Relativity question, photon and proton

[Prodigium]

Relativity question, photon and proton...

Homework Statement

Proton of gamma factor 10^12 and a photon set off from one side of our galaxy to the other take the distance to be 9.3x10^20m.
What is the time interval that the photon precedes the proton?

Homework Equations

\gamma = sqrt(c/2epsilon). (epsilon being c-v)

The Attempt at a Solution

Worked out epsilon to be 1.5x10^(-18)Not sure where to go from there anyhelp/hints would be most appreciated.

 

[vela]

You should try to form ratios that are dimensionless. In this case, you could write
v = c - \Delta v = c\left(1 - \frac{\delta v}{c}\right) = c(1-\varepsilon)where \varepsilon = \Delta v/c. Why? Because it avoids any complication due to units, for one thing. Regardless of which set of units you use, \varepsilon defined this way will always turn out to have the same value. Moreover, an expression is often more naturally expressed in terms of these unitless ratios.

You can show that
\gamma = \frac{1}{\sqrt{1-(v/c)^2}} = \frac{1}{\sqrt{1-(1-\varepsilon)^2}} \cong \frac{1}{\sqrt{2\varepsilon}}

Now write down expressions for the time each takes to travel the given distance.

 

[Prodigium]

\Delta t = \frac{(1- \epsilon )x-x}{c (1- \epsilon) }


after putting t_1 = \frac{x}{c} and t_2 = \frac{x}{c(1- \epsilon )}, then to find \Delta t = t_1 - t_2

but I can't seem to get it , did I do this correctly?

 

[vela]

\Delta t = \frac{(1- \epsilon )x-x}{c (1- \epsilon) }


after putting t_1 = \frac{x}{c} and t_2 = \frac{x}{c(1- \epsilon )}, then to find \Delta t = t_1 - t_2

but I can't seem to get it , did I do this correctly?

Because ε is so small, you can approximate t₂ well with a series expansion to first order.

 

[Prodigium]

Because ε is so small, you can approximate t₂ well with a series expansion to first order.

so t_2= \frac{x}{c} \cdot (1- \epsilon )^{-1} = \frac{x}{c} \cdot (1)

or, because that completely gets rid of \epsilon :-

t_2= \frac{x}{c} \cdot (1- \epsilon )^{-1} = \frac{x}{c} \cdot (1+ \epsilon )

That after finding \Delta t gave me and answer of -1.55 \cdot 10^{-12} s

only problem is you have to find the time as being in the reference frame of the proton later and need t_2.