# Can the relative speed of an object meet/exceed the speed of light?

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## Main Question or Discussion Point

I was curious if the relative speed of an object can exceed the speed of light. Specifically, I am curious about the following thought experiment. I am not a physicists (and if I were asking the following would make me a poor one) and it has been 20 years since college physics.

If a vessel is traveling at speed X (less than the speed of light) on a vector heading of A away from the fixed object M and object Q is traveling within the vessel on the same heading of vector A at a speed of Y (less than the speed of light), can the speed of Q relative to object M (X+Y) equal or exceed the speed of light? While I not assuming the vessel has to be a space vessel or that the vessel or object have to be in a vacuum, that is my natural inclination as it makes it easier to dismiss the need to muddy the experiment with friction and gravity. As for how object Q is being propelled within the vessel, I have no specific constraints. However, to avoid muddying the water, I would assume however it is being propelled within the vessel that the force is negligible on the vessels speed away from M. So, if object Q is being propelled on a rail with a rocket, the size of the vessel is too large for the velocity of the vessel to slow in any meaningful fashion. Perhaps we can assume that Q isn't fixed in any way to the vessel (and gravity is a non factor) but instead is highly magnetic and a massive electromagnet sits just ahead of the vessel (but not attached to the vessel) and when it is turned on it attracts Q toward it in the same vector heading as the vessel (and we assume the electromagnet has negligible effect on the vessel).

If it is possible in some sort of scenario that X+Y = C relative to M, let me expand this to ask one more question. From what I remember/understand, light instantaneously reaches C the moment a photon comes into being and C does not change relative to another objects speed (i.e. it is not additive). Let's assume that a laser (emitting all photons in vector heading A) is fixed to object M and a human is seated at the front of M looking back toward the laser light. Once emitted, relative to M, the photons are traveling along vector A at C. If X+Y=C, then relative to M, the person in front of the light is also traveling a C away from M on vector heading A. It would seem that the light would not catch up with the person (until object Q slowed/stopped). If this is this is so, wouldn't the person have to be traveling at the speed of light? And, if this is true, the person would only be traveling at the speed of light because of an additive property allowing for the speed of the vessel and relative speed of the object Q to equal or exceed C. Lastly, if this were somehow possible, what does the person see when Q stops and the photons move "forward?" From what I understand, photons can occupy the same space (I reason this from my understanding photons can pass through one another) and during the period Q is at C, as each new photon is emitted (since everything is traveling on vector heading A ... and let's assume the laser is so perfect each photon can be emitted with such precision), it will occupy the same space. So, when Q stops, will a person see an intense (depending on duration of Q traveling relative to the vessel) "photon" of light.

I imagine there are many fallacies to my thought experiment and would love to learn what I a missing that doesn't make this possible. Thanks.

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PeroK
Homework Helper
Gold Member
I was curious if the relative speed of an object can exceed the speed of light. Specifically, I am curious about the following thought experiment. I am not a physicists (and if I were asking the following would make me a poor one) and it has been 20 years since college physics.
What you need is the relativistic velocity additiona formula. See here, for example:

http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html

Ibix
Ibix
You can also google "relativistic velocity addition", which answers your immediate question. Velocities do not add linearly, so if the craft has velocity ##u## and the object inside it has speed ##v## as measured by the craft, an external observer will measure velocity$$\frac{u+v}{1+uv/c^2}$$This never exceeds ##c##.

Thanks. I will do more reading on that to expand my understanding.

berkeman
Ibix
Fine apart from the final paragraph, which is wrong. The speed of a light pulse as measured by another light pulse is undefined, not ##c##. The velocity addition formula apparently works, but is derived from the Lorentz transforms which require at least one of the objects to be moving slower than ##c## in order for there to be a definable reference frame.

(Note that it's using a different sign convention from me, which is why its version of the velocity addition formula has minus signs).

Well, I just knew that ##c## it's a constant. It is the same in every frame of reference.

Ibix
Well, I just knew that ##c## it's a constant. It is the same in every frame of reference.
Yes, but you can't define a frame of reference travelling at ##c##. So there's no way to define what speed a light pulse would measure anything else travelling at, including another light pulse.

mcastillo356
I think the quote you mention is unfortunate.

Ibix
It's rather poor. Given a quick look at the author's publications I don't doubt that he knows better. Unfortunately in some contexts some people seem to decide to provide some sort of answer to an invalid question, instead of challenging the incorrect assumptions that underlie it. To be fair, that kind of thing is easier to do in a forum like this one where there's space for some back-and-forth discussion with the person who initially asked the question, but still...

Let's say I am in an inertial frame of reference and I note two objects moving towards each other. Object 1 has velocity v1 to the right and object 2 has velocity v2 to the left. In my frame of reference, their speed of approach (relative speed according to me) is v1 + v2 which can be > c. But this is not the speed of either object in any inertial reference frame, it's just a calculated quantity.

Ibix
Ibix