# B Very basic Questions on Relativity and Speed of Light

#### nick J

Can anyone help me with a very basic question please.
Is it really not possible for anything to travel faster than the speed of light, as dictated by relativity theory ?
I am a complete beginner to Physics and only have very basic mathematics knowledge and I'm trying to understand the basics of relativity.
I have tried to find the answer on this forum but its mostly way above my head.
I have also been reading up as much as possible and watching various youtube videos and lectures.
In one lecture it was explained that 'relativity' actually refers to relativity of motion. The way I see it, if all motion is 'relative' then that means EVERYTHING is moving faster than the speed of light. Because something somewhere in the universe must be moving in one direction at least 60% of the speed of light and there must something else which is moving in the other direction at least 60% of the speed of light as well. Because everything's motion is relative to everything else, therefore everything has to be moving faster than the speed of light (relatively). I do understand that the speed of light is constant to all observers, but as I see it, that doesn't mean that the speed of light is a limit to how fast things can actually go.
A simpler way of looking at it, is two different light beams going in opposite directions - relative to each other they are both travelling faster than the speed of light (to an external observer).
So can someone please tell me if I am understanding this correctly ?

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#### DrGreg

Gold Member
Because something somewhere in the universe must be moving in one direction at least 60% of the speed of light and there must something else which is moving in the other direction at least 60% of the speed of light as well.
In Newtonian physics, you would add 60% to 60% to get 120%. But in relativity, the correct calculation is
$$\frac{0.6 + 0.6}{1 + 0.6 \times 0.6} = 0.88$$
88% of the speed of light.

To see why that's the right formula you'd need to read a book on relativity. (Look up relativistic velocity composition or velocity addition.)

#### phinds

Gold Member
I do understand that the speed of light is constant to all observers, but as I see it, that doesn't mean that the speed of light is a limit to how fast things can actually go.
Of course it does. Think about it. No matter how fast you are going (relative to ANYTHING), if you turn on a flashlight, the beam moves away from you at the speed of light so clearly you are not going at the speed of light. In fact, no matter how fast you are moving relative to me, if I shine a flashlight past you as you go past, you will see the beam moving at the speed of light. I will see you moving at less than c and the beam of light moving at c

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#### nickjones

The relative velocity calculation formula is interesting and seems like a good starting point for me to try and learn more.

There is still something bothering me though - and please forgive my naivety - despite these explanations, I still can't see why everything isn't already moving faster than the speed of light. I do accept that to any observer in a particular frame of reference, the speed of light is constant and its not possible for them to observe anything moving faster, but isn't that just because it's 'relative' to that observer ?

If you take for instance a rocket-ship which has another smaller rocket-ship attached to it's back (like the space shuttle with it's booster rocket, except where the booster is another vessel), then both together lift off from a certain place (point A) and reach a velocity 60% of the speed of light. At some place somewhere in inter stellar space (point B) the ships separate, so both are now side by side but have a velocity of 0 (relative to each other). Then the second rocket-ship with its magic never ending fuel supply boosts off into the distance, again at 60% of the speed of light leaving the first one behind. So now the second Rocket is travelling 120% the speed of light if you consider it's original starting point. If velocity is always relative, then why can the pair reach a speed of 60% the speed of light from the original starting point, but then Rocket-ship 2 (after separation) is not allowed to ? Because there should be no difference in the laws of nature between point A and Point B (if all velocities are relative to each other then Point B can be considered as a 'stationary' point in just the same way as Point A).

I accept that to an observer at point A, then the mathematical formula from DrGreg's post must hold true. But it doesn't change the fact that rocket-ship 2 has twice accelerated to 60% the speed of light on two occasions going in the same direction, and on each occasion the 'starting' points can be considered as stationary.

I am sure I am missing some very fundamental points here but it would be great if there is a simple answer which I could understand to explain the basic concepts (am guessing there probably isn't and the only option I have is to hit the text books........)

#### Pencilvester

I accept that to an observer at point A, then the mathematical formula from DrGreg's post must hold true. But it doesn't change the fact that rocket-ship 2 has twice accelerated to 60% the speed of light on two occasions going in the same direction, and on each occasion the 'starting' points can be considered as stationary.

I am sure I am missing some very fundamental points here but it would be great if there is a simple answer which I could understand to explain the basic concepts (am guessing there probably isn't and the only option I have is to hit the text books........)

Consider 100 nested ships, and each one detaches and begins accelerating once its parent ship reaches 1% the speed of light. It will take MORE than 60 detachments to get any ships up to 60% the speed of light relative to the home planet that the ship(s) were launched from due to the nature of velocity addition in special relativity. If you extrapolate to infinite nested ships you find that the speed they can reach relative to home planet has a limit: the speed of light. This is just a way of saying you can apply a force indefinitely on a thing (say, using magic rocket fuel), but it will never reach the speed of light no matter how long you apply that force for.

