Justin Hunt said:
@Dale Thank you for your informative reply! I have a further question though based on your response.
Suppose that there is no inertial frame that exists over the entire region of interest. But, there are locally inertial frames at one end and at the other end. an observer at either end would measure a constant speed of light C in their local environment, but since there is no inertial frame that exists over the entire region, the coordinate speed of light is not C. (My example of this case would be a spaceship in deep space versus a spaceship on the surface of a very massive planet). We know that gravitational time dilation exists.
So, my question is would the coordinate speed of C on the planet be different than then the coordinate speed in deep space not because the speed of light C is different, but because time is passing at different rates? speed is Distance over time and they wouldn't agree on time t. In particular light would appear to be moving slower on the planet from deep space perspective and faster than C from the planet perspective of the ship in deep space. is this correct? (I am assuming there is no relative velocity between the two objects, just the large gravitational potential difference).
First, let's take an example from SR. There is a star 10 light years away. You accelerate towards the star and, after a year say, have reached a significant relative velocity for which the gamma factor is 2. Now, due to length contraction, the star is less than 5 light years away.
If you do a raw calculation, then the star has moved more than 5 light years in a year: greater than the speed of light.
The moral is that if you want to understand the invariance of the speed of light, you have to be precise in how you measure distances and times. In this case, as you accelerate you are continuously changing your inertial reference frame.
Let's take another example. If there is a spaceship moving through the solar system at, say, ##\frac35 c##. The spaceship clock will be dilated in your reference frame (assume you are on the Earth) by a factor of ##\frac54##. Now, you have two clocks with which you can measure time: your clock on Earth and the spaceship clock. Both of these give you a time coordinate of every event. You then measure things moving through the solar system (including light). If you use your clock, you get one set of coordinate speeds. But, if you use the spaceship clock, you get a different set of coordinate speeds. Including a speed of ##> c## for light.
The moral is that if you want to calculate the speed of light and get ##c##, you must use a clock that is at rest in your inertial reference frame.
Now, let's move to a GR sitiuation, where light is moving close to a large mass, perhaps a black hole. Let's assume you know the distance between two points at rest relative to the black hole. Perhaps there is a spaceship there, of known length, hovering at rest relative to the black hole (and at rest relative to Earth). A light beam moves across the spaceship. Again, you can measure time using your clock back on Earth or the spaceship clock which is local to the event. The spaceship clock, to you, will be gravitationally time dilated. So, you will get two different speeds of light depending on what clock you use. I.e. depending on what you use as your time coordinate.
The moral is that "coordinate" speed of anything depends on your choice of coordinates! In particular, you can choose to measure time according to any clock in the universe. And, if all those clocks are moving and/or at different gravitational potential relative to each other, then they will all give you a different time coordinate.
There is no such thing as
the coordinate speed of light. There is
a coordinate speed of light for
each and every set of coordinates. There is
no physical significance in changing your coordinates and getting a different coordinate speed of light.
In summary:
In SR: if you use coordinates based on an inertial reference frame, then the speed of light is always ##c##. In practice, this would mean a clock and a metre stick at rest in the inertial reference frame. And, because inertial reference frames are global, you can in fact use one clock to give your time coordinate everywhere.
In GR: if you use local coordinates: a local metre stick and a local clock (at rest with respect to each other), then the measured speed of light is always ##c##. And that is what is meant by "the speed of light locally is ##c##".
The final moral is to understand that, in GR, there is no physical significance in using global coordinates and getting a coordinate speed of light other than ##c##.