Question

Assume that, at some event, an energy tensor \(T^{\mu v}\) satisfies \(T^{\mu \nu} V_{\mu} V_{v}>0\) for all timelike and null vectors \(V^{\mu}\). By reference to (43.13) show that for an electromagnetic energy tensor this inequality does not necessarily hold. But if it holds, prove that there is at most one 'rest frame' for \(T^{\mu v}\), i.e. a frame in which \(T^{0 i}=0\). [Hint: the fourvelocity of the required frame satisfies \(T^{\mu v} U_{v}=k U^{\mu}\) for some nonzero \(k\) (why?); if there were two such timelike 'eigenvectors' of \(T^{\mu \nu}\) then there would also be a null eigenvector.] Note: one can also show that the inequality implies the existence of at least one rest frame, and, conversely, that a unique rest frame implies the inequality.

Short answer

Based on the step by step solution and the assumptions made, it can be shown that for an electromagnetic energy tensor, the inequality \(T^{\mu\nu}V_\mu V_\nu>0\) does not necessarily hold for all timelike and null vectors \(V^\mu\). However, assuming the inequality holds, it was proved that there can be at most one 'rest frame' for the energy tensor \(T^{\mu\nu}\) if the inequality holds. This unique rest frame is characterized by having \(T^{0i}=0\), and its four-velocity satisfies \(T^{\mu\nu}U_\nu=kU^\mu\) for some nonzero scalar \(k\).

Step-by-step solution

Showing the inequality does not necessarily hold for electromagnetic energy tensor

Let's first recall the electromagnetic energy tensor (43.13): $$ T^{\mu\nu} = F^{\mu\alpha}F^\nu_{\ \alpha}-\frac{1}{4}\eta^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta} $$ where \(F^{\mu\alpha}\) represents the electromagnetic field tensor. Now, if we multiply by both \(V_\mu\) and \(V_\nu\) for some timelike or null vector \(V^\mu\), and check if the inequality holds: $$ T^{\mu\nu}V_\mu V_\nu = V_\mu F^{\mu\alpha}F^\nu_{\ \alpha}V_\nu-\frac{1}{4}V_\mu\eta^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}V_\nu $$ We can't guarantee this inequality is always positive as it depends on the components of the electromagnetic field tensor \(F^{\mu\alpha}\). Thus, the inequality \(T^{\mu\nu}V_\mu V_\nu>0\) does not necessarily hold for the electromagnetic energy tensor.

Assume the inequality holds

We will now assume that the inequality \(T^{\mu\nu}V_\mu V_\nu>0\) holds for all timelike and null vectors \(V^\mu\). We are asked to prove that there can be at most one 'rest frame' for the energy tensor \(T^{\mu\nu}\). In this rest frame, \(T^{0i}=0\).

Using the hint

The hint suggests that the four-velocity \(U^\mu\) of the required rest frame satisfies the following condition: $$ T^{\mu\nu}U_\nu=kU^\mu $$ for some nonzero scalar \(k\). If there were two such timelike eigenvectors of \(T^{\mu\nu}\), then there would also be a null eigenvector. To show this, consider two timelike eigenvectors \(U_1^\mu\) and \(U_2^\mu\) satisfying: $$ T^{\mu\nu}U_{1\nu}=k_1U^\mu_1 $$ and $$ T^{\mu\nu}U_{2\nu}=k_2U^\mu_2 $$ Now, let's consider their difference: $$ T^{\mu\nu}(U_{1\nu}-U_{2\nu})=k_1U^\mu_1-k_2U^\mu_2 $$ If both \(k_1\) and \(k_2\) are nonzero, we can divide both sides by \((k_1-k_2)\) and define a "normalized" difference vector \(V^\mu=\frac{U^\mu_1-U^\mu_2}{k_1-k_2}\). This would lead to a null eigenvector because the difference vector \(V^\mu\) would have to be orthogonal to both timelike eigenvectors. But we assumed that the inequality \(T^{\mu\nu}V_\mu V_\nu>0\) holds for all timelike and null vectors.

