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Ch. 4 Problems

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Basic

Medium

Advanced

Basic

B-1. Conversion to Natural Units and Restoring \(c\)

Rewrite the following expressions given in SI units into natural units (\(c = 1\)). Conversely, convert expressions in natural units back to SI units.

(a) Write the SI expression \(E^2 = p^2 c^2 + m^2 c^4\) in natural units.

(b) Write the natural units expression \(\gamma = 1/\sqrt{1 - v^2}\) in SI units.

(c) In natural units, a statement reads "energy \(E = 5\)." Assuming the rest mass \(m\) (in kg) is nonzero, express the corresponding energy in SI units (joules) in terms of \(m\).

(d) Express the rest energy of the electron with rest mass \(m_e \approx 9.11 \times 10^{-31}\) kg in both SI units (joules) and natural units (kg).

Hint

(a)(b) Insert or remove factors of \(c\) so that the dimensions of all terms match. (c) Multiplying the energy in natural units by \(c^2\) gives the energy in SI units. (d) In natural units, \(E = m\). In SI units, \(E = mc^2\).

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B-2. Time and Length in Natural Units

In natural units (\(c = 1\)), time and length are measured in the same units. Answer the following questions.

(a) Express 1 second in units of length (meters). How many meters is it?

(b) Express 1 meter in units of time (seconds). How many seconds is it?

(c) The distance from the Earth to the Sun is approximately \(1.5 \times 10^{11}\) m. Express this in units of time (seconds). Physically, what does this value mean?

(d) Take the walking speed of a person to be \(v \approx 1\) m/s. What is the numerical value of this speed in natural units (where the speed of light equals 1)?

Hint

(a) The distance light travels in 1 second is approximately \(3 \times 10^8\) m. Therefore, in natural units, "1 second = \(3 \times 10^8\) m." (c) This corresponds to the time it takes light to travel from the Earth to the Sun (approximately 8 minutes and 20 seconds). (d) \(v/c \approx 1 / (3 \times 10^8) \approx 3.3 \times 10^{-9}\). Everyday speeds are extremely small compared to the speed of light.

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B-3. Calculating the Minkowski Inner Product

For the 4-vectors \(A^\mu = (5,\, 3,\, 0,\, 0)\) and \(B^\mu = (2,\, 1,\, 0,\, 0)\), calculate the Minkowski inner product

\[ \eta_{\mu\nu}\,A^\mu\,B^\nu \]

using Einstein's summation convention (in units where \(c = 1\)).

Hint

Since \(\eta_{\mu\nu}\) is a diagonal matrix, only the terms with \(\mu = \nu\) survive. Note that \(\eta_{00} = -1\).

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B-4. Components of a Covariant Vector

Given the 4-vector \(A^\mu = (E,\, p_x,\, p_y,\, p_z)\), express each component \(A_0,\, A_1,\, A_2,\, A_3\) of the covariant vector \(A_\mu = \eta_{\mu\nu}\,A^\nu\) in terms of \(E,\, p_x,\, p_y,\, p_z\).

Hint

Apply \(\eta_{00} = -1\), \(\eta_{11} = \eta_{22} = \eta_{33} = +1\) to each component.

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B-5. Expansion of the Spacetime Interval into 16 Terms

Write out all 16 terms of the spacetime interval \(ds^2 = \eta_{\mu\nu}\,dx^\mu\,dx^\nu\) with respect to \(\mu\) and \(\nu\), and show that since the off-diagonal components of \(\eta_{\mu\nu}\) are zero, one obtains \(ds^2 = -dt^2 + dx^2 + dy^2 + dz^2\) (with \(c = 1\)).

Hint

Since \(\eta_{\mu\nu} = 0\) for \(\mu \neq \nu\), only the 4 terms with \(\mu = \nu\) survive.

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B-6. Relabeling Dummy Indices

Practice relabeling dummy indices. Verify that the following equality is correct under the summation convention by writing out the sums explicitly.

\[ \eta_{\mu\nu}\,A^\mu\,B^\nu = \eta_{\alpha\beta}\,A^\alpha\,B^\beta \]
Hint

Expand both sides using \(\sum\) and show that they yield the same sum of 16 terms.

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B-7. Verification of the Four-Velocity Normalization Condition

For a particle with three-velocity \(\mathbf{v} = (v,\, 0,\, 0)\), verify by direct calculation that the four-velocity

\[ U^\mu = \gamma(1,\, v,\, 0,\, 0) \]

satisfies \(\eta_{\mu\nu}\,U^\mu\,U^\nu = -1\) (with \(c = 1\)).

Hint

Simplify \(\gamma^2(-1 + v^2)\) using the definition of \(\gamma\).

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B-8. Low-Speed Limit of Relativistic Energy

Regarding the low-speed limit of relativistic energy \(E = \gamma mc^2\), show the following.

(a) Using the approximation \((1 + x)^n \approx 1 + nx\) (for \(|x| \ll 1\)), show that when \(v \ll c\):

\[ E \approx mc^2 + \frac{1}{2}mv^2 \]

That is, in the low-speed limit, the total energy becomes the sum of the rest energy and the Newtonian kinetic energy.

