Subtle Distinction: Speed of Light for All Observers

[actionintegral]

Friends,

In a recent post I said

"The speed of light is the same for all observers".

I meant unequivocally. With no exceptions. Always the same. No tricks or gimmicks.

But I was quickly corrected in triplicate!

'To be exact: The speed of light is the same for all inertial observers. There is a significant difference.'

If the difference is significant, then I need to understand it. Does this mean the speed of light can change for an observer?

 

[pmb_phy]

Does this mean the speed of light can change for an observer?

Yes. When it is said that the speed of light is observer dependant it refers to the "coordinate speed of light." Consider an observer in a uniform g-field (or, equivalently, a uniformly accelerating frame of reference). The speed of light in the field will be a function of the location of a light packet in the field. If the observer is located at z = 0 then, for a g-field pointing in the -z direction, the speed of light will be a function of z. Einstein derived the expression for this speed of light and published it in 1907. As measured at z = 0 the speed of light is that measured by an inertial observer in flat spacetime.

Pete

 

[actionintegral]

Thank you for your reply. I will learn about that.

 

[George Jones]

Whenever I see the unqualified statement "The speed of light is the same for all observers.", I take it to mean the physical speed as measured by the measuring apparatus of the observer. With this take on this statement, it is true for all observers in both special and general relativity.

As Pete has pointed out, of the coordintate speed of light can change, but while coordinate speed is often easier to compute than physical speed, coordinate speed is less natural to measure than physical speed.

 

[pervect]

I tend to agree with George. The coordinate speed of light can change with regard to the obsever - the coordinate speed being defined as the rate of change of the distance coordinate with respect to the time coordinate in some chosen coordinate system.

However, when you measure the speed of light in a vacuum using properly set-up local clocks and local rulers (clocks and rulers located near the same point that the light is), you always get the same value, 'c'.

"Properly set up" means that the frame defined by the rulers isn't rotating, the rulers are all at right angles to each other, and that the rulers are Minkowski-orthogonal to the time vector defined by the clocks.

This follows from the fact that the geometry of space-time is always locally "flat", i.e. has the familiar Minkowski metric in some small area around a given point.

MTW expressses this by saying that the geometry of space-time is locally Lorentzian.

 

[bernhard.rothenstein]

Friends,

In a recent post I said

"The speed of light is the same for all observers".

I meant unequivocally. With no exceptions. Always the same. No tricks or gimmicks.

But I was quickly corrected in triplicate!

'To be exact: The speed of light is the same for all inertial observers. There is a significant difference.'

If the difference is significant, then I need to understand it. Does this mean the speed of light can change for an observer?

Have a look please at Edward A. Desloge and R.J.Philpott Am.J.Phys. 55 252 1987 Ch.VII

 

[Robert.Hipple]

Have a look please at Edward A. Desloge and R.J.Philpott Am.J.Phys. 55 252 1987 Ch.VII

Hi Bernhard!

Fantastic!

I have been looking for a paper like this for quite some time. It forms a perfect pedagogical bridge between SR and GR.


Robert

 

[MeJennifer]

By the way, a common confusion is the difference between the speed of light and c.

The speed of light is not constant in all frames of reference but c is!

 

[selfAdjoint]

By the way, a common confusion is the difference between the speed of light and c.

The speed of light is not constant in all frames of reference but c is!

Excuse me but what "frames of reference" are you speaking of here? Are you talking about media? Light can be made to "sit up and beg" with the right nonlinear media, but frames of reference?

 

[MeJennifer]

Excuse me but what "frames of reference" are you speaking of here? Are you talking about media? Light can be made to "sit up and beg" with the right nonlinear media, but frames of reference?

For instance in a non-inertial frame of reference the speed of light is different, however c remains constant.

This is sometimes very confusing in discussions about the speed of light, in these cases it is often better not to use c at all.

 

[pervect]

This doesn't have anyting to do about the issue of the speed of light in a media vs 'c'.

This issue has to do with how velocities are measured. People who rely on coordinates are usually the ones confused by this issue.

The physics is actually very simple - when you use local rulers and local clocks, the speed of light over a short path is always 'c'. This is true even in accelerated coordinate systems, though one has to pay attention to the "short" part of the above statement.

People often use coordinate systems to measure the "speed of light" and think of the speed as being measured by the rate of change of a distance coordinate with respect to a time coordinate. This is a fundamentally different defintion, for it uses coordinates to measure the speed, rather than local clocks and rulers.

[edits below]

An analogy might help. Suppose someone tells you "We have a naval ship, and it always travels in the sea at a constant velocity". Now,if you set up a lattitude/longitude coordinate system on the Earth, and measure how many degrees of longitude the ship moves in one hour. you would find that the ship moves faster and fasater as it gets closer and closer to the north pole. This is the coordinate based defintion of speed - one measured the coordinate of the ship (its longitude) and how fast it changes with time.

However, someone actually on the ship, measuring its velocity relative to the water, fnds that the speed of the ship really is constant, as you were informed of in the first place. So the confusion arised because of differences in defintions.

For this example, we are ignoring the issue of ocean currents - assuming that they are negligible. In relativity, of course, there are no "ether currents" to worry about.

In our naval example, the issue is how one defines distances and velocities. Using coordinates, as one gets closer and closer to the north pole, a minute of longitude (a coordinate measure) is not the same as a physical nautical mile in one's local coordinate system. The curvature of the Earth is responsible for this confusion, just as the curvature of space-time is what makes the "coordinate speed" of light vary.

One can set up a metric that gives physical distances on the Earth as a function of changes in coordinates - i.e. distances as a function of $\delta$ lattitude and $\delta$ longitude. This metric for the curved surface of the Earth is conceputally the same as Einstein's metric for space-time, except that Einstein's metric gives the invariant Lorentz interval, rather than the distance.

One can see that when the metric coefficients are diagonal and all equal to 1, physical distances correspond to coordinate distances.

You might ask "Can we set up a coordinate system so that the metric of the Earth is everywhere an identity matrix"?. The answer is no, and the reason this can't be done is because the Earth's surface is curved. Relativity says the same thing about space-time - in the presence of curvature near large masses, you can't make the metric coefficients equal to an identity matrix everywhere, though at any particular point you can "normalize" them so that coordinate distances are equal to physical distances (ruler distances) and coordinate times are equal to clock times.