Lorentz Transformation - Exponential factor, why not Proportional?

1. Jan 4, 2005

Gamish

Lorentz Transformation --- Exponential factor, why not Proportional?

When using the Lorentz Transformation, the increase in mass, or the decrease in space-time is an Exponential, why can't it be Proportional? What is the logic behind that? For instance, if SR had a proportional decrease in time, then if you traveled .5c, time would be cut in half (t=(t/2)). But, according to the Lorentz Transformation, to cut time in half, you have to travel about .86c, why is that? Why can't v = .5c, then time = .5t? Is this just how nature works, or is there an actual reason?

2. Jan 4, 2005

dextercioby

I believe that mother Nature has an awkward habit of following the principles of Relativity:1).The laws of physics are invariant to general coordinate transformations (change of noniertial reference frames).2)The speed of light in vacuo is constant and independent of the noniertial reference frame it is measured.
Since LT are a consequence of a weaker version of the priciples stated,i guess the question should be equivelent to
"Why Nature behaves according to the Principles of Relativity??"and that's a question for the phylosophers...

Daniel.

3. Jan 4, 2005

Gamish

So, the exponental factor is just something of nature, we cannot explaine why it operates that way? We just observe it, and say that it is true?

4. Jan 5, 2005

pervect

Staff Emeritus
The factor isn't really exponential, it's 1/sqrt(1-(v/c)^2).

(c * time interval)^2 - (distance interval)^2

This quantity is the same for all observers. As a relatively simple formula, and as it's an invariant, this quantity is extremely interesting.

The Lorentz transforms are the only linear transforms that leave this interval invariant. So you might say that the specific factors are "because" the Lorentz interval was invariant. Of course you can always ask "why" again ("Why is the Lorentz interval invariant) and wind up at a point where there is no answer. One can always do this - asking the question "why" more than 2-3 times in a row is going to reach a point where there is no answer.

5. Jan 5, 2005

AVNguyen

Yes, because physics is a science of nature which is based on the experiment and observation. All we can do is created the best theory to predict things and making the module which is suited the theory. If the result came from the observation is diferrent from the module then we will have to elliminate that module and the theory and replace them by the new one. I really don't like the idea of somebody that considering physics is just a bunch of algebra and theory. That will ruin the purpose of the whole subject.

6. Jan 5, 2005

Gamish

So....

I can say that the lorenz transformation is an exponental factor, because you have to get extremely close to c for a significant change to take place. So, with that said, let me ask this. How did we discover the effects of the lorenz transformation? The scientist (Im assuming he goes by the name lorenz) must have observed it somehow, and developed a math equation to express it. But how did he observe it?

7. Jan 5, 2005

jcsd

1) It's interesting to me as on one hand it seems sensible to me thta the fundamental theories of nature respect the principle of diffeomorphism invariance (i.e. general covariance), but on the otherhand I wonder is this an empricial requirement (whilsts Lornetz covariance woul most definitely seem to be an empriical requirement I'm not sure if we can confidentally assert the same for the general case) or is it a philosophical requirement (it seems to me that it was a philosphical wish for general covariance that drove Einstein to formualte genral relativity).

2) Shurely you mean that to be true only in the local case?

8. Jan 5, 2005

Gamish

It seems that no one has a real answer, but how did we observe the effects of the lorenz transformation in the first place? Maybe if I know how we observed it, I can conclude why it may work that way.

9. Jan 5, 2005

Staff Emeritus
The first experiment that was explained in terms of what were later understood to be Lorentz Transformations was the Michaelson-Morley experiment, which failed to find a relative motion between the earth in its orbit and the supposed ether. Fizgerald conjectured that if x' observed in a frame at motion with respect to the observer behaved as the observer's x multipled by $$\frac{1}{\sqrt{1 - (\frac{v}{c})^2}}$$ then the M-M null effect would result.

10. Jan 5, 2005

Gamish

11. Jan 5, 2005

Staff: Mentor

The MM experiment was designed to measure the earth's motion through the "aether" - since earth is moving in several ways that could be measured (rotating about its axis and revolving around the sun, for starters), it stood to reason that this motion could be detected by measuring the speed of light in different directions through this motion.

