# Non inertial frame: Speed of light

## Main Question or Discussion Point

The speed of light is not constant in a non-inertial frame; the light accelerates with the acceleration of the observer in reverse direction.
Consider the following problem:
If a light pulse is created in an inertial frame at some time, t<0 (say t=-10); the light pulse moves at velocity c with respect to the observer. If the observer starts accelerating with some acceleration a1 for time interval, t=0 to t=t1, the light pulse will seem to accelerate with acceleration a1 since the observer is in non-inertial frame. At time, t=t1, the velocity of light pulse be v1 where,
v1 = c + a1*t1. If the observer stops accelerating after t=t1, the observer will now be in inertial frame.

Question: What will be the speed of light pulse for time t>t1 for the observer

A) Velocity c since the observer is in inertial frame for t>t1
B) Velocity v1 because the speed of light cannot change abruptly at t=t1 when the observer jumps from non-inertial frame into inertial frame. Remember, the speed of light didnt change suddenly at t=0 when the observer jumps from inertial frame into non-inertial frame
C) Any other option. Please explain with reason

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Drakkith
Staff Emeritus
What do you mean by "speed of the light pulse"? The speed of light is always c. The duration of the light pulse when measured by the observer would change however.

Who said that speed of light is non-constant in non-inertial reference? Have you consider time dilation? Though special relativity doesn't apply to non-inertial frames, just take a limit. I don't think the postulate is correct.

Statements about the speed of light should be qualified by a description of how the speed is measured by some observer. Otherwise the spped is just an abstract quantity, that could have any value.

The postulate of SR is that all inertial observers will measure the same speed in their labs. The simplest way is to bounce a light beam off a mirror and have one clock that starts when the light is emitted and stops when the reflected light arrives back.

In order to show that this speed will be measured differently in an accelerating lab, try analysing this experiment under those conditions.

Who said that speed of light is non-constant in non-inertial reference? Have you consider time dilation? Though special relativity doesn't apply to non-inertial frames, just take a limit. I don't think the postulate is correct.
This is a common knowledge in general relativity that speed of light is non-constant in non-inertial frame. The einstein elevator thought experiment describes that the light will follow a curved parabolic path in an accelerating non-inertial frame.

Statements about the speed of light should be qualified by a description of how the speed is measured by some observer. Otherwise the spped is just an abstract quantity, that could have any value.

The postulate of SR is that all inertial observers will measure the same speed in their labs. The simplest way is to bounce a light beam off a mirror and have one clock that starts when the light is emitted and stops when the reflected light arrives back.

In order to show that this speed will be measured differently in an accelerating lab, try analysing this experiment under those conditions.
The problem at hand includes non-inertial frame so special relativity may be of little use. General relativity deals with accelerating frame but I have little grasp of its understanding. I hope you get the problem now.

This is a common knowledge in general relativity that speed of light is non-constant in non-inertial frame. The einstein elevator thought experiment describes that the light will follow a curved parabolic path in an accelerating non-inertial frame.
The curved path implies a change of velocity, not speed. Velocity is a 4-vector, and its (invariant) norm is c, a scalar.

The problem at hand includes non-inertial frame so special relativity may be of little use. General relativity deals with accelerating frame but I have little grasp of its understanding. I hope you get the problem now.
SR can deal with acceleration as long the spacetime is flat. There are several other threads in this forum dealing with this issue.

bcrowell
Staff Emeritus
Gold Member
FAQ: Is the speed of light equal to c even in an accelerating frame of reference?

The long answer is that it depends on what you mean by measuring the speed of light.

In the SI, the speed of light has a defined value of 299,792,458 m/s, because the meter is defined in terms of the speed of light. In the system of units commonly used by relativists, it has a defined value of 1. Obviously we can't do an experiment that will remeasure 1 to greater precision. However, it could turn out to have been a bad idea to give the speed of light a defined value. For example, it would have been a bad idea to give the speed of sound a defined value, because the speed of sound depends on extraneous variables such as temperature.

