Objects travelling relatively to each other faster than light?

Question

When we say that something is travelling a certain speed, it's really travelling that speed relative to the Earth. When saying the speed of anything, it is, for the most part, relative to something else. That being said. If I have an object moving at half the speed of light, and another moving at just above half the speed of light in the opposite direction, would the second object be moving faster than the speed of light to the first one?

Note: I know this question is similar to some other questions like this one. However, with my limited physics knowledge (taking AP Physics class next year) I found the explanation a bit confusing. So, even though this might be a bit similar to other questions, I'm looking for a simpler explanation that could help me understand this and its foundation.

Answer 1

In Special Relativity, we use Lorentz Transformations to add speed. The relevant formula here is $$u = \frac{u^{'}+v}{1+\frac{u^{'}v}{c^2}} $$ where $u^{'}$ and $v$ are the speeds of the objects and $u$ (what you are looking for) is the speed that the object at $u$ sees the object at $v$.

This embodies Einstein's postulate that no information can be transferred faster than the speed of light in vacuum. Now using this formula, we can put $u^{'} = 0.5c$ and v = $0.6c$ and still get that $u = \frac{1.1c}{1.3} = 0.85c$. Note that even if $u^{'}=v=c$ we get $u=c$, which tells us that the speed of light in vacuum is the same for all observers (which is really the more-precise text of Einstein's conjecture)

Note what you have learned that you can just add the two speeds up is only a good approximation when $v<<c$.

P.S. See e.g. here for a simple derivation of the formula, which we get from using the lorentz transformations of time and position. It is only after we realise that time and space can be a combination of each other that we arrive at this.

Answer 2

A key point to bear in mind when trying to understand Special Relativity is that you have to put aside your common sense ideas about distance, time and speed. It turns out that while those ideas apply to a very high degree of approximation at the kinds of low speeds that we experience as humans on Earth, they are totally wrong and misleading in the general case.

We now know that it is impossible for two massive objects (by which I mean objects with mass, as opposed to massless objects such as photons), to move relative to each other at a speed greater than or equal to c. The reason is baked into the geometry of spacetime.

As a very very distant analogy, imagine you are moving on the surface of a sphere- no matter how fast you move, you can never get any nearer to the horizon or further from the centre. The has nothing to do with energy, power, mass etc- it is a straightforward consequence of the geometry of the sphere.

Spacetime turns out to have a particular type of geometry called hyperbolic geometry, which is quite unlike the Euclidean geometry that is drummed into us at school. If you apply yourself, you can figure out some of the properties of spacetime by focussing on a key experimental fact, which is that the speed of light is the same relative to everyone, no matter how quickly or slowly people are moving relative to each other. From that single fact, you can work out some of the fundamental ideas of special relativity.

If you consider the example you mention in your question, of two objects moving towards each other, each going at just above half the speed of light relative to the Earth, then from the perspective of someone on Earth the two objects are closing on each other at a speed greater than c. However, from the perspective of either of the objects, the other is approaching at a speed less than c.

It turns out that the reason why that happens is because of something called the relativity of simultaneity, which is a property of the geometry of spacetime. Loosely speaking, what that means is that 'now' means different thing to people who are moving relative to each other. Someone on Earth asking where are the two objects 'now' will be referring to the positions of the objects at a particular time, whereas if someone moving with one of the objects asks where is the other object 'now', they will be considering the position of the other object at some different time.

More generally, people moving relative to each other will disagree about what time it is at any given point in space, and the size of the disagreement increases with distance. Again that's a property of the geometry of spacetime. 'Now' for each person means a flat slice through spacetime at right angles to their time axis. However, when people are moving relative to each other, their respective time axes are tilted relative to each other, so what is a flat slice through spacetime for one person is a sloping slice to the other person, and vice versa.

If you start with a good understanding of the relativity of simultaneity, you will find that all of the key effects of special relativity- such as time dilation and length contraction- become much easier to understand correctly.

Answer 3

so actually you could simplify the equation a lot.

A better way to frame the question to someone might be as follows

"If two objects are moving in opposite directions relative to a third object at just under the speed of light (c), how fast are the two objects moving relative to each other?"

The question doesn't involve trigonometry. It can be solved using simple multiplication and addition. The formula is as follows:

A(c)+B(c)= X, where A is the relative speed of the first object compared with the third, B is the relative speed of the second object compared to the third, c equals a speed just under the speed of light, and X is the total relative speed.

If this equation isn't correct, please explain!