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Appendix C Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Klein-Gordon \(\partial \mathcal{L}/\partial \phi\)

$$\mathcal{L} = -\frac{1}{2}\,\eta^{\mu\nu}(\partial_\mu \phi)(\partial_\nu \phi) - \frac{m^2}{2}\,\phi^2 $$ is given. Find the partial derivative with respect to \(\phi\): \(\dfrac{\partial \mathcal{L}}{\partial \phi}\).

Hint

The only term containing \(\phi\) is the mass term \(-\frac{m^2}{2}\phi^2\). The derivative term depends on \(\partial_\mu\phi\) and does not depend on \(\phi\) itself.

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B-2. Klein-Gordon \(\partial \mathcal{L}/\partial(\partial\phi)\)

Hint

When differentiating \(\eta^{\alpha\beta}(\partial_\alpha\phi)(\partial_\beta\phi)\) with respect to \(\partial_\mu\phi\), there are contributions from both \(\alpha = \mu\) and \(\beta = \mu\). Use the symmetry of \(\eta^{\alpha\beta}\).

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B-3. Partial Derivatives of the String Lagrangian

For $$\mathcal{L} = \frac{\rho}{2}\left(\frac{\partial \psi}{\partial t}\right)^2 - \frac{\mathcal{T}}{2}\left(\frac{\partial \psi}{\partial x}\right)^2 $$ find \(\dfrac{\partial \mathcal{L}}{\partial \psi}\), \(\dfrac{\partial \mathcal{L}}{\partial(\partial_t \psi)}\), and \(\dfrac{\partial \mathcal{L}}{\partial(\partial_x \psi)}\) respectively.

Hint

\(\psi\) itself does not appear explicitly in \(\mathcal{L}\). Treat each derivative term as independent.

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B-4. \(\partial \mathcal{L}/\partial \phi\) in \(\phi^4\) Theory

For the Lagrangian density $$\mathcal{L} = -\frac{1}{2}\,\eta^{\mu\nu}(\partial_\mu \phi)(\partial_\nu \phi) - \frac{\lambda}{4!}\,\phi^4 $$ find \(\dfrac{\partial \mathcal{L}}{\partial \phi}\).

Hint

Differentiating \(\phi^4\) with respect to \(\phi\) gives \(4\phi^3\). Pay attention to how this combines with the \(4!\) factor.

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B-5. Explicit Form of the d'Alembert Operator

Hint

Substitute \(\eta^{00} = -1\), \(\eta^{11} = \eta^{22} = \eta^{33} = +1\) and expand the sum over \(\mu, \nu\).

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B-6. Euler–Lagrange Equation for a 2-Dimensional Scalar Field

Write down the field Euler–Lagrange equation for $$\mathcal{L} = \frac{1}{2}(\partial_t \phi)^2 - \frac{1}{2}(\partial_x \phi)^2 - V(\phi) $$ where \(V(\phi)\) is an arbitrary function of \(\phi\).

Hint

In 2 dimensions, the sum over \(\partial_\mu\) consists of two terms: \(\mu = t\) and \(\mu = x\). The derivative of \(V(\phi)\) can be written as \(V'(\phi) = dV/d\phi\).

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B-7. \(\sqrt{-g}\) of the Minkowski Metric

Hint

The determinant of a diagonal matrix is the product of its diagonal elements.

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B-8. \(\sqrt{-g}\) for the Schwarzschild Metric

$$ds^2 = -!\left(1 - \frac{2M}{r}\right)dt^2 + \left(1 - \frac{2M}{r}\right)^{-1}dr^2 + r^2 d\theta^2 + r^2\sin^2!\theta\, d\varphi^2 $$ For the metric above, compute \(g = \det(g_{\mu\nu})\) and find \(\sqrt{-g}\).

Hint

Since the metric is diagonal, \(g = g_{tt}\,g_{rr}\,g_{\theta\theta}\,g_{\varphi\varphi}\). Note that the product \(g_{tt}\,g_{rr}\) simplifies nicely.

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Medium

M-1. Euler–Lagrange Derivation of the Wave Equation for a String

Apply the field Euler–Lagrange equation to

\[\mathcal{L} = \frac{\rho}{2}\left(\frac{\partial \psi}{\partial t}\right)^2 - \frac{\mathcal{T}}{2}\left(\frac{\partial \psi}{\partial x}\right)^2 \]

and derive the wave equation

\[\rho\,\frac{\partial^2 \psi}{\partial t^2} = \mathcal{T}\,\frac{\partial^2 \psi}{\partial x^2} \]

Also, express the wave propagation speed \(v\) in terms of \(\mathcal{T}\) and \(\rho\).

Hint

Use the two-dimensional Euler–Lagrange equation \(\frac{\partial \mathcal{L}}{\partial \psi} - \partial_t\!\left(\frac{\partial \mathcal{L}}{\partial(\partial_t \psi)}\right) - \partial_x\!\left(\frac{\partial \mathcal{L}}{\partial(\partial_x \psi)}\right) = 0\). Rewrite the wave equation in the form \(\partial_t^2 \psi = v^2 \partial_x^2 \psi\).

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M-2. Equation of Motion for \(\phi^4\) Theory

Apply the field Euler–Lagrange equation to the Lagrangian density $$\mathcal{L} = -\frac{1}{2}\,\eta^{\mu\nu}(\partial_\mu\phi)(\partial_\nu\phi) - \frac{\lambda}{4!}\,\phi^4 $$ and derive the equation of motion for \(\phi\). Explain how the resulting equation differs from the massless Klein–Gordon equation \(\Box\phi = 0\), including the physical meaning of the difference.

