Does the Lorentz equation preserve causality?

[silenzer]

Homework Statement

Observer S' moves at a speed u compared to observer S in x-direction. Two events happen on the x-axis of S, with the coordinates x1 and x2 at times t1 and t2 measured in S. Let L = x2-x1 and T = t2-t1.

a) If the events happen at the same time in S, does the same apply to S'? b) What relationship must there be between L and T for the order of the events become switched? c) Let there be a causal relationship between the events. Can the order seem different in S'?

Homework Equations

The Lorentz equation t = (gamma) * ( t' + x' * u / c^2 )

The Attempt at a Solution

a) The answer to this is no, because they happen at varying units of x. If they had happened at the same location, the answer would be yes.

b) Not really sure, other than to have the product on the right in the parentheses be larger than t', resulting in a negative time for t. Only thing is, I don't know what that means in terms of the Lorentz equation...

c) Obviously no.

 

[voko]

Let's assume that T > 0. Then, in (b), T' < 0. Find what T' is, and see what it takes for it to be negative, assuming T is positive.

 

[Chestermiller]

You need to use both Lorentz equations, not just one of them. And you need to use the version where x' and t' are expressed in terms of x and t (rather than the other way around).

Chet

 

[vela]

c) Obviously no.

I think you're meant to show that the Lorentz transformation preserves causality.

 

[HalfLostAndSearching]

I would like to clarify that the Lorentz equation does indeed preserve causality. Causality refers to the principle that an effect cannot occur before its cause. In the context of the Lorentz equation, this means that the order of events cannot be reversed in different reference frames.

a) In the Lorentz equation, t and t' represent the times measured in the two different reference frames. If the events happen at the same time in S, then t1 = t2 and the same applies to S'. This is because the Lorentz transformation only affects the spatial coordinates, not the time coordinate.

b) The relationship between L and T for the order of events to be switched is that L must be greater than T. This means that the events are happening at different locations in S, but at the same time in S'. In this case, the Lorentz transformation will result in a negative time for t, indicating that the event at x2 happened before the event at x1 in the frame S.

c) If there is a causal relationship between the events in S, the same order of events will be observed in S'. This is because the Lorentz transformation preserves the temporal ordering of events. If event A causes event B in S, then event A will also cause event B in S'. The Lorentz equation does not change the causal relationship between events, it only changes the way they are measured in different reference frames.

Overall, the Lorentz equation is a fundamental equation in special relativity that preserves causality and allows us to understand the effects of time dilation and length contraction in different reference frames. It is an important tool for understanding the behavior of objects moving at high speeds and has been confirmed by numerous experiments and observations.