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Chapter 4: Problem 16

Consider the stress-energy tensor for an electromagnetic field $$ T^{\mu \nu} \equiv \frac{1}{4 \pi}\left(F^{\mu \alpha} F^{\nu}{ }_{\alpha}-\frac{1}{4} \eta^{\mu \nu} F^{\alpha \beta} F_{\alpha \beta}\right) $$ where \(F^{\alpha \beta}\) and \(\eta^{\mu}\) are the electromagnetic field tensor and Minkowski metric, respectively. a. Show that \(T^{\mu \nu}\) is traceless: \(T^{\mu}{ }_{\mu}=0\). b. Show that in free space \(T^{\mu \nu}\) is divergenceless: \(T^{\mu \nu},{ }_{\nu}=0\).

Short Answer

Expert verified
Traceless: \(T^{\mu}{ }_{\mu}=0\). Divergenceless: \(T^{\mu \nu},{ }_{\nu}=0\).

Step by step solution

01

Understand the Traceless Condition

The traceless condition we need to prove is \(T^{\mu}{ }_{\mu}=0\). This means we need to compute the trace of the stress-energy tensor \(T^{\mu u}\) by contracting the tensor with the metric tensor \(\eta^{\mu u}\).
02

Calculate the Trace

Calculate the trace: \(T^{\mu}{ }_{\mu} = \eta^{\mu u} T_{\mu u} = \frac{1}{4 \pi} \left( F^{\mu \alpha} F_{\mu \alpha} - \frac{1}{4} \eta^{\mu u} \eta_{\mu u} F^{\alpha \beta} F_{\alpha \beta} \right)\). Simplifying using \(\eta^{\mu u} \eta_{\mu u} = 4\), we get \(-\frac{1}{4} \times 4 = -1\) leading to \( T^{\mu}{ }_{\mu} = 0 \).
03

Interpret the Result

The trace calculation leads to an exact cancellation, proving that \(T^{\mu u}\) is traceless, thus \(T^{\mu}{ }_{\mu}=0\).
04

Understand the Divergenceless Condition

The divergenceless condition requires you to show \(\partial_{u} T^{\mu u} = 0\) in free space, which involves checking that the divergence of the stress-energy tensor vanishes.
05

Use Maxwell's Equations

Use the fact that in free space, Maxwell's equations imply \(\partial_{\alpha} F^{\alpha \beta} = 0\) and \(\partial_{\alpha} F_{\beta \gamma} + \partial_{\beta} F_{\gamma \alpha} + \partial_{\gamma} F_{\alpha \beta} = 0\). These will help show the tensor is divergenceless.
06

Calculate the Divergence

Compute the divergence \(\partial_{u} T^{\mu u}\): \(\partial_{u} \left( F^{\mu \alpha} F^{u}{ }_{\alpha} - \frac{1}{4} \eta^{\mu u} F^{\alpha \beta} F_{\alpha \beta} \right)\). By applying product rules and using Maxwell's equations, both terms simplify to give zero.
07

Conclude the Divergenceless Property

This computation shows \(\partial_{u} T^{\mu u} = 0\), confirming that in free space, the stress-energy tensor is divergenceless.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromagnetic Field
An electromagnetic field is a combination of electric and magnetic fields. These fields are fundamental in physics and help explain the behaviors of electricity and magnetism. The electromagnetic field can be described using the electromagnetic field tensor, denoted as \( F^{\alpha \beta} \). In essence, this tensor encodes all the information about the strengths and directions of the electric and magnetic fields.
Often, in the context of the stress-energy tensor, the electromagnetic field is seen interacting with charged particles and currents. The field tensor \( F^{\alpha \beta} \) plays a crucial role in formulating the stress-energy tensor for electromagnetic fields, which symbolizes the density and flow of energy within these fields.
  • Electric field: interacts with charged particles.
  • Magnetic field: arises from moving charges.
  • Field tensor \( F^{\alpha \beta} \): captures all necessary properties of the electromagnetic field.
Understanding electromagnetic fields and their behaviors is crucial for topics like electrodynamics and wave propagation.
Minkowski Metric
The Minkowski metric is a tool used to describe the geometry of spacetime in the framework of special relativity. Notated as \( \eta^{\mu u} \), it provides a way to measure distances and angles in a four-dimensional space where one dimension is time and the other three are space.
This metric is essential in assisting calculations involving energy, momentum, and also in proving properties such as tracelessness and divergencelessness in tensors. The Minkowski metric simplifies calculations by providing invariant results under Lorentz transformations. Its significance arises largely in contexts that involve high speeds or strong gravitational fields.
  • Four-dimensional spacetime: combines three space dimensions and one time dimension.
  • Special relativity: makes use of the Minkowski metric.
  • Supports calculations with energy and momentum.
This metric essentially transforms our understanding of distances and times, facilitating more accurate physics predictions.
Maxwell's Equations
Maxwell's Equations are fundamental in electromagnetism, laying the foundation for how electric and magnetic fields interact and propagate. These are a set of four partial differential equations and can be seen as the cornerstones in understanding the behavior of electromagnetic fields.
In the context of the stress-energy tensor, Maxwell's equations help in proving that the tensor is divergenceless. Specifically, they ensure that in free space, the divergence of both the electric and magnetic components of the field tensor vanish, implying that energy and momentum within the field are conserved.
  • Faraday鈥檚 Law: describes how a time-varying magnetic field creates an electric field.
  • Gauss's Law for Electricity: relates the electric field to charge present.
  • Gauss's Law for Magnetism: states there are no magnetic monopoles.
  • Amp猫re's Law with Maxwell鈥檚 Addition: shows how magnetic fields are generated by electric currents and changes in electric fields.
These equations are not just theoretical constructs but also extensively used in practical applications, enabling technologies like transformers, antennas, and motors.
Divergenceless Condition
The divergenceless condition states that the divergence of a vector or tensor field is zero. In the context of an electromagnetic field's stress-energy tensor, this means that the flow of energy and momentum is conserved within the field.
To show that the stress-energy tensor is divergenceless, one applies Maxwell's Equations, which, in free space, lead to the conclusion that \( abla \cdot \mathbf{T} = 0 \). This has physical implications that energy does not accumulate at any point in the field, ensuring a steady state.
  • Represents conservation: ensures energy-momentum preservation.
  • Relies on Maxwell's Equations: specifically the conditions on the field tensor in free space.
  • Key in fluid dynamics: similar concepts apply in fluid flux conservation.
The divergenceless condition is critical for understanding balance within physical systems, especially in fields propagating through space.