Conclusion

Since we have shown that two timelike eigenvectors of the energy tensor \(T^{\mu\nu}\) would lead to a null eigenvector, which contradicts our initial assumption, we can conclude that there can be at most one 'rest frame' for the energy tensor \(T^{\mu\nu}\) if the inequality holds.

Key concepts

Electromagnetic Energy Tensor

The electromagnetic energy tensor plays a vital role in vast domains of physics, especially in the context of relativity. Often denoted as \(T^{\muu}\), it describes the flow of electromagnetic energy and momentum through space and time. This tensor is given by a specific formula:\[T^{\muu} = F^{\mu\alpha}F^u_{\ \alpha} - \frac{1}{4}\eta^{\muu}F^{\alpha\beta}F_{\alpha\beta}\] Here, \(F^{\mu\alpha}\) represents the electromagnetic field tensor, while \(\eta^{\muu}\) is the metric tensor in Minkowski spacetime.
This construction captures electromagnetic interactions and relates to notions of energy density, momentum density, and stress by characterizing how the field interacts with itself and the surrounding space.
It is important to note that while the electromagnetic energy tensor is robust, it does not necessarily satisfy the positivity condition \(T^{\muu}V_\mu V_u > 0\) for all timelike and null vectors \(V^\mu\). This complexity arises from the fact the electromagnetic fields can have varying configurations, sometimes leading to non-positive values which implies limitations in directly associating this tensor with physical quantities like energy densities.

Rest Frame in Relativity

The concept of a 'rest frame' is crucial in understanding various physical phenomena, particularly when working with energy tensors in relativity. A rest frame is essentially a coordinate system where an object, or energy system like \(T^{\muu}\), appears at rest. In terms of an energy tensor, this frame is where the spatial components vanish, specifically \(T^{0i}=0\).
This implies that the energy is fully aligned with the time axis in this frame, eliminating any spatial momentum components. Determining a rest frame helps simplify complex equations and make predictions more manageable. It should be noted that under the assumption of \(T^{\muu}V_\mu V_u>0\), the presence of at most one rest frame is ensured. This uniqueness means it can serve as a consistent reference for analyzing energy and momentum distributions.
A rest frame provides a perspective where the tensor's structure unveils hidden symmetries or characteristics, aiding in deeper physical interpretations.

Timelike and Null Vectors

Timelike and null vectors are fundamental in the realm of relativity as they characterize different types of trajectories in spacetime. A timelike vector symbolizes an observer's path through time; it represents a worldline that can be connected by a clock or an object with mass. Mathematically, it satisfies \(V^\mu V_\mu < 0\) using the metric tensor's signature, indicating a trajectory limited by the speed of light.
Conversely, a null vector signifies the path of light in spacetime and fulfills \(V^\mu V_\mu = 0\). These trajectories connect points that light can traverse, marking the boundary between causal and spacelike separations.
In many problems, including those involving energy-momentum tensors, such vectors help define essential properties like causing divergence issues when accounting for energy flow in spacetime. By evaluating \(T^{\muu}V_\mu V_u\), one can understand how the energy fields affect objects moving at various speeds, offering insights into both the dynamics and potential interactions within spacetime.

Eigenvectors in Relativity

In the framework of relativity, eigenvectors provide a crucial insight into an energy tensor's inherent properties, such as \(T^{\muu}\). They are vectors that remain unchanged in direction after a linear transformation is applied, characterized by eigenvalue equations like \(T^{\muu}U_u = kU^\mu\), where \(k\) is an eigenvalue.
In relativistic physics, finding eigenvectors often leads to identifying distinct timelike or null vectors that reveal underlying symmetries or conservation laws in the tensor's description of a field. These vectors, particularly when pertaining to an energy-momentum tensor, can showcase how the energy is oriented or if unique rest frames exist. It’s intriguing that timelike eigenvectors can define rest frames by aligning with the energy's dominant direction in spacetime.
When evaluating problems with multiple eigenvectors, one may inadvertently derive null eigenvectors, reflecting different poses in multidimensional spacetime analysis. This intricacy plays a part in proving theories related to uniqueness or existence of rest frames and contributes to a thorough understanding of relativistic energy distributions.