# Time Dilation and Length Contraction

Gecko
how can you explain that a rocket ship traveling at 99.999% of c will see a beam of light pass by it at c with respect to the space ship? what exactly is the definition of time dilation and length contraction and what formula's are used to explain the phenomenon(sp?) thanks ## Answers and Replies

zefram_c
One does not "explain" this. The constancy of the speed of light is, after all, a postulate of SR. If you want to make a different postulate, such as "the Lorentz transformations are the correct formulas for converting quantities between frames of reference" or (my favorite) "Spacetime is described by the Minkowski metric", then the constancy of the speed of light (i.e. a universal speed that has the property of being invariant in all reference frames) follows mathematically. But in short, I cannot give you a satisfactory answer, other than variations on "this is the way things are". That particular experiment has been carried out using pions moving at high speeds wrt the lab and the results agree with SR.

Gecko
but there has to be some thing that explains why the people on the ship and people looking at the beam of light and the ship see different rates in the passage of the beam of light(as in the rate that the beam of light passes, to the people on the ship its c, to the people looking from afar at the light and ship, they see the light passing the ship at a rate of .001% of c). and by explain i mean tell mathmatically, why the speed of light is always the same from any reference point, which in this case is the ship traveling at 99.999% of c.

RandallB
Gecko said:
i mean tell mathmatically, why the speed of light is always the same from any reference point

Mathematically you can not "Tell" anything - You can show or extend from known information to new conclusions. Conclusions that can be tested to prove the original "known information" was correct.

As an example:
Let us assume we "know" that sound travels only at 1 foot per millisecond regardless of the speed of the observer!
Well then we can make a 'sound clock' that bounces sound between two sound mirrors 3ft vertical apart - moving or not it will always take 6 msec for one round trip. But have an observer watch it move horizontally fast enough that goes 8ft before the round trip is completed. That means the sound "BEAM" traveled 10 ft in 10 msec for the external observer but still only 6ft for the moving clock for 6 msec. 4 msec slower!

Just simple math here where 3*3 + 4*4 = 5*5 for a 3 by 4 by 5 right triangle OK.

But boy can we start cranking out some great formulas now - we are off and running & going to be famous!!
One problem - all the math is right, even looks good, but I can not get any of the resulting formulas to explain why I'm hearing Sonic Booms out there. Say it isn't so - pilots catching up with and going though there own sound! BUMMER for what I thought I knew about sound being the same for all observers (maybe the speed is wrong too). Should I loose confidence in the math or just my idea?

BUT if you want to make a bold leap - and use the same mirrors, and one nano second per foot for the speed of a bouncing light beam - - now your cooking up some really nutty conclusions. Extend them to even more truly nutty ideas and you'll be telling folks light will bend around the gravity of the sun!! And clocks need to be adjusted in orbit!!
The math doesn’t prove it but helps you create the formulas that need to be tested and how to test them. We still need to convince people to perform the tests that confirm them.
Draw confidence from never looking up into the sky and seeing any "sonic" light booms.
I think we have a winner here.

But for being able to build the foundation of the formulas yourself from your own 'Thought Experiments' the light clock is a good one and the geometry and math is easy.

Randall B

Gecko
ok, then can you tell me formula's that have been tested that support the constancy of c, even when its passing a rocket thats traveling at .99999c so that i can test this specific problem. was there any point at all to your post above except "The math doesn’t prove it but helps you create the formulas that need to be tested and how to test them"? jesus.

Gold Member
Gecko said:
ok, then can you tell me formula's that have been tested that support the constancy of c, even when its passing a rocket thats traveling at .99999c so that i can test this specific problem. was there any point at all to your post above except "The math doesn’t prove it but helps you create the formulas that need to be tested and how to test them"? jesus.

What your missing is that the second postulate of relativty is "the speed of light is constant in all inertial refrence frames", so the formulas are derived from this fact not vice versa. The second postulate of SR come from experimental observation.

Gold Member
Though what your probaly after is the compostion o velocities law which is:

$$w = \frac{u + v}{1 + uv}$$

where u and v are the velcoties of the two observers in the original frame.

Gecko
ok, thanks for clearing that up. and in the formula you posted, $$w = \frac{u + v}{1 + uv}$$, would you put .999c and c for u and v? i saw one similar to this but it looked like $$u = \frac{v + w}{1 + \frac {vw}{c^2}}$$. i was told that one only worked for speeds under the speed of light. thanks.

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Staff Emeritus
Gold Member
The formula that jscd suggested is the same as yours. The only difference is that he uses natural units, so c=1. The formula works when the speeds are less than or equal to the speed of light.

Let's use his version of the formula. Let u be the speed of the rocket ship relative to Earth, and v the speed of the beam of light relative to Earth (i.e. v=1). w is the speed of the beam of light relative to the rocket ship.

$$w=\frac{u + v}{1 + uv}=\frac{u+1}{1+u\cdot 1}=\frac{u+1}{u+1}=1$$

(u would be 0.999 here, but as you can see, it doesn't really matter what u is).

The others who answered you told you that the constancy of the speed of light is a postulate of SR, and not a consequence of the postulates, and I agree, because I think that what zefram c called his favorite postulate is by far the best way to formulate SR mathematically. But I think it should be mentioned that it's possible to take the velocity addition law as the fundamental postulate and derive the constancy of the speed of light from there. This would not make SR any less "strange" though.