# Maxwell's equation for non inertial observer

Applying Maxwell's equation we can prove that light will move at the speed of light for every inertial frame, is it true as well for non-inertial frames? How light moves slowly near a black hole??

These are two separate questions.

1) Does light move at c in non-inertial frames?

First to get a bit pedantic, I assume you mean the coordinate velocity. As light does not have a well defined proper velocity, this seems to be the most reasonable way to interpret your question here.

While rewriting Maxwell's equations in a non-inertial frame, and then solving this potential mess can be difficult, we can easily take a solution in an inertial frame and just do the coordinate transformation to get the solution in some non-inertial frame. In particular, if we have the position of some light wave packet vs time in an inertial frame, we can just apply the coordinate transformation to find the new coordinate labels for the points on its path.

Consider the inertial frame with coordinates $t,x,y,z$. A simple example of a non-inertial coordinate system $t',x',y',z'$ is be given by the transformation: $$t'=t,\ x'=x+vt,\ y'=y,\ z'=z$$ This Galilean transformation gives us a non-inertial frame, and the coordinate velocities will just change by $v$ in the $x$ direction as is usual with Galilean transformations. So if in the inertial frame the velocity of the light packet was $c$ in the $x$ direction, the velocity in this new frame would be $c+v$. This is probably the easiest counter example to see.

2) How light moves slowly near a black hole?

Now you are talking about curved spacetime as well. In curved spacetime, there is no choice of coordinate system in which the coordinate speed of light will everywhere be c.

However, locally we can still choose a coordinate system in which that is the case. A free falling frame is such an example, and so if you fall into a black hole nothing strange will be seen with your local measurements of the speed of light. When people say light "slows" while approaching a blackhole, they are again talking about the coordinate speed of light, and they are implicitly assuming a coordinate system (usually the Schwarzschild coordinates). Light slowing to a stop in the limit of approaching the event horizon is just an artifact of a coordinate singularity in this particular choice of coordinate system. This singularity is only an issue with the coordinate choice and can be completely removed if another coordinate system is used. A very similar coordinate singularity occurs even in flat space-time in a common choice of coordinate system with the spatial origin having a constant proper acceleration (Rindler coordinates).