What does the Lorentz factor actually mean?

ghwellsjr

Arindamsinha is well aware of the many situations where the Lorentz factor is used. He started his first post with:
Nevertheless, I don't see anything wrong with his question:
After Einstein derived the Lorentz transformation in section 3 of his 1905 paper introducing Special Relativity where he assigned β to the Lorentz factor, he went on in section 4 entitled "Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks" to show that τ = t√(1-v²/c²) where τ is the time on the inertially moving clock and t is the coordinate time for the "stationary" frame (where both were at time zero at the origin of the frame). Using his nomenclature for the Lorentz factor, this becomes τ = t/β, or in modern terminology, τ = t/γ. As the ratio of the rates between the moving clock and the coordinate time, this becomes Δτ/Δt = 1/γ or dτ/dt = 1/γ. This is what I showed in post #4 and explained in more detail in post #15.

Obviously, this is not the only physical meaning of the Lorentz factor, but it is one of the easiest to explain and understand, and if it was OK for Einstein to explain it this way, I don't see why we can't either.

arindamsinha

Yes, I can see what you are saying. I was taking the underlying assumption that one of them is moving at a certain velocity w.r.t. the coordinate clock.

I suppose we can extend my definition a bit more and say:

• The Lorentz factor is 'the ratio of time passage rates (or time-speeds) between two observers, one of whom is stationary and the other is inertially moving with a certain velocity'

This includes the possibility that the velocity is 0, in which case, the Lorentz factor will be 1.

Does that sound more like it?

You are right, I am not saying this is the only definition (intuitive or otherwise) of the Lorentz factor, but one possible intuitive description.

ghwellsjr

No, that is still not right.

It's between one observer and one reference frame.

Please go back and read my posts.

DaleSpam

and hence the problem. Why pick that one application of the factor and crown it and say this one is the real meaning? It isn't. It is simply one application of the Lorentz factor, whose meaning is given by the definition.

If I were going to talk about the meaning of the Lorentz factor as something different from its definition then I would talk about its derivation, not its applications.

arindamsinha

I have read your posts in complete detail, even before my earlier responses. Still, I don't understand the difference. We are talking about one observer who is stationary w.r.t. the coordinate frame and one who is traveling. I believe we are saying the same thing.

Let me know where specifically you are disagreeing on this.

ghwellsjr

In arindamsinha's first post, he quoted a definition of Lorentz factor from wikipedia:
This definition had an extension that ended in dt/dτ. I'm just trying to explain what that extension means in the context of a simple, understandable, intuitive, physical way, which is what he asked for. Einstein categorized this explanation as a physical meaning of the equations along with length contraction, as well as numerous other applications throughout the rest of the paper. If arindamsinha had asked about length contraction, then we would be talking about that and not about time dilation. We're just focusing on time dilation because that is what the equation he quoted is focused on.

ghwellsjr

You're talking about two observers and one coordinate frame. We calculate the time dilation of both observers in exactly the same way. We take their speed and plug it into the equation you posted on your first post and from that we calculate gamma. Then we can use the formula I posted on post #4, dτ/dt = 1/γ, to calculate the Proper Time of each observer with respect to the coordinate time. The Proper Time for your first stationary observer will pass at the same rate as the coordinate time. The Proper Time for your second traveling observer will pass at a slower rate than the coordinate time.

Do you see that the Proper Time for each observer can be easily calculated from the formula no matter what the speed is? And do you see that a coordinate frame does not require any observer to be stationary nor does it require any particular number of observers, not even one? Finally, do you see that if you use a definition for the Proper Time (or for time dilation) that does not include a specified coordinate frame, but rather is just between two observers, then it won't work because whatever you say about the passage of time for one of them with respect to the other one can also be said about the two observers if you interchange them and that would create a dichotomy. You can't say that the ratio of the times between A and B is the same as the ratio of the times between B and A unless both ratios are one.

arindamsinha

Yes, but one observer is at rest w.r.t. the coordinate reference frame. So what's the issue? I do not mean observers as in 'human beings who happen to be at that location', but just the point of view.

ghwellsjr

Suppose you take your two observers, one at rest and one traveling, and you transform to a new reference frame using the Lorentz transformation process such that now both observers are traveling in the opposite direction at the same speed. How will you interpret their respective Proper Times or their respective time dilations?

arindamsinha

OK, thanks. I got what you mean.

