Deriving Lorentz Factor

[SionR25]

Hi everyone,

As something to put down on my personal statement for university I have been reading why does E=mc² by Brian Cox and Jeff Forshaw. It is a great book but the problem is they do not go into the maths very much and merely state and equation and you have to take it as it is.

I usually don't have problem with it as I can see where the equation has come from and can derive it from that but I can't for the life of me derive the Lorentz factor.

I will try and describe the situation. It is using a moving train and a light clock. Train moves at speed=v and time=T. So, if you imagine a right angled triangle, the base is vT. The height is classed as 1 to keep it simpler. The hypotenuse is classed cT (c=speed of light). Using Pythagoras you get (cT)²=1²+(vT)², that bit is fine, nice and simple .

That bit is fine but then it just says that using complex maths you arrive at the equation T²=1/(c²-v²). I just can't see how it gets to this point so does anybody mind explaining it ?

Sion

[George Jones]

Using Pythagoras you get (cT)²=1²+(vT)², that bit is fine, nice and simple .

That bit is fine but then it just says that using complex maths you arrive at the equation T²=1/(c²-v²). I just can't see how it gets to this point so does anybody mind explaining it ?

Sion

Take the (vT)^2 term on the right over to the left side of the equation, and factor T^2 out of both terms now on the left.

[SionR25]

Thanks for that, thought it might be something simple but just had a mental block .

Sion

Edit: I actually feel a bit stupid now , such a simple thing

[Hugechris]

Sion, this is not the Lorentz factor you have mised a factor of c^2. The Pythagorous expression you need for the distances traveled is

(c*T'/2)^2 = (cT/2)^2+(vT'/2)^2

where T' is in the frame of the person outside the train and T is the elapsed time in the train to complete the up and down bounce of light.

solving you get T = T'*sqrt((1-(v/c)^2)) which is the time dilation formula (it looks as though the clocks on the train are running slower)

[SionR25]

Sion, this is not the Lorentz factor you have mised a factor of c^2. The Pythagorous expression you need for the distances traveled is

(c*T'/2)^2 = (cT/2)^2+(vT'/2)^2

where T' is in the frame of the person outside the train and T is the elapsed time in the train to complete the up and down bounce of light.

solving you get T = T'*sqrt((1-(v/c)^2)) which is the time dilation formula (it looks as though the clocks on the train are running slower)

Yes, I should have put that in. I could get how to do the rest of the steps of it, just not that first bit .

Sion

[George Jones]

Sion, this is not the Lorentz factor you have mised a factor of c^2.

Yikes! I completely missed this.

T = T'*sqrt((1-(v/c)^2)) which is the time dilation formula (it looks as though the clocks on the train are running slower)

Sorry for being overly picky, but ...

Actually, clocks on the train don't necessarily look (at least not directly) like they are running slower, and they never look like they are running slower to the tune of the time dilation formula (for motion directly towards or away from the observer). This is unfortunate, because "time dilation" in general relativity does mean something that is seen directly by observing a clock, i.e., "time dilation" has different operation definitions in special and general relativity .

Okay, here is an attempt at outlining some of the predictions that special relativity makes with respect to clocks. Sorry about the length of the post.

Assume that Alice is moving with constant speed directly towards Ted. When Ted uses his telescope to watch Alice's wristwatch, he sees her watch running at a faster rate than his watch. Ted sees Alice's moving watch running fast, not slow! Ted sees this because of the Doppler shift. Because Alice moves towrds Ted, the light that Ted sees from her watch is Doppler-shifted to a higher frequency. But the rate at which a clock or watch runs is like frequency, i.e., a second-hand revolves at a certain frequency, and all frequencies are Doppler-Shifted.

To explain what "A moving clock runs slow." means, I first have to explain how Ted (with helpers Bob and Carol) establishes his frame of reference.

Starting from Ted, a series of metre sticks, all at rest with respect to Ted, are laid end-to-end along the straight line oining Alice and Ted. At each joint between two consecutive metre sticks, a small clock is placed. The metre sticks and clocks are at rest with respect to Ted. Initially none of the clocks are running; before turning them on, the clocks have to be synchronized. To do this, Ted directs a laser pointer along the line joining Ted and Alice, and then sends a flash of light. Each clock is turned on when the flash of light reaches it. The speed of light is not infinite, so the time taken for the light to travel from Ted to each clock has to be taken into account. To do this, the clocks' hands are set initially as follows. The clock one metre away from Ted is set to the time taken for light to travel one metre; the clock two metres away from the tower is set to the time taken for light to travel two metres; ... .

This whole setup of metre sticks and clocks establishes Ted's reference frame.

Now, As Alice moves toward Ted, Ted uses his telescope to watch Alice's wristwatch, and to watch his clocks. First, he watches one of the distant clocks in his reference frame. The time he sees on the clock is the time at which the light he views set out, so Ted sees an earlier time on the distant clock than he sees on his wristwatch. He does, however, see the distant clock running at the same rate as his watch. Similarly, Ted's sees all the clocks in his frame running at the same rate as his watch.

As Alice approaches Ted, she whizzes by clock after clock of Ted's reference frame. Using his telescope, Ted sees that Alice is beside a particular clock, and he notes the time on her watch and the time on the clock adjacent to her. Some time later, Ted sees Alice beside a different clock, and he again notes the time on her watch and the time on the clock adjacent to her.

Ted checks his notes, and he finds that the time that elapsed on Alice's watch as she moved between these two clocks of his frame is less than the difference of the readings of the two clocks. This what is meant by "A moving clock runs slow."

[SionR25]

Reading further into E=mc²is this because everything moves through spacetime at the same speed? If you are stationary, to achieve the same speed as someone moving your clock needs to run faster. Vice Versa, if someone is moving their clock has to run slower because they are moving through space and the time needs to be slower. Simplified a bit (I know the real equation is a bit different but don't have book on me atm) movement in spacetime= movement through time + movement through space. So as one gets smaller the other gets larger and vice versa.

Sion

[Hugechris]

Reading further into E=mc²is this because everything moves through spacetime at the same speed? If you are stationary, to achieve the same speed as someone moving your clock needs to run faster. Vice Versa, if someone is moving their clock has to run slower because they are moving through space and the time needs to be slower. Simplified a bit (I know the real equation is a bit different but don't have book on me atm) movement in spacetime= movement through time + movement through space. So as one gets smaller the other gets larger and vice versa.

Sion

As George Jones indicated with special relative the relative positions as well as times of people in a frame of reference influence the exact 'time dilation' or 'length contraction' that is observed. There is something called the metric that relates distances (in both time and space) from one frame of reference to another. These metrics are universal in the sense that all observers (sticking to simple cases in relativity will come up with the same metric), these give the relations between time and space in one frame to time and space in another frame of reference. The wiki article is a university level intro to the subject http://en.wikipedia.org/wiki/Special_relativity

There are some great old books on the big ideas of the 20th centuary such as Mr Tomkins in paperback which takes a look at relativity quantum mechanics and other ideas. Einsteins own book is also acessible for first year undergrad / A-level student and gives a good intro to the idea of making measurements in different frames of reference.

Relativity: The Special and the General Theory Second Edition - A. Einstein