The way the experiment works is very much like a person swimming across a river: the person swims at a constant speed (or so he thinks) straight across the river, and can calculate his speed based on the distance across the river and the time it took to cross it. To a person on either bank, however, the swimmer swam at a diagonal because of the motion of the river. Thus, someone on the bank calculates a different speed for the swimmer. The MM experiment found that the swimmer (a beam of light) always went at the same speed regardless of how the speed was measured.

The Lorentz transformations accurately model this phenomena.

Last edited: Jan 5, 2005
12. Jan 5, 2005

Galileo

He's referring to the result of the Michelson and Morley experiment.
It was an (extremely accurate) experiment conducted in 1887 which tried to determine the speed of light relative to the ether, which was assumed to be the medium in which light propagates.
You can find lots of it on the internet. e.g. here:
http://scienceworld.wolfram.com/physics/Michelson-MorleyExperiment.html

The null result meant that no difference in the speed of two lightbeams was found when the sources had different velcoties with respect to this ether.
Lorentz and Fitzgerald showed that they could explain this null-result by saying that an object moving with respect to the ether got shorter by a factor of $1/\sqrt{1-(v/c)^2}$. That's where the Lorentz-Fitzgerald contraction comes from. It's just called Lorentz contraction now.
The interpretation was wrong, but the equation was still correct. Later, with Einstein's theory of relativity the same factor occurred.

From a theoretical standpoint, Lorentz contraction is a result of the 2 postulates of special relativity. Like dextercioby said. You can derive it from these 2 principles.

13. Jan 5, 2005

Gamish

Thanks for all your replies, I understand it more. But, I still don't get one thing. They were able to observer that c is constant, so why didn't they make the lorenz transformation a proportional increase in mass, or decrease in length/time? They must have observed somehow, that the mass must get extreamly close to c for a significant relavistic effect to take place.

I think it is, the math just expresses the nature of it, and the nature is that you must get extremely close to c for the relavistic effects to kick in.

Last edited: Jan 5, 2005
14. Jan 5, 2005

krab

The Lorentz factor is not exponential. "Exponential" is a mathematical term. It means that as the independent variable is incremented, for each increment, the functional effect is to multiply by some factor. Or in mathematical symbols, $f(x)\propto e^{kx}$

15. Jan 5, 2005

Staff Emeritus
I said before that Michaelson and Morley did NOT find a relative speed between the earth and the ether. This is therefore called a null effect. M-M is just an abbreviation for Michealson and Morley, pretty common on this forum.

What Fitzgerald said (I have not read his paper, I am quoting what I have read about him) was that if every length in the ether, which the earth was moving through, was seen from the earth as multiplied by that particular nonlinear factor, then when you plugged these changed lengths into the math used by the experimenters, the relative velocity would be cancelled out. Even though it was "really there", because of the nonlinear length change, it would appear to be zero. He had worked out the math to show that was so. I do not know if Fitzgerald thought the length was really shortened or if it only appeared to be.

16. Jan 5, 2005

Gamish

OK, for those who do not understand my question, I will rephrase it.

The decrease in space and time, and the increase in mass, are not proportional to the percentage of the speed of light which you are traveling, v/c. So, it is not porportional, I just used the phrase "exopnental", because it is in a way, not getting too technical. So, can someone explain why if I travel .5c, my mass does not increase to .5/c, and my time does not slow to .5l, and my length does not contracT to 5t? Why is it not proportional to the increase in velocity? We all know what the lorenz transformation tells us, but can someone explaine why it works that way, and not porportional to v/c?

I hope I explained it simple enough

Last edited: Jan 5, 2005
17. Jan 5, 2005

yogi

Maybe you can get same inkling as to why the transformation formulas involve a second order term
(v/c)^2
If you consider that in nature we are dealing with things that can be expressed best on a graph with time at right angles to space. - and this geometry involves the theorom of Pythagoras - when we put time on a graph it is usually in the direction of the y axis - and space for example may be along the x axis - now when you are combining time and space measurment in what is called the interval which is constant for every frame- then you say that x squared plus y squared equals some constant squared for every reference frame - the constant is the velocity of light c appropriately scaled to make the units come out right. Does this make any sense - propably not!

