Lorentz Transformation Limit: Proving U=c

[SprucerMoose]

G'day,

I'm just doing some physics homework and decided to attempt to prove something. This is not a homework problem, I'm just unsure how to evaluate the limit.

Using the equation for transformation of velocity U=(U'+V)/(1+(VU'/c²)), I'm trying to show that if V=-c, as U' approaches c, U should approach c. This is the case when something travels at c in one direction and shines a light in the opposite direction, to an observer on the ground, where U will still be c.

 

[PAllen]

I'm confused what your issue is. Rearrange your limit expression, and for any u' < c, it is -c. Therefore the limit is -c, as you are looking for.

[Edit: ok I see, you are looking for it to come out c, not -c. There are several problems here. First, the velociy addition law (not the Lorentz Transform) is really only valid for speeds < c. It will work directly, or in the limit, for c, for many cases, but that is not strictly valid - it is derived by doing two Lorentz boosts, and there is no such thing as a Lorentz boost by c.

In the case of c, -c, the direct formula is undefined. That is telling you something: that the answer will depend on what limiting process you use. The way you have set it up, what you are computing is that no matter how fast u' becomes, the light (V) will still be seen as going -c. You can fake it out to get the result you want by letting U' be c, and taking the limit V goes to -c; now you get c as the limit. This is saying no matter how fast V goes 'left', light emitted to the right still goes c.

I think another confusion here is your use of U on the left. What is really being computed is more like U'+V, either:

a) A sees B going U'; B sees C going V; how does A see C?
b) A sees B going V; B sees C going U'; how does A see C? ]
]

 

[mathman]

(u'-c)/(1 - cu'/c²) = (u' - c)/(1 - u'/c) = -c.

No limit is needed.

 

[PAllen]

(u'-c)/(1 - cu'/c²) = (u' - c)/(1 - u'/c) = -c.

No limit is needed.

A limit is still needed - your simplification is valid only on assumption that u' < c, else you have zerodivide. Technically, you are still taking the limit of -c as u'->c.