Addition of Velocities in Special Relativity

In summary, The conversation is about determining whether a given set G is a group under a specific operation in Special Relativity. The group is defined as (-c,c) and the operation is addition of velocities. The speaker initially thought that the set was not a group due to potential issues with associativity or closure. However, after trying some values and proving associativity, they realized that the set could be a group. The speaker is currently having trouble with proving closure, but noticed a potential similarity to a known proof. The conversation concludes with the speaker realizing that factoring may be the key to solving the problem.
  • #1
Bleys
74
0
The problem is in an Algebra book but the jist of it is about calculus/analysis. If G is the interval (-c,c) and the operation is addition of velocities in Special Relativity, (that is:
[tex]x \circ y = \frac{x+y}{1+\frac{xy}{c^2}}[/tex]) then decide whether G is a group. At first I thought it wasn't; figured either associativity or closure would fail. But when I tried some strange values it actually worked so I went about proving it. Associativity was fine. The identity is zero, inverse of x is -x. Closure is what I'm having trouble with. Everytime I try to use the fact |x|<c and |y|<c I lose information and I get upper bounds that are way too high to be of any use. I noticed things looked 'somewhat' (which is an overstatement) similar to what i was trying to prove if I used [tex] (x+y)^2 = x^2 + 2xy + y^2 [/tex], played around with it, but again once I started using inequalities i get too far. Is there some piece of information other than -c<x,y<c that I'm missing which would be key to solving this?
 
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  • #2
Note, that [itex]x \circ y < c \Leftrightarrow (c-x)(c-y) > 0[/itex], which is always true for [itex] -c < x,y < c[/itex].
 
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  • #3
ah thanks. Every time I had c(x+y) < c^2 + xy it never occurred to me to factorise it!
 

1. What is the concept of addition of velocities in special relativity?

In special relativity, velocities do not simply add up as they do in classical physics. Instead, they follow a more complex formula that takes into account the speed of light and the relative motion of two observers.

2. How is the addition of velocities different in special relativity compared to classical physics?

In classical physics, the addition of velocities is a simple sum. However, in special relativity, velocities add up using the Lorentz transformation equations, which take into account the effects of time dilation and length contraction.

3. Can velocities in special relativity exceed the speed of light?

No, according to the theory of special relativity, the speed of light is the maximum speed that can be achieved in the universe. Therefore, the addition of velocities cannot result in a final velocity greater than the speed of light.

4. How does the addition of velocities affect the perception of time and space for observers in relative motion?

The addition of velocities in special relativity leads to the concepts of time dilation and length contraction. Time appears to pass slower for an observer in motion compared to a stationary observer, and objects appear shorter in the direction of motion.

5. Are there any real-life applications of the concept of addition of velocities in special relativity?

Yes, the concept of addition of velocities in special relativity is crucial in fields such as space travel, particle accelerators, and GPS technology. It is also used in understanding the behavior of particles at high speeds, such as in particle collisions.

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