Last edited: Jan 5, 2005
18. Jan 6, 2005

Gamish

yogi, I will think on your reply over night (relavent to locationon earth, lol), maybe while I am lucid dreaming, I will travel near c, and figure it all out. Just kidding. Maybe I will make a graph in MS word, and try to see some sort of a relationship, which will yield why the lorenz transformation is not porportional to v. I will reply in about 9 hours.

19. Jan 6, 2005

Integral

Staff Emeritus
We do not choose the laws of nature, they are what they are. It is our task to figure them out. One of Einsteins accomplishments was to show that the Lorentz transforms followed logically from his 2 postulates. In his 1905 paper he used the fact that the speed of light is constant to derive a difference equation he took a limit to arrive at a partial differential equation. The solution to this equation is the Lorentz transforms.

20. Jan 6, 2005

Gamish

Equation

OK, I have developed the module equation which will illistraight the lorenz transformation, keeping the speed of light constant, and keeping all the relavistic effects in SR, but it will be porportional to v. Look...

1-v/c

If we want to do time dilation, just do

t=t*1-v/c

For length contraction

l=l*1-v/c

For mass increasment

m=m/(1-v/c)

So, why can't this equation work. I am in no wat purposing that this equation will actually work, but how did they figure out exactly how the relavistic effects took place at speeds near c, they could have been guessing, although I'm sure they had a reason.

21. Jan 6, 2005

Integral

Staff Emeritus
Because it doesn't and because it is not a solution to the governing differential equation as derived by AE.

22. Jan 6, 2005

Gamish

So, can someone please explaine exactly why my equation will not work. I know that it doesn't work, I just want to learn how they figured out exponental relavistic effects while traveling near c.

23. Jan 6, 2005

Galileo

I'm unsure as to what you are exactly asking.

Are you asking for a derivation of this factor?
If so, then that is easily done, if we assume that the speed of light is c for all observers. (I think you already know this though)
Take a train of height h travelling with speed v wrt the ground and let a light beam in the train be emitted straight down.
Inside the train the time it takes for the light to hit the bottom of the train is:
$$t'=h/c$$
For someone on the ground the beam will not travel straight down.
The height of the train squared is (by the pythagorean theorem):
$$h^2=c^2t^2-v^2t^2$$
inserting $h=ct'$ we find:

$$ct'=t\sqrt{c^2-v^2}$$
or
$$t=\frac{t'}{\sqrt{1-v^2/c^2}}$$
so it follows from the fact that c is the same for both an observer in the train and for one on the ground.

$$w = (u + v)/(1 + uv/c2)$$

which has the property that it is always smaller or equal to c. (equal to c when u or v is c).
If t=t'(1-v/c) you would probably violate this principle.

24. Jan 6, 2005

krab

This is just a really strange thing to ask. It's like asking why I need twice the force to get the same acceleration for double the mass; like, why not triple or quadrupole? It's just that it doesn't. Your formulas are in a sense completely arbitrary, relating to a taste for proportional effects? Because that makes the math simpler? Nature doesn't work according to some arbitrary standard of what's simple. Anyway, I'm just totally guessing what it is you are asking.

Consider also that if your formulas were correct, you would get negative times, lengths, masses for v>c. None of that's observed of course. The correct formula actually prohibits v>c, and in a really neat way.

Are you asking how this was arrived at? One can prove from E&M that these are the right factors. Subsequently, it was derived, as stated here, to explain the null MM effect. So then it seemed to be even more fundamental than just applying to E&M. Einstein put it all together with his two postulates. Using just those postulates, you come to the Lorentz factor, and no other possibility exists. Since 1905, hundreds of accelerators have been built that use these formulas and they all work exactly as designed.

Electron accelerators have reached v=0.99999999995c. You said:
So "they" did not have to travel near c to figure out the relativistic effects. Physicists first figured out relativistic effects, long before such accelerators were built. And once they had developed the theory, it turned out to work beautifully and flawlessly.

25. Jan 6, 2005

Gamish

I know that my math equations will not work, I am just saying why? This is the conclusion I have come up with so far. The relavistic effects near c are not porportional to the % of c because that is the way nature is. Either one could have happend. Or am I wrong, there is a reason why relavistic effects are not porportional to the % of c?