(Relativity) A flash of light is emitted at x1 and absorbed at x1 + l

[PhDeezNutz]

Homework Statement

A flash light is emitted at $x_1$ and absorbed at $x_1 + \ell$.

In a reference frame moving $v = \beta c$ along the x-axis:

a) What is the spatial separation $\ell'$ between the point of emission and absorption?

b) How much time elapses between the emission and absorption?

Relevant Equations

As I have interpreted the formulas


$\ell_{travel} = \frac{\ell_{stationary}}{\gamma}$

$\gamma = \frac{1}{\sqrt{1 -\left(\frac{v}{c}\right)^2}}$

$distance = v \Delta t$

The answers are according to the back of my book (Special Relativity by AP French) the answers are

$\ell \sqrt{\frac{\left(1 - \beta \right)}{\left(1 + \beta \right)}}$

and

$\frac{\left(1 - \beta \right) \gamma \ell}{c}$

My answers are quite different and here are my attempts

a) What is the spatial separation $\ell'$ between the point of emission and absorption? (In the frame going $v = \beta c$)

From what I know is that moving causes lengths to be contracted according to the formula

$\ell_{travel} = \frac{\ell_{stationary}}{\gamma}$ and in our case $\gamma = \frac{1}{\sqrt{1 - \beta^2}}$ so

$\ell_{travel} = \sqrt{1 - \beta^2} \ell$

b) How much time elapses between the emission and absorption?

$d = v \Delta t$

$\Delta t = \frac{d}{v} = \frac{\sqrt{1 - \beta^2} \ell}{\beta c} = \frac{\ell}{\gamma \beta c}$

Apparently I am wrong but I cannot see why.

Any help would be greatly appreciated.

Edit: sorry for the latex mistakes, it should be fixed now.

 

[George Jones]

Apparently I am wrong but I cannot see why.

Length contraction and time dilation do not apply in all situations.

One way to do this problem is to use the Lorentz transformation equations to calculate the primed coordinates of the emission and reception events.

 

[PeroK]

From what I know is that moving causes lengths to be contracted according to the formula

Apparently I am wrong but I cannot see why.

Any help would be greatly appreciated.

To add to what George has said:

A "length", by definition, is the spatial distance between two points measured at the same time. If you have a train that is $100m$ long, then that means that the rear of the train (at some time $t$) and the front of the train (at the same time $t$) are $100m$ apart. If you measure the position of the two ends at different times, you do not get the length of the train.

Note: length contraction only applies, therefore, to simultaneously measurements.

Time dilation only applies when you consider clocks or objects that are at a fixed position in one frame, when measured in another frame. It does not apply to the time intervals between events that are also separated in space.

Note that, in addition to length contraction and time dilation, you also have the relativity of simultaneity - which is possibly the most important aspect of SR and certainly the most often neglected.

The general rule for coordinate transformations is, as above, the Lorentz Transformation.

Has AP French not emphasised this?

 

[PhDeezNutz]

Length contraction and time dilation do not apply in all situations.

One way to do this problem is to use the Lorentz transformation equations to calculate the primed coordinates of the emission and reception events.

To add to what George has said:

A "length", by definition, is the spatial distance between two points measured at the same time. If you have a train that is $100m$ long, then that means that the rear of the train (at some time $t$) and the front of the train (at the same time $t$) are $100m$ apart. If you measure the position of the two ends at different times, you do not get the length of the train.

Note: length contraction only applies, therefore, to simultaneously measurements.

Time dilation only applies when you consider clocks or objects that are at a fixed position in one frame, when measured in another frame. It does not apply to the time intervals between events that are also separated in space.

Note that, in addition to length contraction and time dilation, you also have the relativity of simultaneity - which is possibly the most important aspect of SR and certainly the most often neglected.

The general rule for coordinate transformations is, as above, the Lorentz Transformation.

Has AP French not emphasised this?

Thank you very much for these posts. I was confusing myself and these posts have done a lot to address the misconceptions I have/had. I will keep all of these in mind as I'm doing the rest of this homework assignment.

AP French has emphasized this but I wasn't thinking straight.

 

[PhDeezNutz]

Thank you again, I just got my answers to match the book's answers. I guess the biggest trip up I had was distinguishing between "spatial separation" and "length measured"/"temporal separation" and "time measured". Badly worded question in my opinion, but now I know what to watch out for.