtonian equation does this equation reduce?

Hint

(a) \(G_{\mu\nu}\) is a rank-2 tensor with indices, while \(G\) is a scalar constant without indices. The use of the same symbol is a historical coincidence. (b) The coefficient is determined by requiring consistency with Newton's Poisson equation. (c) The Poisson equation \(\nabla^2 \Phi = 4\pi G\rho\).

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Advanced

A-1. Quantities Derived from the Metric Tensor

The connection coefficients \(\Gamma^\mu_{\alpha\beta}\) and the Einstein tensor \(G_{\mu\nu}\) are both quantities constructed from the metric tensor \(g_{\mu\nu}\). State what order of derivatives of the metric each is constructed from, and explain what each quantity physically "determines."

Hint

\(\Gamma\) is constructed from first-order derivatives of the metric and determines particle trajectories through the geodesic equation. \(G_{\mu\nu}\) is constructed from second-order derivatives of the metric (via the Riemann tensor and Ricci tensor) and represents the curvature of spacetime itself.

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