Lorentz Transformation: Solving Homework Statement

[yellowputty]

Homework Statement

Two events occur at the same place in an inertial reference fram S, but are separated in time by 3 seconds. In a different frame S', they are separated in time by 4 seconds.

(a) What is the distance between the two events as measured in S'?
(b) What is the speed of S relative to S'?

Homework Equations

I'm presuming:

t' = gamma*(t-ux/c^2)

The Attempt at a Solution

I have the answer, and a hint saying to use the interval S^2, but I have no idea what that means, and where I start. When I look and the relevant Lorentz equations, they involve velocity, and I do not have a velocity here.

Could you please point me in the right direction?

Thank you in advance.

 

[G01]

The hint is implying that you use the invariant space-time interval to solve this problem. The following quantity is called the space time interval:

$$(\Delta s)^2= (\Delta x)^2 + (\Delta y)^2 + (\Delta z^2) - (c\Delta t)^2$$

This quantity is a Lorentz scalar and is thus invariant over Lorentz transformations (it is the same in all inertial frames). So, in one dimension this means:

$$(\Delta x)^2- (c\Delta t)^2=(\Delta x')^2- (c\Delta t')^2$$

Can you use this the solve the problem?

 

[yellowputty]

The hint is implying that you use the invariant space-time interval to solve this problem. The following quantity is called the space time interval:

$$(\Delta s)^2= (\Delta x)^2 + (\Delta y)^2 + (\Delta z^2) - (c\Delta t)^2$$

This quantity is a Lorentz scalar and is thus invariant over Lorentz transformations (it is the same in all inertial frames). So, in one dimension this means:

$$(\Delta x)^2- (c\Delta t)^2=(\Delta x')^2- (c\Delta t')^2$$

Can you use this the solve the problem?

Do I find $$(\Delta t)$$ by doing SQRT[(4^2)-(3^2)] = ROOT 7

Then at they are both at the same coordinates in the inertial reference frame, we can ignore x , y and z. Therefor S equals the root of (c^2)*(ROOT 7) = 7.9x10^8m

Is this correct?

 

[G01]

Do I find $$(\Delta t)$$ by doing SQRT[(4^2)-(3^2)] = ROOT 7

Then at they are both at the same coordinates in the inertial reference frame, we can ignore x , y and z. Therefor S equals the root of (c^2)*(ROOT 7) = 7.9x10^8m

Is this correct?

You should only have one factor of c in your final line, since you take the square root of c^2 when solving for the answer.