Addition of Velocities in Special Relativity

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SUMMARY

The discussion focuses on the addition of velocities in Special Relativity, specifically the operation defined as x ⊕ y = (x+y)/(1+xy/c²). The interval G is defined as (-c, c), and the participants explore whether G forms a group under this operation. Key properties such as associativity and the existence of an identity element (zero) are confirmed, while closure remains problematic. The conclusion emphasizes the necessity of understanding the relationship between the bounds of x and y to resolve closure issues.

PREREQUISITES
  • Understanding of Special Relativity and its principles
  • Familiarity with group theory concepts, particularly closure and associativity
  • Knowledge of algebraic manipulation and inequalities
  • Basic calculus, specifically operations involving limits and bounds
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  • Research the properties of groups in abstract algebra
  • Study the implications of velocity addition in Special Relativity
  • Learn about the significance of the Lorentz transformation in physics
  • Explore advanced algebraic techniques for handling inequalities
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Students of physics, mathematicians interested in group theory, and anyone studying the implications of Special Relativity on velocity addition.

Bleys
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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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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].
 
Last edited:
ah thanks. Every time I had c(x+y) < c^2 + xy it never occurred to me to factorise it!
 

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