Editing Electrogravitics (section)

== Stress-Energy Tensor for Electromagnetic Field in Vacuum ==

The '''stress-energy tensor''' for an electromagnetic field in vacuum is a fundamental concept in [[General Relativity]] and [[Electromagnetism]]. It describes the distribution of energy, momentum, and stress associated with electromagnetic fields in empty space (vacuum). This tensor plays a crucial role in the [[Einstein Field Equations]] of general relativity, where it contributes to the curvature of spacetime.

=== Definition ===

The stress-energy tensor <math>T^{\mu\nu}</math> is given by:

<math>
T^{\mu\nu} = \frac{1}{\mu_0} \left( F^{\mu\lambda} F^\nu{}_\lambda - \frac{1}{4} g^{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} \right)
</math>

Where:
* <math>T^{\mu\nu}</math> is the stress-energy tensor,
* <math>F^{\mu\nu}</math> is the electromagnetic field tensor,
* <math>g^{\mu\nu}</math> is the metric tensor describing spacetime geometry,
* <math>\mu_0</math> is the permeability of free space,
* <math>F_{\alpha\beta}</math> represents the components of the electromagnetic field tensor arranged differently.

==== <math>\mu</math> - mu ====
The symbol <math>\mu</math> represents one of the indices in the stress-energy tensor. It ranges from 0 to 3, representing the four dimensions of spacetime.

==== <math>\nu</math> - nu ====
The symbol <math>\nu</math> represents one of the indices in the stress-energy tensor. It also ranges from 0 to 3, representing the four dimensions of spacetime.

==== <math>\alpha</math> - alpha ====
The symbol <math>\alpha</math> represents one of the indices in the electromagnetic field tensor. It ranges from 0 to 3, representing the four dimensions of spacetime.

==== <math>\beta</math> - beta ====
The symbol <math>\beta</math> represents one of the indices in the electromagnetic field tensor. It also ranges from 0 to 3, representing the four dimensions of spacetime.

=== Components ===

The components of the stress-energy tensor describe various aspects of the electromagnetic field's influence on spacetime, including energy density, momentum density, and stress.

==== Other Versions ====

There are alternative formulations of the stress-energy tensor for specific applications or contexts. These versions may involve different physical quantities or mathematical expressions depending on the problem at hand. Examples include formulations for specific materials, boundary conditions, or energy-momentum distributions.

===== Examples =====

* Stress-energy tensor for a material medium, incorporating the effects of material properties such as conductivity, permittivity, and permeability.
<math>
T^{\mu\nu} = \varepsilon_0 \left( E^\mu E^\nu - \frac{1}{2} g^{\mu\nu} E_\alpha E^\alpha \right) + \frac{1}{\mu_0} \left( B^\mu B^\nu - \frac{1}{2} g^{\mu\nu} B_\alpha B^\alpha \right)
</math>

* Stress-energy tensor for an electromagnetic field in the presence of matter, accounting for the interaction between electromagnetic fields and matter fields.
<math>
T^{\mu\nu} = \frac{1}{\mu_0} \left( F^{\mu\lambda} F^\nu{}_\lambda - \frac{1}{4} g^{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} \right) + T^{\mu\nu}_{\text{matter}}
</math>

* Stress-energy tensor for an electromagnetic field in a curved spacetime, considering the gravitational effects on the electromagnetic field.
<math>
T^{\mu\nu} = \frac{1}{\mu_0} \left( F^{\mu\lambda} F^\nu{}_\lambda - \frac{1}{4} g^{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} \right) - \frac{1}{4\pi} \left( R^{\mu\nu} - \frac{1}{2} g^{\mu\nu} R \right)
</math>

* Stress-energy tensor for an electromagnetic field in a non-inertial frame of reference, incorporating effects such as acceleration and rotation.
<math>
T^{\mu\nu} = \frac{1}{\mu_0} \left( F^{\mu\lambda} F^\nu{}_\lambda - \frac{1}{4} g^{\mu\nu} F_{\alpha\beta} F^{\alpha\beta} \right) - \frac{1}{c^2} \left( F^{\mu\lambda} a_\lambda^\nu + F^{\nu\lambda} a_\lambda^\mu \right)
</math>

These equations demonstrate the versatility of the stress-energy tensor and its adaptability to different physical scenarios.


=== Significance ===

The stress-energy tensor for an electromagnetic field in vacuum provides crucial information about how electromagnetic fields interact with the fabric of spacetime. It contributes to the curvature of spacetime according to general relativity, influencing the behavior of matter and energy on cosmic scales.

=== See Also ===

* [[Maxwell's Equations]]
* [[Einstein Field Equations]]
* [[General Relativity]]
* [[Electromagnetism]]