#### Arkalius

I accept that to an observer at point A, then the mathematical formula from DrGreg's post must hold true. But it doesn't change the fact that rocket-ship 2 has twice accelerated to 60% the speed of light on two occasions going in the same direction, and on each occasion the 'starting' points can be considered as stationary.

I am sure I am missing some very fundamental points here but it would be great if there is a simple answer which I could understand to explain the basic concepts (am guessing there probably isn't and the only option I have is to hit the text books........)
Relative to the original frame, the first part of the ship is flying at 0.6c, and the second part is flying away at 0.88c. Using the formula $\frac {0.6 + 0.6} {1 + 0.6*0.6} \approx 0.88$. So, the first ship sees Earth flying away in one direction at 0.6c and the second ship flying in the other at 0.6c, but Earth sees the second ship flying away at around 0.88c. Note that it is not a violation of relativity to observe two things in motion to be separating from each other at a speed greater than c (never equal to or greater than 2c, though). The speed of light limit only applies relative to something stationary.

#### phinds

Gold Member
...The speed of light limit only applies relative to something stationary.
"Stationary" is a tricky term. To be fully correct, your statement needs more qualification. I mean, it is clear what you mean to someone who KNOWS what you mean, but can be confusing to newcomers to SR.

#### Mister T

Gold Member
Then the second rocket-ship with its magic never ending fuel supply boosts off into the distance, again at 60% of the speed of light leaving the first one behind. So now the second Rocket is travelling 120% the speed of light if you consider it's original starting point.
No, that way of combining speeds only works as an approximation when the percentages are small compared to 100%. The speeds we're likely to encounter are much much smaller than 100%, so it's a very very good approximation. But other people are used to speeds that are much much faster and to them the correct way of combining speeds is a part of their every day lives.

To put it another way, if that second rocket ship were to race a beam of light, it would lose the race.

Take a couple of wedges and stack them to make a steeper wedge. If you look at the math used to find the slope of this steeper wedge you will find that you cannot just add the slopes of the two individual wedges to get the slope of the stacked wedge. You instead have to add the angles. But you will find that if the slopes are very very small, adding the slopes gives you a good approximation to the slope of the stack. Figuring all this out is a good exercise in trigonometry. Note that the slope equals the tangent of the angle.

Likewise, there is a parameter called the rapidity that you can add to get the rapidity of that second rocket. You use hyperbolic trigonometry. $$\tanh^{-1}{(0.6)} \approx 0.693$$ Which means that the rapidity of something moving at a speed of $0.6\ c$ is about 0.693. If we double that we get a rapidity of $1.386$. And the corresponding speed is $$\tanh{(1.386)} \approx 0.88$$ Or about 88% of the speed of light.

#### Vitro

If you take for instance a rocket-ship which has another smaller rocket-ship attached to it's back (like the space shuttle with it's booster rocket, except where the booster is another vessel), then both together lift off from a certain place (point A) and reach a velocity 60% of the speed of light. At some place somewhere in inter stellar space (point B) the ships separate, so both are now side by side but have a velocity of 0 (relative to each other). Then the second rocket-ship with its magic never ending fuel supply boosts off into the distance, again at 60% of the speed of light leaving the first one behind. So now the second Rocket is travelling 120% the speed of light if you consider it's original starting point..
With the observation that @DrGreg has already answered this in #2, take a full stop here and analyze the conclusion in your last sentence. Why do you think your setup implies your conclusion, what implicit assumptions are you making without even realizing it, and how can you test experimentally (not theoretically) that your conclusion is indeed correct?

Start with step 1, the rockets start from point A and accelerate to 60% of the speed of light. How do you determine that speed? Do you measure it directly somehow or are you integrating the acceleration over time? The best way is the first, direct measurement. If you use the second way then you're implicitly making assumptions about that integration (about how acceleration, time and speed are related) and those assumptions might be incorrect. So let's say you measure it directly somehow and it's indeed 60%.

Then the second rocket accelerates away from the first and confirms by direct measurement that it's going at 60% of the speed of light compared to the first rocket.

Now you want to know how fast the second rocket is going compared to the initial starting point. You could do it mathematically using the two speeds you already know, like you are trying to do, but you have to make an assumption about how to combine the two speeds. How do you choose a formula, like 60 + 60 = 120? What's it based on? How do you know it's correct? Well, you can use the speed measurement procedure used above and directly measure the speed of rocket 2 relative to the start point A and discover that the speed is only 88% and not 120% as you expected. This should tell you that your assumption was wrong and speeds don't add up the way you thought, the correct way is per post #2.

To understand why that's the case, paraphrasing @DrGreg, you need to study relativity. Otherwise the only short answer is because that's how the world works, the observed relationship between space and time does not match Galilean/Newtonian mechanics but instead Einstein's relativity.

• Pencilvester