I can’t wrap my head around it. But maybe it might be a very silly question with a simple answer.
In: Physics
There is a formula from relativity for adding speeds, that formula is:
(v_1 + v_2) / ( 1 + (v_1*v_2)/c^2 )
where v_1 and v_2 are the speeds. If either, or both, of the speeds are much smaller than the speed of light, c, then the denominator is close to 1 and we just add speeds like we do with standard Newtonian physics. If the two speeds are both close to c, what is commonly called ‘relativistic’ speeds, then the denominator gets close to 2 and instead of adding the two speeds, we average them.
For example, if both rockets are going 0.9c or 90% of the speed of light, we get:
(0.9 + 0.9)/(1+0.9×0.9/1^2) = (0.9 + 0.9) ∕ (1 + ((0.9 × 0.9) ∕ (1^2 ))) = 0.99447514c or 99.45% of the speed of light. That is the speed each passenger sees the other rocket approaching at.
It really depends on how the physics of the particular light speed travel your ship is capable of works. Assuming all the particles within the ship also accelerate along with it, but the particles outside don’t then whatever you’re traveling away from would just disappear instantly into the vanishing point.
It won’t be going twice the speed of light or anywhere near it. It will still be less than the speed of light.
The way we think speed adds together isn’t really true.
In school we are taught that if we add 10 mph + 10 mph we get 20 mph.
However that is not true.
The truth is that the real way to calculate this is far more complicated than that.
At low velocities the truth is close enough that it makes no difference.
The close we get to the speed of light the more difference it makes.
90% of the speed of light + 90% of the speed of light is not 180% of the speed of light but something closer to 99.5% of the speed of light.
No matter how fast two objects move away from each other from the point of view of the object it will never be faster than the speed of light.
Additionally a flashlight shining out of a spaceship will emit light going at the speed of light in all directions and that will be true no matter which point of view you look at it from.
Velocity does not work at all the way we normally think of it.
Your assumptions are wrong. The occupants of each rocket will not see the other moving “almost twice the speed of light.” Each will see the other moving close to but not up to *c* since nothing in either’s reference frame can exceed it. Someone else has already given the equation for the exact relative observed speed
This is neglecting the effect on light itself of each’s perceptions. Those will be severely messed up in ways that depend on the rockets’ orientation to each other
You have learnt that you can arrive at a combined speed between two parties by adding their speeds together. That is a *simplified* formula that is approximately correct at low speeds, certainly any speeds you will encounter in your lifetime.
There is a more complex formula for combining speeds that are close to the speed of light that is more correct and you’ll never get a nonsense result such as “twice the speed of light” when using that formula. There’s no reason to believe that your everyday experience is a good yardstick by which to measure high-speed interactions.
For both rockets time will have slowed down to the point where from their viewpoint the other rocket isn’t going above the speed of light.
Addition of velocities doesn’t work like that near the speed of light
In a classical reference frame, u=v+u’
In a relativistic reference frame, u=(v+u’)/(1+vu’/c^(2))
They just see the other rocket go by at very close to the speed of light, much closer than they actually are from a static reference frame.
Let’s say v=.9c and u’=.9c
u=1.8c/(1+.81c^(2)/c^(2)) = .99c (approximately)
And this is what you would see if you were on one rocket observing the other
Space, time and speed don’t work the way we think they do, or are used them working.
Our intuitive ideas about these things only work for small relative speeds (so tens of thousands of miles per hour). When things get to relative speeds that are a decent factor of the speed of light things work differently. Or rather, the approximations we use normally no longer work.
To use an analogy, we know the Earth is a sphere. But if you only look locally (dozens of miles) it looks flat, and we can treat it as flat. The surface of the Earth isn’t flat, but we can pretend it is. If we look on larger scales (or want really, really precise measurements) we have to add corrections for the Earth being curved.
The key rule in Special Relativity is that the “speed of light” (noting that it is the speed that is special, not the light part – light (sometimes) travels at this speed because the speed is important) is the same for all inertial observers. If something is travelling at *c* relative to you, it is travelling at *c* relative to everyone else, no matter how fast they are moving compared with you. *c* is the ‘hinge’ around which space, time and speed rotate and twist.
As things get faster compared with you, their time and space get twisted together; their time passes slower (from your point of view) and their distances get squished (from your point of view). The result of this is that from your point of view they’re not moving as fast as they “should” be.
So if we are in a rocket that is still, and another rocket that is travelling at 0.9c towards us, its time will be slowed down (and it will be squished) by a factor of about 2 when viewed from our point of view. If we have another rocket travelling in the other direction towards us at the same speed, it will also be squished by that factor (from our point of view).
But the second rocket will see things differently; they will see our rocket coming towards them and 0.9c, and it will be *our* time that is running slow, and us that are squished. And then they’ll see a second rocket travelling towards them at about 0.99c (who will be slowed down by a factor of about 7, and squished by that factor).
Times, distances and speeds all look different from the different perspectives. And each perspective is equally valid.
You know that shit where people say “time is the 4th dimension”? This situation is why people say that.
Time does funky shit. From the perspective of one spacecraft, they observe the “head” of the other spacecraft to be “in the past” where it has not traveled that far, and the “tail” of the other spacecraft to be “in the future” where it has traveled even further.
The effect is that the other spacecraft “appears to” compress along its length. Because it’s shorter, it doesn’t travel as far per unit time as it would add its “true” length so it goes slower than the speed of light.
>Relative to each other one will seem stationary while the other will look like it’s going almost twice the speed of light.
You’re right that they’ll always appear stationary within their own reference frames.
Your guess at the speed they perceive the other ship to be moving would apply in basic (Galilean) relativity, but does not bear out in reality. We haven’t seen anything travelling faster than light. Special relativity can address this. A full discussion of special relativity goes well beyond the scope of the question and gets very mind-bendy.
Regarding what passengers see:
– they each perceive the other ship travelling clos_er_ to the speed of light, but still below the speed of light
– if they see a clock on the other ship, they’ll perceive it’s time to be passing more slowly than their own (yes both ships will perceive each other’s time to be slower than their own)
– if in identical ships they perceive the other ship to be shorter than their own (both will perceive each other’s ship to be shorter than their own)
– they’ll perceive the other ship to be blue-shifted as it approaches, and red-shifted after it passes
You might look at the claims above and assume there’s some stationary observational frame of reference from which we can measure the _true_ time, distance, velocity, etc. After all, the external observer, and observers on each of the ships are all measuring different times, lengths, and velocities than each other, and may even disagree on orders of events. Heres the real kicker… each of their frames of reference are equally valid.
It isn’t surprising that you can’t wrap your head around it. Most people are accustomed to Galilean relativity, and we don’t tend to have much experience observing relativistic objects.
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