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Ch. 2 Solutions

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Determining the Tensor Rank of Physical Quantities

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Problem:

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 \(U^\mu\)

Solution:

Quantity Rank
(a) Temperature \(T\) Rank 0 (scalar)
(b) Force \(\vec{F}\) Rank 1 (vector)
(c) Mass \(m\) Rank 0 (scalar)
(d) Velocity \(\vec{v}\) Rank 1 (vector)
(e) Spacetime interval \(ds^2\) Rank 0 (scalar)
(f) Metric tensor \(g_{\mu\nu}\) Rank 2
(g) Energy-momentum tensor \(T_{\mu\nu}\) Rank 2
(h) Four-velocity \(U^\mu\) Rank 1

Key point: The rank is determined by the number of indices. Zero indices means an invariant (a quantity whose value does not change under coordinate transformations), one index means a vector-like quantity, and two indices means a matrix-like quantity.

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

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Problem:

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.

Answer:

What Newton's \(\vec{F} = m\vec{a}\) lacks:

  1. It is a 3-dimensional vector in space only. In special relativity, time and space are mixed together (Lorentz transformation), so unless it is rewritten using 4-dimensional "4-vectors" that include the time component, the form is not preserved under transformations between inertial frames.

  2. It treats gravity as a "force." Newton's gravity \(F = -GMm/r^2\) is an action-at-a-distance that propagates instantaneously, and even if we reformulate it as a 4-vector, the problem of "instantaneous propagation of gravity" remains. In Einstein's framework, this problem is fundamentally resolved by describing gravity not as a "force" but as "curvature of spacetime."

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

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Problem:

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 ? ?

Answer:

Newton Einstein
Particle motion \(\vec{F} = m\vec{a}\) (equation of motion) \(\dfrac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\dfrac{dx^\alpha}{d\tau}\dfrac{dx^\beta}{d\tau} = 0\) (geodesic equation)
Field equation \(\nabla^2 \Phi = 4\pi G\rho\) (Poisson equation) \(G_{\mu\nu} = \dfrac{8\pi G}{c^4}T_{\mu\nu}\) (Einstein equation)

Roles:

  • Particle motion: Determines how a particle moves within a given spacetime (the potential \(\Phi\) in Newton's case, or the metric \(g_{\mu\nu}\) and the \(\Gamma\) derived from it in Einstein's case).
  • Field equation: Determines how matter (mass density \(\rho\) or energy-momentum \(T_{\mu\nu}\)) generates spacetime (\(\Phi\) or \(g_{\mu\nu}\)).

Medium

M-1. Understanding the Geodesic Equation

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Problem:

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?

Solution:

(a) Meaning of the right-hand side being zero:

Right-hand side equals zero = no force is acting on the particle. The "force" that appears on the right-hand side of Newton's \(\vec{F} = m\vec{a}\) is absent in Einstein's geodesic equation.

In Einstein's framework, gravity is not a force but is absorbed into the geometry of spacetime. The connection coefficients \(\Gamma^\mu_{\alpha\beta}\) in the second term on the left-hand side carry the effect of spacetime curvature, so the gravitational "force" disappears from the equation. Therefore, a "geodesic" means "the worldline of a particle on which no forces other than gravity act" = "the worldline of a particle on which no forces act at all."

(b) What determines the connection coefficients:

The connection coefficients \(\Gamma^\mu_{\alpha\beta}\) are determined from the first derivatives of the metric tensor \(g_{\mu\nu}\). Specifically,

\[ \Gamma^\mu_{\alpha\beta} = \frac{1}{2} g^{\mu\nu} \left( \partial_\alpha g_{\nu\beta} + \partial_\beta g_{\nu\alpha} - \partial_\nu g_{\alpha\beta} \right) \]

This will be studied in detail from Ch. 6 onward.

(c) Charged particle in an electromagnetic field:

In an electromagnetic field, the particle experiences an electromagnetic force, so that force appears on the right-hand side:

\[ m\frac{d^2 x^\mu}{d\tau^2} + m\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = qF^{\mu}{}_{\nu}\frac{dx^\nu}{d\tau} \]

An equation with a force on the right-hand side is not called the "geodesic equation" but simply the "equation of motion." A particle whose path is deflected by a force no longer follows a geodesic.

M-2. Coefficient of the Einstein Equation

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Problem:

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?

Solution:

(a) Difference between \(G_{\mu\nu}\) and \(G\):

  • \(G_{\mu\nu}\) is called the Einstein tensor, a rank-2 tensor (with two indices). It represents the curvature of spacetime and is constructed from second derivatives of the metric tensor \(g_{\mu\nu}\).
  • \(G\) is Newton's gravitational constant (a scalar constant with no indices). \(G \approx 6.67 \times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}\).

The use of the same symbol \(G\) is a historical coincidence. They are distinguished by the presence or absence of indices.

(b) Determination of the coefficient \(8\pi G/c^4\):

This coefficient is not a number Einstein chose freely; it is determined by requiring consistency with Newton's Poisson equation \(\nabla^2 \Phi = 4\pi G\rho\). The condition that the Einstein equation reduces to the Poisson equation in the weak-gravity, slow-velocity limit requires the coefficient to be \(8\pi G/c^4\). The factor \(4\pi\) originates from the surface area of a sphere, and the \(8\pi\) arises from an additional factor of 2 due to the tensor structure. Details are covered in Ch. 13.

(c) Weak-gravity limit:

In the weak-gravity, slow-velocity limit, the Einstein equation reduces to Newton's Poisson equation \(\nabla^2 \Phi = 4\pi G\rho\). In this limit, the time-time component of the metric \(g_{00}\) is related to the gravitational potential \(\Phi\) by \(g_{00} \approx -(1 + 2\Phi/c^2)\).

Advanced

A-1. Quantities Derived from the Metric Tensor

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Problem:

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."

Answer:

Number of derivatives from the metric tensor \(g_{\mu\nu}\) and their physical meanings:

Quantity Derivative of metric Physical meaning
Connection coefficients \(\Gamma^\mu_{\alpha\beta}\) 1st derivative Determines particle trajectories through the geodesic equation
Einstein tensor \(G_{\mu\nu}\) 2nd derivative Represents the curvature of spacetime itself (constructed from the Riemann tensor)

Flow:

g_{μν} (metric)
  ├─ 1st derivative → Γ^μ_{αβ} (connection) → geodesic equation → particle trajectories
  └─ 2nd derivative → R^μ_{ναβ} (Riemann) → R_{μν} (Ricci) → G_{μν} (Einstein) → spacetime curvature

\(\Gamma\) is the quantity used to determine "how particles move in this spacetime." \(G_{\mu\nu}\) is the quantity that diagnoses "how much this spacetime is curved" (such as identifying singularities inside black holes). Both start from the metric \(g_{\mu\nu}\), and it is this metric that is the protagonist determined by the Einstein equations.