One such extraneous variable might be the direction in which the light travels, as in the Sagnac effect, which was first observed experimentally in 1913. In the Sagnac effect, a beam of light is split, and the partial beams are sent clockwise and counterclockwise around an interferometer. If the interferometer is rotating in the plane of the beams' path, then a shift is observed in their interference, revealing that the time it takes light to go around the apparatus clockwise is different from the time it takes to go around counterclockwise. An observer in a nonrotating frame explains the observation by saying that the beams went at equal speeds, but their times of flight were unequal because while they were in flight, the apparatus accelerated. An observer in the frame rotating along with the apparatus says that clearly the beams could not have always had the same speed c, since they took unequal times to travel the same path. If we insist on letting c have a defined value, then the rotating observer is forced to say that the same closed path has a different length depending on whether the length is measured clockwise or counterclockwise. This is equivalent to saying that the distance unit has a length that depends on whether length is measured clockwise or counterclockwise.

Silly conclusions like this one can be eliminated by specifying that c has a defined value not in all experiments but in local experiments. The Sagnac effect is nonlocal because the apparatus has a finite size. The observed effect is proportional to the area enclosed by the beam-path. "Local" is actually very difficult to define rigorously [Sotiriou 2007], but basically the idea is that if your apparatus is of size L, any discrepancy in its measurement of c will approach zero in the limit as L approaches zero.

In a curved spacetime, it is theoretically possible for electromagnetic waves in a vacuum to undergo phenomena like refraction and partial reflection. Such effects are far too weak to be detected by any foreseeable technology. Assuming that they do really exist, they could be seen as analogous to what one sees in a dispersive medium. The question is then whether this constitutes a local effect or a nonlocal one. Only if it's a local effect would it violate the equivalence principle. This is closely related to the famous question of whether falling electric charges violate the equivalence principle. The best known paper on this is DeWitt and DeWitt (1964). A treatment that's easier to access online is Gron and Naess (2008). You can find many, many papers on this topic going back over the decades, with roughly half saying that such effects are local and violate the e.p., and half saying they're nonlocal and don't.

Sotiriou, Faraoni, and Liberati, arxiv.org/abs/0707.2748

Cecile and Bryce DeWitt, "Falling Charges," Physics 1 (1964) 3

Gron and Naess, arxiv.org/abs/0806.0464v1

The curved path implies a change of velocity, not speed. Velocity is a 4-vector, and its (invariant) norm is c, a scalar.

SR can deal with acceleration as long the spacetime is flat. There are several other threads in this forum dealing with this issue.
Speed is just the magnitude of velocity without the direction. If the light is accelerating then its velocity is changing at every instant then. Thus, the speed of light doesnt remain constant at every instant.

For more detail about non-inertial frame: http://arxiv.org/abs/gr-qc/9909081v7

Speed is just the magnitude of velocity without the direction. If the light is accelerating then its velocity is changing at every instant then. Thus, the speed of light doesnt remain constant at every instant.

For more detail about non-inertial frame: http://arxiv.org/abs/gr-qc/9909081v7
This paper is talking about the coordinate speed of light. That can vary in a gravitational field. But coordinate speed is not what is measured by a local observer, so we're talking about two different things.

This paper is talking about the coordinate speed of light. That can vary in a gravitational field. But coordinate speed is not what is measured by a local observer, so we're talking about two different things.
For all matter of discussion, assume speed of light= dx/dt (where x= position of light) as observed by the observer.

pervect
Staff Emeritus
An observer could use a large number of different coordinate systems. While we could take some guesses as to which ones you might mean, it turns out in my experience that if it's not precisely specified, people generally have something different in mind and it impairs communications.

This is a common problem with specifying things in terms of coordinates, one has to define exactly the coordinates one is using for any communication to occur. Coordinate-free language doesn't have that particular issue, one reason I prefer it.

Going back to your example, for instance, you could say (dx/dt) as measured in fermi-normal coordinates....

Dale
Mentor
Question: What will be the speed of light pulse for time t>t1 for the observer

A) Velocity c since the observer is in inertial frame for t>t1
B) Velocity v1 because the speed of light cannot change abruptly at t=t1 when the observer jumps from non-inertial frame into inertial frame. Remember, the speed of light didnt change suddenly at t=0 when the observer jumps from inertial frame into non-inertial frame
C) Any other option. Please explain with reason
The question is ill-posed as stated. There is no unique coordinate system associated with a non-inertial observer, so you need to specify exactly the coordinate system you are referring to. Then calculating the coordinate speed is simply a matter of calculating dx/dt.