# Additive speed of light for three bodies

I know this question, or similar ones have likely been asked before, but I have tried reading several, and they just don't properly explain what I'm trying to understand.

The quick version of the question is, if 3 bodies, each relative to the previous are moving at a speed where their combined velocity would be greater than the speed of light, what would be the final velocity of the 3rd body during the fastest time.

For this I'll be using 3 bodies, A Galaxy, Solar System & Planet.

To make the numbers simple, I'm going to round c to 3 million m/s

If the galaxy was moving at 1.5m m/s (0.5 c)
A solar system in that galaxy was moving around the galaxy at 1.2m m/s (0.4 c)
And a planet was moving around that sun at 600k m/s (0.2 c)

Relative to a static observer, during the ideal time, when the planet is moving away from the observer and the solar system is moving away from the observer, the combined velocities of the 3 would be (1.5 + 1.2 + 0.6 = 3.3m (1.2 c ) which is impossible according to general relativity.

Now, I know that some of the math here will move into special relativity, which I'm not quite as familiar with, but as they are all traveling at a relatively slow velocity (relative to their frame of reference), how would the planet be affected as it moves into the part of its rotation where to the static observer it would be moving faster than the speed of light?

I guess what really confuses me is, I know that c is a constant, and that relative speeds are not the same. I also have a limited knowledge of special relativity which may be what is hampering my understanding in this case.

What if someone tried to launch a space ship off the planet while its going 3.3m m/s (from the static reference frame)

• – Dhruv Saxena Oct 17 '18 at 23:03
• I understand it when it comes to 2 bodies, (or particles) relative to each other, but in my example there are multiple relative frames of reference, which during a certain phase, would align their forward velocities relative to a static point of reference. All the examples talk about two relative particles, I'm talking about objects relative to an observer – Kindar Conrath Oct 17 '18 at 23:13
• And if the speed of light is constant and a maximum, what would happen to the planet as its total velocity (galaxy + solar system + planet) approached c – Kindar Conrath Oct 17 '18 at 23:18
• "Relative to a static observer," - what does that mean? There is no absolute rest so what is a static observer? – Alfred Centauri Oct 17 '18 at 23:19
• The more I think about this, the more I think I am missing something. 1)Something can not travel faster than the speed of light. 2)All objects are moving 3) some are moving relative to others, their velocity could be described as a combined v1 + v2 vector. If several objects combined vectors were larger than c would they be traveling faster than c . Is c relative to a frame of reference? or is it absolute? – Kindar Conrath Oct 17 '18 at 23:28

## 1 Answer

One can use the rapidity to sum like this:

$$v_\text{total} = c \tanh(\zeta_1 + \zeta_2 + \zeta_3)$$

where

$$\zeta_i = \tanh^{-1}\left(\frac{v_i}{c}\right)$$

• Ahh, so the velocity sum doesn't work quite the same – Kindar Conrath Oct 17 '18 at 23:38
• @KindarConrath, we do expect that you do some basic research on the subject of the questions that you ask. Have you not researched the relativistic addition of velocities? – Alfred Centauri Oct 18 '18 at 1:07
• I have done some basic research into it, but the huge amount of information stored in those pages makes it difficult to understand some times. I usually find it easier to understand when I talk to someone rather than to read huge amounts of formulas wrapped in generalized descriptions. When I was in university I tried to take physics courses, but because I wasn't in a physics related course, like engineering, I didn't qualify for them. – Kindar Conrath Oct 18 '18 at 1:44
• @KindarConrath I agree that the maths can be a bit intimidating, but the concept is actually quite simple. In special relativity, the speed of a body is the slope of its worldline (distance/time), measured in natural units where $c$, the speed of light, is 1. – PM 2Ring Oct 18 '18 at 9:29
• @KindarConrath In normal Euclidean geometry, if you want to calculate the combined slope of 2 wedges you can't just add their slopes together, you need to add their angles (the arctan of the slopes), although adding the slopes is an ok approximation when the wedges are thin. The same thing applies in SR, except we use the hyperbolic tangent to get the worldline angle, because we're working in Minkowski spacetime, not Euclidean space. – PM 2Ring Oct 18 '18 at 9:30