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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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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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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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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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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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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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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.