Question

From the form \((49.5)\) of its energy tensor and the cquation \(T^{\mu v} . v=0\) prove directly that every portion of an incoherent fluid, subject to no external forces, moves with uniform velocity, i.e. that \(A^{\mu}\) \(=\mathrm{d} U^{\mu} / \mathrm{d} \tau=0 .\left[H i n t:\right.\) expand \(\left\\{\left(\rho_{0} U^{\mu}\right) U^{\mu}\right\\}_{, \nu}=0\) and use \(U^{\mu},{ }_{, v} U^{v}\) \(\left.=A^{\mu}, U_{\mu} A^{\mu}=0 .\right]\)

Short answer

Based on the given information about the form of the energy tensor and the equation \(T^{\mu v} . v=0\), we were asked to prove that every portion of an incoherent fluid, subject to no external forces, moves with uniform velocity. By expanding the given equation, relating terms with the equations of motion, rearranging equations, and utilizing the provided conditions, we demonstrated that the four-acceleration \(A^{\mu}\) must be equal to zero. This means that every portion of an incoherent fluid, subject to no external forces, moves with uniform velocity, as desired.

Step-by-step solution

Write the four-velocity, four-acceleration and equation of motion

The four-velocity \(U^\mu\) and four-acceleration \(A^\mu\) of the incoherent fluid are given by: \(U^{\mu} = \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}\) and \(A^{\mu} = \frac{\mathrm{d} U^{\mu}}{\mathrm{d} \tau}\). The given equation of motion for the energy tensor \(T^{\mu v}\): \(T^{\mu v} . v = 0\).

Start by expanding the given equation

Begin by expanding the equation \(\left\{ (\rho_0 U^{\mu}) U^{\mu} \right\}_{,\nu} = 0\) using the chain rule and product rule of differentiation: \(\frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu} = \frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu} + \rho_0 U^{\mu}\frac{\partial U^{\mu}}{\partial x^\nu}\).

Relate the terms with equations of motion

Using the equations of motion from Step 1, we can relate the term \(\frac{\partial U^{\mu}}{\partial x^\nu}\) to the four-acceleration \(A^\mu\): \(\frac{\partial U^{\mu}}{\partial x^\nu} = U_{,\nu}^{\mu} = A^{\mu}\). Now substitute this back into the expanded equation from Step 2: \(\frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu} = \frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu} + \rho_0 U^{\mu}A^{\mu}\).

Use the equation \(T^{\mu v} . v=0\)

Given the equation \(T^{\mu v} . v=0\), we can rewrite the above equation as: \(\frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu} = T^{\mu v} . v + \rho_0 U^{\mu}A^{\mu}\). Now rearrange the equation for \(A^\mu\): \(\rho_0 U^{\mu}A^{\mu} = - T^{\mu v} . v - \frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu}\).

Show that \(A^\mu = 0\)

Since we are given that \(U_{\mu} A^{\mu} = 0\), we can multiply both sides of the above equation by the four-velocity \(U_\mu\): \(\rho_0 U_{\mu} U^{\mu} A^{\mu} = - U_{\mu} T^{\mu v} . v - U_{\mu} \frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu}\). Now use the fact that \(U_{\mu} U^{\mu} = -1\) and \(U_{\mu} A^{\mu} = 0\): \(-\rho_0 A^{\mu} = - U_{\mu} T^{\mu v} . v - U_{\mu} \frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu}\). Since \(U_{\mu} T^{\mu v} . v = 0\), we can simplify the equation further: \(-\rho_0 A^{\mu} = - U_{\mu} \frac{\partial}{\partial x^\nu}\left(\rho_0 U^{\mu}\right)U^{\mu}\). Now we can observe that the right side of the equation has the same expression as on the left side, but with opposite signs. This implies that: \(A^{\mu} = 0\). Hence, we have shown that every portion of an incoherent fluid, subject to no external forces, moves with uniform velocity (i.e., \(A^{\mu} = \mathrm{d} U^{\mu} / \mathrm{d} \tau = 0\)).

Key concepts

Incoherent Fluid

In the context of special relativity, an incoherent fluid is a type of fluid without any internal thermodynamic forces affecting its motion. This means the particles within this fluid don't exert forces on each other, unlike substances with structured interactions. Such a fluid is often used for theoretical models since it behaves predictably under the laws of physics.

Incoherent fluids are useful in studying the basics of how matter interacts with space-time without the complexity of additional forces at play. They provide a clean slate for understanding how objects move in a relativistic framework.

When analyzing incoherent fluids, it becomes clear that their motion is straightforward. Essentially, because there are no internal forces, their velocity remains constant unless external forces are applied. This characteristic serves as a foundation for exploring more complex fluid dynamics in relativity.

Four-Velocity

Four-velocity is a concept arising from the framework of special relativity, forming a crucial part of understanding motion in a four-dimensional space-time.

Defined as \(U^{\mu} = \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}\), four-velocity combines the spatial and temporal components of velocity into a unified parameter.
Here, \(\tau\) represents the proper time, which is the time measured in the frame co-moving with the object.

This concept ensures that velocity respects the intricate geometry of space-time described by Einstein's theory. In essence, four-velocity maintains a constant norm, typically \(-1\) for time-like intervals, preserving the relativistic consistency.

Understanding four-velocity allows us to grasp how objects move through both space and time, emphasizing its base importance in relativistic equations of motion.

Four-Acceleration

In the realm of relativity, four-acceleration is defined as the rate of change of four-velocity with respect to proper time. Mathematically, it is given by \(A^{\mu} = \frac{\mathrm{d} U^{\mu}}{\mathrm{d} \tau}\).

Unlike classical acceleration, four-acceleration applies to objects moving at relativistic speeds, incorporating both changes in velocity and direction within a four-dimensional space-time backdrop.

A distinctive feature of this construct is that it’s always orthogonal to the four-velocity. That is, \(U_{\mu} A^{\mu} = 0\).
This property maintains the magnitude of the four-velocity at \(-1\), adhering strictly to the structure of space-time in relativity.

Four-acceleration provides critical insight into how forces translate into motion within a relativistic framework, illustrating subtle differences from everyday Newtonian interpretations of acceleration.

Energy Tensor

The energy tensor, also known as the energy-momentum tensor or stress-energy tensor, serves as a key element in the equations governing general relativity.

In simple terms, it encapsulates the distribution and flow of energy and momentum in space-time. This tensor ties together the energies, pressures, and stresses within a physical system.

Expressed generally as \(T^{\mu u}\), each component of the tensor relates to specific physical properties:

• The terms \(T^{00}\) represent energy density.
• The terms \(T^{0i}\) relate to momentum flow or energy flux.
• The space-space components \(T^{ij}\) relate to pressures and stresses.

Within the context of special relativity, the energy tensor's simplicity allows us to model the behavior of systems like incoherent fluids effectively, sparking deeper insights into the complex interplay of matter and spacetime.