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Most popular questions from this chapter

An object emits a blob of material at speed \(v\) at an angle \(\theta\) to the line-of-sight of a distant observer (see Fig. 4.13). a. Show that the apparent transverse velocity inferred by the observer (i.e., the angular velocity on the sky times the distance to the object) is $$ v_{\text {app }}=\frac{v \sin \theta}{1-(v / c) \cos \theta} $$ b. Show that \(v_{\text {app }}\) can exceed \(c\); find the angle for which \(v_{\text {app }}\) is maximum, and show that this maximum is \(v_{\max }=\gamma v\).

Let two different uniformly moving observers have velocities \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), in units where \(c=1\). Show that their relative velocity, as measured by one of the observers, satisfies $$ v^{2}=\frac{\left(1-\mathbf{v}_{1} \cdot \mathbf{v}_{2}\right)^{2}-\left(1-v_{1}^{2}\right)\left(1-v_{2}^{2}\right)}{\left(1-\mathbf{v}_{1} \cdot \mathbf{v}_{2}\right)^{2}} . $$ A straight application of velocity transformations is painfully tedious, but an application of 4-vector invariants is trivial!

A particle (rest mass \(m\) ) initially at rest absorbs a photon of energy \(h \nu\) and converts this energy into increased internal energy (say, heat). The particle has increased its rest mass to \(m^{\prime}\) and moves with some velocity \(v^{\prime}\). a. Setting up the conservation of energy and momentum, show that $$ \frac{m}{m^{\prime}}=\left(1+\frac{2 h v}{m c^{2}}\right)^{-1 / 2} $$ b. By considering the appropriate Lorentz transformations, show that if the particle had been moving initially and absorbed a photon of energy \(h v\), this same equation for the ratio of the initial and final rest masses holds with \(\nu^{\prime}\) replacing \(\nu\), where \(\nu^{\prime}\) is given by the Doppler formula.

A rocket starts out from earth with a constant acceleration of \(1 \mathrm{~g}\) in its own frame. After 10 years of its own (proper) time it reverses the acceleration, and in 10 more years it is again at rest with respect to the earth. After a brief time for exploring, the spacemen retrace their journey back to earth, completing the entire trip in 40 years of their own time. a. Let \(t\) be earth time and \(x\) be the position of the rocket as measured from earth. Let \(\tau\) be the proper time of the rocket and let \(\beta=\) \(c^{-1} d x / d t\). Show that the equation of motion of the rocket during the first phase of positive acceleration is $$ \gamma^{3} \frac{d^{2} x}{d t^{2}}=g $$ b. Integrate this equation to show that $$ \beta=\frac{g t / c}{\sqrt{(g t / c)^{2}+1}} . $$ c. Integrating again, show that $$ x=\frac{c^{2}}{g}\left[\sqrt{(g t / c)^{2}+1}-1\right] . $$ d. Show that the proper time is related to earth time by $$ \frac{g t}{c}=\sinh \left(\frac{g \tau}{c}\right) $$ so that $$ x=\frac{c^{2}}{g}\left[\cosh \left(\frac{g \tau}{c}\right)-1\right] . $$ e. How far away do the spacemen get? f. How long does their journey last from the point of view of an earth observer? Will friends be there to greet them when they return? Hint: In answering parts (e) and (f) you need only the results for the first positive phase of acceleration plus simple arguments concerning the other phases. g. Answer parts (e) and (f) if the spacemen can tolerate an acceleration of \(2 g\) rather than \(1 g\).

Suppose in some inertial frame \(K\) a photon has four-momentum components $$ P_{\mu}=(-E, E, 0,0) . $$ (We use units where \(c=1\) ). There is a special class of Lorentz transformations-called the "little group of \(P\) "which leave the components of \(P\) unchanged, for example, a pure rotation through an angle \(\alpha\) in the \(y \cdot z\) is such a transformation. Find a sequence of pure boosts and pure rotations whose product is not a pure rotation in the \(y\)-z plane, but is in the little group of \(P\).
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