Hint

The contribution from the \(\phi^4\) term appears as a nonlinear self-interaction term. Verify that in the limit \(\lambda = 0\), the equation reduces to \(\Box\phi = 0\).

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M-3. Massless Scalar Field in Curved Spacetime

For the action $$S = \int d^4x\,\sqrt{-g}\left[-\frac{1}{2}\,g^{\mu\nu}(\partial_\mu\phi)(\partial_\nu\phi)\right] $$ derive the equation of motion following the steps below:

(a) Take the variation \(\phi \to \phi + \delta\phi\) and compute \(\delta S\). Note that \(\sqrt{-g}\) does not depend on \(\phi\).

(b) Perform integration by parts and derive the equation of motion for \(\phi\) from \(\delta S = 0\). Verify that in flat spacetime, \(\partial_\mu(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi) \to \eta^{\mu\nu}\partial_\mu\partial_\nu\phi = \Box\phi\).

Hint

In integration by parts in curved spacetime, total derivative terms of the form \(\partial_\mu(\sqrt{-g}\,f^\mu)\) appear. The equation of motion takes the form \(\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi) = 0\).

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M-4. Derivation of the Energy-Momentum Tensor

$$T_{\mu\nu} = -\frac{2}{\sqrt{-g}}\,\frac{\delta S_m}{\delta g^{\mu\nu}} $$ Using this definition, derive \(T_{\mu\nu}\) from the Lagrangian density of a free scalar field:

$$\mathcal{L}m = -\frac{1}{2}\,g^{\mu\nu}(\partial\mu\phi)(\partial_\nu\phi) - \frac{m^2}{2}\,\phi^2 $$ When varying \(S_m = \int d^4x\,\sqrt{-g}\,\mathcal{L}_m\) with respect to \(g^{\mu\nu}\), you may use the identity \(\dfrac{\delta(\sqrt{-g})}{\delta g^{\mu\nu}} = -\dfrac{1}{2}\sqrt{-g}\,g_{\mu\nu}\).

Hint

The variation with respect to \(g^{\mu\nu}\) acts in two places: the direct contribution to \(g^{\mu\nu}\) in \(\mathcal{L}_m\), and the contribution through \(\sqrt{-g}\). Compute each separately and add them together.

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Advanced

A-1. Maxwell Equations from the Electromagnetic Field Lagrangian

The Lagrangian density of the electromagnetic field in 4-dimensional Minkowski spacetime is given by

\[\mathcal{L}_{\text{EM}} = -\frac{1}{4}\,\eta^{\mu\alpha}\,\eta^{\nu\beta}\,F_{\mu\nu}\,F_{\alpha\beta} \]

where \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\) is the electromagnetic field tensor (Faraday tensor), and \(A_\mu\) is the electromagnetic four-potential.

(a) Verify that \(\mathcal{L}_{\text{EM}}\) can be written as \(\mathcal{L}_{\text{EM}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\) using \(F^{\mu\nu} \equiv \eta^{\mu\alpha}\eta^{\nu\beta}F_{\alpha\beta}\).

(b) Apply the field Euler–Lagrange equation with respect to \(A_\nu\) to derive the vacuum Maxwell equations (source-free):

\[\partial_\mu F^{\mu\nu} = 0 \]

(c) Show that the \(\nu = 0\) component and the \(\nu = i\) (\(i = 1,2,3\)) components correspond to Gauss's law \(\nabla \cdot \mathbf{E} = 0\) and the Ampère–Maxwell law \(\nabla \times \mathbf{B} = \frac{\partial \mathbf{E}}{\partial t}\) (source-free), respectively.

Hint

In (b), use the fact that \(\frac{\partial \mathcal{L}}{\partial A_\nu} = 0\) (since \(A_\nu\) does not appear explicitly) and exploit the antisymmetry of \(F_{\alpha\beta}\) when computing \(\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}\). In (c), use the correspondences \(F^{0i} = -E^i\) and \(F^{ij} = -\epsilon^{ijk}B_k\).

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A-2. Einstein Equation with Cosmological Constant

Consider the action obtained by adding a cosmological constant \(\Lambda\) to the Einstein–Hilbert action:

\[S = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,(R - 2\Lambda) + \int d^4x\,\sqrt{-g}\,\mathcal{L}_m \]

(a) Vary the \(\sqrt{-g}\,\Lambda\) term with respect to \(g^{\mu\nu}\). You may use \(\dfrac{\delta(\sqrt{-g})}{\delta g^{\mu\nu}} = -\dfrac{1}{2}\sqrt{-g}\,g_{\mu\nu}\).

(b) Vary the full action above with respect to \(g^{\mu\nu}\) and set \(\delta S = 0\) to show that the Einstein equation with cosmological constant

$$G_{\mu\nu} + \Lambda\, g_{\mu\nu} = 8\pi G\, T_{\mu\nu} $$ is obtained. You may use as known that the variation of \(\sqrt{-g}\,R\) with respect to \(g^{\mu\nu}\) gives \(\sqrt{-g}\left(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R\right) = \sqrt{-g}\,G_{\mu\nu}\).

(c) In the vacuum case (\(T_{\mu\nu} = 0\)), discuss the physical effect that \(\Lambda > 0\) has on spacetime, based on the structure of the Einstein equation.

Hint

(a) Since \(\Lambda\) is a constant, the variation acts only on \(\sqrt{-g}\). (b) Sum the variations of each term and set the result to zero. (c) Moving the \(\Lambda g_{\mu\nu}\) term to the right-hand side, note that even in vacuum a contribution resembling an energy-momentum tensor remains.

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