# What are the Maxwell's equations for gravitational waves?

Maxwell's four equations can be used to describe the propagation of electromagnetic waves. What is the equivalent for gravitational waves - if that question makes sense?

• You mean the equations from which gravitational waves can be predicted? Mar 10, 2018 at 14:48
• @YuzurihaInori yes
– Alf
Mar 10, 2018 at 15:00

Gravitational waves are a prediction of linearised gravity in General Relativity, analogous to that of electromagnetic waves in Electromagnetism. The equations predicting gravitational waves can be written as :

$$\partial^b\overline{\gamma_{ab}}=0$$ $$\partial^c\partial_c\overline{\gamma_{ab}}=-16\pi T_{ab}$$

where $\gamma_{ab}$ is the 'small' deviation from a flat spacetime $\eta_{ab}$ and $T_{ab}$ is the stress-energy tensor.

The above equations are similar to that of the maxwell equations :

$$\partial^aA_a=0$$ $$\partial^a\partial_aA_b=-4\pi j_b$$

where $A^a$ is the vector potential and $j_b$ is the current density. [The first equation is the Lorenz gauge condition and the second is the combined Maxwell's equation]

Cheers!!

• The overbar must denote trace reversal. The analog of the EM field strength tensor would be the affine connection. Mar 10, 2018 at 18:13
• Unanimously Agreed. Mar 10, 2018 at 18:17
• math.stackexchange.com/questions/552347/… Mar 11, 2018 at 9:19
• @Alf This might help you out in details. You need to understand the difference between covariant and contravariant (subscript is covariant, superscript is contravariant). Mar 11, 2018 at 9:20
• @Alf I was studying the classical ED again, and came to know that it was called Lorentz gauge but somehow it later came to be known as Lorenz gauge. The reason is just that it was found by Lorenz (but was not accepted initially), but the gauge is a Lorentz invariance condition, so the term was used. Apr 30, 2018 at 18:02