What exactly are Lorentz transformations?

[aychamo]

Hey guys

What exactly are the Lorentz transformations? In the "Was Einstein a genius" thread, it looks like the transformations were known before Einstein had SR/GR?

 

[Hurkyl]

Lorentz came up with them; that's why they're called the Lorentz transformations.

One of the basic principles of science is that the laws of physics should be the same in all inertial reference frames. This is called relativity, and we had long before Einstein came along.

One day, Maxwell came along and derived his theory of electrodynamics, which derives the existence and speed of electromagnetic waves.

This doesn't sound too bad yet; however the catch was that this wasn't a conditional speed. It does not derive the speed of an electromagnetic wave through a medium, it does not derive the speed of an electromagnetic wave with respect to the source; it simply derives the speed of electromagnetic waves.

Now, according to relativity, Maxwell's theory of Electrodynamics is valid in every inertial reference frame, thus the speed of an electromagnetic wave is the same in every inertial reference frame!

In order to reconcile the classical notion of the reference frame with this fact, some people suggested that the "Aether" could compress rods and slow clocks which are moving with respect to it, which would give the illusion of a constant speed of light, because our metersticks are smaller when we're in motion.

So, people set about to derive transformation equations for these changes that would create the illusion of a constant speed of light in all reference frames, and we wound up with Lorentz transformations.

 

[quantumdude]

Originally posted by Hurkyl
So, people set about to derive transformation equations for these changes that would create the illusion of a constant speed of light in all reference frames, and we wound up with Lorentz transformations.

Not only that, but the Lorentz transformation (LT) preserves the form of Maxwell's equations. That point is worth stressing, because it is so often overlooked by critics of SR. If you do a Galilean transformation (GT) on the EM wave equation, you get something that is not a wave equation at all. It would seem that everyone here proves the validity of the LT each time they listen to the radio in their moving cars, because if the GT were valid, your car's antenna should not receive an EM wave at all.

 

[aychamo]

So if Lorentz got the equations, what exactly did Einstein do?

*Please don't hate me!*

 

[quantumdude]

Einstein was the one who discovered that the equations are applicable to our universe, and he was the one who recognized that their application to our universe rendered superfluous the notion of an aether, which was thought to be an existent frame of absolute rest.

 

[aychamo]

So was the idea that time slows down when you move at high fractions of c an Einstein idea or was that pre-Einstein?

 

[quantumdude]

This, I'm not sure of. I have not read Lorentz' papers in detail, but I know that he incorporated length contraction into his theory. I do not know if he also incorporated time dilation into it, or if that was unique to Einstein's 1905 work. Someone else may be able to help out here.

 

[pmb_phy]

Originally posted by aychamo
Hey guys

What exactly are the Lorentz transformations? In the "Was Einstein a genius" thread, it looks like the transformations were known before Einstein had SR/GR?

Lorentz transformations are a mapping of events in one Lorentz coordinate system to events in another Lorentz coordinate system which is in standard configuration with the first.

A "mapping" I mean a function from (t,x, y, z) to (t',x', y', z'). Think of it as a vector function.

An "event" is a place and a time. E.g. "Fire cracker explodes in the middle of the yard at 12:00pm" is an example of an event. There is a place (x,y,z) and there is a time "t" to it. Mathematically I can represent this as (t, x, y, z).

A "Lorentz coordinate system" is an inertial coordinate system whose spatial axes are Cartesian.

Two systemsm, S and S', are said to be in standard configuration when the xyx axes are parallel to the x'y'z' axes and S' in moving in the +x direction and the origins coincide at t = t' = 0.

Thus if I tell you the (t,x,y,z) of an event in S and I know the Lorentz transformation then I can tell you what the coordinates of that same event is in the (t', x', y') coordinate system.

For details please see
http://www.geocities.com/physics_world/sr/lorentz_trans.htm

Einstein dervived the Lorentz transformations from the postulates of relativity quite independent of considerations of EM as Lorentz did. They then had a more general meaning.

I believe that Lorentz's notions about time dilation were purely mathematical in that he did not attribute them to anything physical, sort of a mathematical "trick". I don't recall the details exactly.

 

[aychamo]

Hmm. May I ask for a real world example of two things moving in inertial frames of reference that would be an event?