# Relative velocity greater than speed of light

Is it possible a relative velocity between 2 objects to be greater than the speed of light?

I know that it is impossible to reach the speed of light considering us to be immobile but is this different?

Velocity is frame-dependent. If by "relative velocity" you mean the difference between the two objects' velocities, then (in any given frame) the speed of each object is limited to $$c$$ so their relative velocity can be anything up to $$2c$$. If you mean to choose a frame in which one of the objects is stationary, then, as the other answers have said, the limit is $$c$$.

If you shoot one object to the east with speed 0.75 c and one object to the west with speed 0.75 c, then in your frame of reference the relative speed of the objects will be 1.50 c.

But in the frames of each of these objects, the other object will have a speed of only $$0.96 c.$$

So your question might be interpreted so that the answer can be yes, but the way I think that you meant, the answer is no.

If we transition to general relativity and an expanding universe, then for objects that are separated with a big enough distance, their relative speed can exceed the speed of light in such a way that they can not send signals between each other.

Is it possible a relative velocity between 2 objects to be greater than the speed of light? I know that it is impossible to reach the speed of light considering us to be immobile but is this different?

Usually the term "relative velocity" means the velocity of one object in the reference frame of the other object. It turns out unsurprisingly that my velocity in your frame is equal and opposite to your velocity in my frame, so our relative velocity is the same regardless of which of us is considered to be immobile. So it is not different: if you consider yourself immobile and me moving or if you consider me immobile and yourself moving our relative velocity is still less than c either way.

• But you could have a third observer standing in the middle and see both objects moving away in opposite directions at more than 0.5c, yet the relative velocity between those objects is still less than c. – Nuclear Wang Jan 4 at 16:34
• Yes. The difference in velocities in a 3rd reference frame is limited by 2c, but the relative velocity is limited by c. – Dale Jan 4 at 16:49

Velocity is definitonally the same as "relative velocity". This is the point of the first postulate of relativity.

The answer you're probably looking for is NO. Suppose you have to spaceships that start at the same location but zip off in opposite directions from each other each traveling at the speed of light (c). The conventional way to calculate the relative velocity would be Vr = V1 - V2 where V2 = (-V1). So Vr = V1 -(-V1) = 2V. But that would be 2c if V = c. Because of Special Relativity this is not possible. The maximum speed at which they could separate would be c. So the actual formula for relative velocity in special relativity is Vr = [v1 - v2]/[1 - (v1*v2/c2)] = 2v/[1 + v2/c2]. Using this formula you'll see the answer becomes just c(the speed of light).

Let's say that due to an explosion, two space ships at a space dock were blown apart from each other and are now moving in opposite directions, with each now moving at a spacial velocity of 0.866c relative to the space dock, but again moving in opposite directions. They will therefore be moving apart from each other at 1.732c.

Now imagine that we had two massively long objects that were side by side, and that one was moving to the left along side of one of these two spaceships, and the other was moving to the right along side of the other spaceship. Despite the fact that these two long objects were relatively in motion a total of 1.732c, if those on board either of the two long objects were to measure the velocity of the other long object, they would measure that they were moving relative to each other at a total of 0.98974c velocity, not 1.732c.

The key to understanding relativity, is to understand exactly why this occurs.﻿

Yes. Yes it absolutely can for any definition of velocity we would like to use.

Wait wait hang on Mr. VgAcid(lovely name I know), doesn't this contradict Einstein's theory of relativity? You look new what do you know?

Well, let us stepback. What is 'relative velocity'? Choose a frame for an observer: measure a physically established object's velocity. There you have it. The problem is... this depends upon your frame.

Now certainly, when it comes to inertial frames, nothing exceeds the speed of light. But when it comes to accelerating frames, it does. In fact, the speed of light isn't even c!

Einstein himself mentioned this! To quote him, "[...]But at the same time it turned out that one of the basic principles of that theory, namely, the principle of the constancy of the velocity of light, is valid only for space-time regions of constant gravitational potential" (Albert Einstein's paper "The Speed of Light and Statics of Gravitational Field", seen here: https://einsteinpapers.press.princeton.edu/vol4-trans/107) [Uses is $$c=1$$ units]

In fact, we know names for coordinates of this: Rindler Coordinates. In this, $$dx/dt=ax$$ where a is the proper constant acceleration relative to an inertial frame.(evaluated here Does the speed of light vary in non-inertial frames?).

Or what about one of the most famous space-times, the Schwarzschild space-time? Let us choose Schwarzschild coordinates... then, we have $$dr/dt = 1-2GM/r$$(evaluated here How does the speed of light vary throughout a black hole?)

However, this is an artifact of our 'coordinates' if you want. But really, what else does measuring velocity amount too? You line up a ruler, find your object, tick the time. The only viable this can be done is locally. Now in THIS sense, can we only find that the speed of light can't be broken and that the speed of light is always c. But then it amounts less to a 'relative velocity' of some distant observer. The problem is that we can't cheat with free vectors like we do in both Galilean relativity and Special Relativity, and there in lies the problem. Velocity is not uniquely defined in General Relativity globally. In a curved space, when you try to 'transport' your vector, it depends on what path you take.

So after all this, we actually come to the conclusion that any sensible measurement of something that acts faster then light can't exist, since the only one that makes sense is the local measurement of light. So actually, relative velocity always less then the speed of light. At the expense of no globally applicable definition. You can try to use a more globally applicable definition of cosmological velocities with that of a 'proper distance' but you then get galaxies that can move at greater then light speed. Personally, I think the trade off is worth it.