(b) For an object with mass \(m = 1\) kg and speed \(v = 100\) m/s (approximately the speed of a bullet train), find the value of \(v^2/c^2\) and the ratio of the kinetic energy \(\frac{1}{2}mv^2\) to the rest energy \(mc^2\). Using this ratio, explain "why Newton's era could not have noticed rest energy."

(c) For a massless particle, substitute \(m = 0\) into \(E^2 = |\vec{p}|^2 c^2 + m^2 c^4\) to obtain \(E = |\vec{p}|c\). Furthermore, combining this with the relation \(E = \gamma mc^2\), argue that a massless particle with finite energy must necessarily travel at the speed of light.

Hint

(a) Substitute \(x = -v^2/c^2\) and \(n = -1/2\) into \(\gamma = (1 - v^2/c^2)^{-1/2}\). (b) Using \(c \approx 3 \times 10^8\) m/s, we get \(v^2/c^2 \sim 10^{-13}\). (c) To realize \(m = 0\) with \(E \neq 0\) in \(E = \gamma mc^2\), we need \(\gamma \to \infty\).

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Medium

M-1. Tensor Contraction and Classification of Indices

The contraction \(T^{\mu\nu}A_\nu\) of a rank-2 tensor \(T^{\mu\nu}\) and a 4-vector \(A_\nu\) is a tensor of what rank? Identify the free index and the dummy index.

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Advanced

A-1. Four-Velocity and Four-Acceleration

Regarding the four-velocity \(U^\mu\) and the four-acceleration \(a^\mu \equiv dU^\mu/d\tau\), show the following.

(a) By differentiating both sides of \(\eta_{\mu\nu}\,U^\mu\,U^\nu = -1\) with respect to proper time \(\tau\), derive \(\eta_{\mu\nu}\,U^\mu\,a^\nu = 0\). That is, the four-velocity and four-acceleration are always orthogonal in the sense of the Minkowski inner product.

(b) In the instantaneous rest frame of the particle (\(U^\mu = (1, 0, 0, 0)\)), show from the result of (a) that \(a^0 = 0\), and explain why the four-acceleration is a purely spacelike vector (\(\eta_{\mu\nu}\,a^\mu\,a^\nu > 0\)).

(c) For one-dimensional motion (in the \(x\)-direction only) with constant proper acceleration \(g\) (\(\eta_{\mu\nu}\,a^\mu\,a^\nu = g^2 = \text{const.}\)), show that the worldline of the particle is described by the hyperbola

\[ x^2 - t^2 = \frac{1}{g^2} \]

(with initial conditions \(x = 1/g\) and \(U^\mu = (1, 0, 0, 0)\) at \(t = 0\)).

Hint

For (c), set \(U^0 = \cosh(g\tau)\), \(U^1 = \sinh(g\tau)\), verify that \(\eta_{\mu\nu}\,a^\mu\,a^\nu = g^2\), and integrate with respect to \(\tau\) to obtain the worldline.

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A-2. General Direction Lorentz Boost

Consider a general Lorentz boost between two inertial frames \(S\) and \(S'\). When \(S'\) is moving with velocity \(\mathbf{v} = (v_x, v_y, v_z)\) (magnitude \(v = |\mathbf{v}|\)) relative to \(S\), the transformation is given by

\[ t' = \gamma\!\left(t - \mathbf{v} \cdot \mathbf{x}\right) \]
\[ \mathbf{x}' = \mathbf{x} + (\gamma - 1)\frac{(\mathbf{v} \cdot \mathbf{x})}{v^2}\,\mathbf{v} - \gamma\,\mathbf{v}\,t \]

(with \(c = 1\)). Show the following.

(a) Verify that this transformation leaves the spacetime interval \(ds^2 = -dt^2 + d\mathbf{x} \cdot d\mathbf{x}\) invariant.

(b) Show that in the case \(\mathbf{v} = (v, 0, 0)\), this reduces to the standard \(x\)-direction boost derived in this chapter.

(c) Discuss why the composition \(\Lambda(\mathbf{v}_2)\Lambda(\mathbf{v}_1)\) of two boosts \(\Lambda(\mathbf{v}_1)\) and \(\Lambda(\mathbf{v}_2)\) in different directions (\(\mathbf{v}_1 \times \mathbf{v}_2 \neq \mathbf{0}\)) is not, in general, a pure boost, but rather a boost plus a spatial rotation. This spatial rotation is called the Thomas rotation (Wigner rotation). For the case where both \(\mathbf{v}_1\) and \(\mathbf{v}_2\) lie in the \(x\)-\(y\) plane, show that the composite transformation contains a spatial rotation by using a symmetry argument for the transformation matrices.

Hint

(c) The matrix of a pure boost is a symmetric matrix, but the product of two symmetric matrices is not symmetric in general. The antisymmetric part corresponds to the rotation.

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