phyti

A very precise measuring system is available using reflected light. The light clock can serve as a standard unit of measure by reflecting light from a mirror to a detector.
In the U frame of reference, one unit of time is defined as the distance light moves from the origin to a mirror m located perpendicular to the x axis and return. If the light source is not moving, the distance is 2d for 1 time unit.
Observer A moves a distance (a) while a light wave composed of multiple photons, moves from the origin a distance d. Because light speed c is constant and independent of the source, the A-clock photon must have an x component equal to (a) to compensate for A's motion. The p component becomes the active part of the A-clock. Since p is less than d, the photon will move a greater distance to reach m, resulting in a tick rate less than that for U.
The clock photon for B will have a greater compensating component b, and thus a smaller active component q, resulting in a tick rate less than that for A.
The tick rate is a non-linear function of (clock speed)/(light speed), or v/c, with each observer clocking a different photon. The observer is not aware of his slow clock rate, because all processes involving photon interaction, including biological/(chemical) are also ocurring at the same rate, i.e. his perception is altered. Gamma(the Lorentz factor) equals d/(the vertical component) with v/c substituted for (the x component)/d. Gamma for A = d/p with v/c substituted for a/d.
The 't' factor has been omitted, because it is common to each vaiable and d, and to emphasize that the clock is in fact counting distances.

ghwellsjr

Your diagram and explanation are very difficult to follow.
You have established that for a stationary light source and observer, the light starts at 0, goes directly up to the mirror at m for a distance of d and then reflects back down for another distance of d for a round-trip distance of 2d. This part I understand.
This part I don't understand. You imply that the light only travels a distance d along the first diagonal but it doesn't reach m and you don't show the return path. You seem to be implying that length contraction is in play here which it is not so I'm very confused about your diagram and your explanation.
I've never heard of "compensating component" and "active component". Are these your terms?

I think what you have discovered is that if we plot the reciprocal of gamma as a function of normalized speed, we get a quarter of a circle. I made a similar plot some time ago to show the normalized age of the traveling twin (compared to the stationary twin) as a function of normalized speed (beta):



This was discussed in this thread, as well as others.

Although your plot might also describe the relationship of 1/gamma to speed, it is not labeled as such and has no discernable connection to a light clock, at least as far as I can understand. Maybe you could explain it some more and show how it relates to your diagram.

bobc2

Great question, arindamsinha. There is a very simple physical explanation. It is a mathematical representation of the way nature has given different observers different instantaneous 3-D cross-section views of the 4-dimensional universe.

DaleSpam

What, no picture?

bobc2

I figured the sketches were getting annoying.

phyti

This post is intended to explain gamma/(the Lorentz factor) in terms of physical
processes and minimal math instead of theoretical statements.
The light clock consists of an integrated light emitter/detector, and a mirror, separated
by a rod of length r. The clock counts a unit of time (t=1 tick) when a photon moves
the length of the rod to the mirror, and returns to the detector.
There are two observers, U who is not moving, and A who is moving at .6c relative to U
on the Ux axis. Each has a copy of the clock with the rod oriented perpendicular to the
x axis.
Since the outbound path equals the inbound path, we only need to consider the first
path.
With U and A at the origin, each clock emits multiple photons (shown as a blue quarter
circle because object motion is restricted to the +x axis.)
For U the photon moves a distance r (.5 tick).
For A the intersection of the circular arc and rod determine which photon becomes part
of the clock. [1]The photon path ct can be resolved into the vt component which
compensates for the motion of A and the st component which is the active part of the
clock. When a photon arrives at the U mirror, the A clock photon has not reached the
mirror because the photon speed relative to the rod is s. If r' equals the path length to
the mirror for the A photon, then t'/t = r'/r = c/s = gamma.

[1] A vector can be expressed in components suitable for the situation.

ghwellsjr

Like I said in my previous post #29, you have discovered a graphical relationship that relates speed to the reciprocal of gamma but it has nothing to do with your explanation of a light clock.

Your problem is that you claim that there is something significant when a photon hits the moving rod at the 80% mark. In fact, there are photons hitting both rods all along their trips to their respective mirrors. So what? There is no significance to the fact that a photon hits a rod at any particular time. What matters is when a photon hits the moving mirror, which you don't show. If you would continue the diagonal line up to the location of where the mirror would be when it hits it and then measure the time it takes for the photon to hit the mirror, you would see that it take 1.25 times as long as it takes for the photon to hit the stationary mirror which gives the correct illustration "in terms of physical processes".