# What does the Lorentz factor actually mean?

1. Nov 5, 2012

### arindamsinha

The Lorentz factor is used ubiquitously in relativity for transformation between frames and in describing various relationships.

Wikipedia describes this as:

----------
The Lorentz factor is defined as:

γ = 1/√(1-v2/c2) = 1/√(1-β2) = dt/dτ

where:
v is the relative velocity between inertial reference frames,
β is the ratio of v to the speed of light c.
τ is the proper time for an observer (measuring time intervals in the observer's own frame),
c is the speed of light.
----------

This is all great mathematically, and well understood in its applications in relativity.

I am wondering if there is a simple and understandable explanation of what the Lorentz factor really is. I mean, is there any intuitive, physical way in which it can be explained? (for example, it is a conversion factor between such and such...)

Any opinions on how we might be able to describe the 'real meaning' of the Lorentz factor in some intuitive and easily understandable way?

2. Nov 5, 2012

### robphy

It is the time-dilation factor...
Suppose inertial observers A and B met at event O.
For any other event Q on B's worldline,
$$\gamma=\frac{\Delta t_{OQ,\ according\ to\ A}} {\Delta t_{OQ, \ according\ to\ B}},$$ as you wrote.
In other words,
$$\gamma=\frac{\mbox{number of A's ticks used to measure an elapsed time on B's worldline}} {\mbox{number of B's ticks used to measure an elapsed time on B's worldline}}.$$

It is analogous to the cosine of the angle between two unit vectors.
Given the 4-velocities $\hat t_A$ and $\hat t_B$ of observers A and B,
$$\gamma=\hat t_A \cdot \hat t_B =\cosh\theta_{between},$$ where $\tanh\theta=v_{AB}$ the relative-velocity.

3. Nov 5, 2012

### Mentz114

My own view is that one has to abandon intuition when dealing with relativity. The idea that clocks do not record some universal time but that each one records the proper time along its own worldline is probably the most counter-intuitive concept ever introduced in physics and was met with a lot of resistance when first mooted. As for the 'real meaning', that mathematical definition is it.

For instance the huge number of words wasted on trying to 'explain' the twins paradox could be saved if people just accepted that clocks record the time along their worldlines - which fact explains exactly why differential ageing happens. There is no simple underlying 'mechanism'. That's just the way the universe works.

4. Nov 5, 2012

### ghwellsjr

Instead of γ = dt/dτ, I like to think of dτ/dt = 1/γ, which applies to the tick rate of a moving clock with respect to the coordinate time of an inertial reference frame. The faster a clock moves, the slower it ticks.

5. Nov 5, 2012

### harrylin

It's not just about time dilation, but a simple physical SR description is as ghwellsjr says: The faster a clock moves, the slower it ticks (according to the used reference system).
- robphy explained how it corresponds to a space-time rotation.
- alternatively it can be described as a conversion factor between measures of duration, length, etc. according to different inertial reference systems.

PS I just found the following presentation that could be helpful:
http://www.astro.ufl.edu/~vicki/AST3019/Special_Relativity.ppt [Broken]

Last edited by a moderator: May 6, 2017
6. Nov 5, 2012

Staff Emeritus
What is the difference between "is" and "really is"? (And is that different from "really truly is"?

7. Nov 5, 2012

### Mentz114

Isn't the number of ticks ( ticks being discrete events) invariant ?

8. Nov 5, 2012

### Staff: Mentor

I don't get the question. You know the definition of the Lorentz factor. The definition is the meaning, that's the whole reason why we make definitions.

What is the difference between the definition and the "real meaning"? Do you somehow think that the standard definition is a facade that some conspiracy publishes and that if you know the secret handshake they will let you in and give you a different definition, the "real meaning"?

Last edited: Nov 5, 2012
9. Nov 5, 2012

### robphy

The number of B's ticks from O to Q along B's worldline is invariant.
All will agree on that number.

What I am referring to is how A and B
make elapsed-time measurements of that duration on B's worldline
using their own respective clocks (i.e. using [proper] time-intervals on their own respective worldlines).

To find the elapsed time according to B from O to Q on B's worldline,
B looks at his own clock at event Q [on his worldline] to get the elapsed-time of Q since O.

To find the elapsed time according to A from O to Q on B's worldline,
A looks at her own clock at the event P on her worldline which she regards as simultaneous with Q to get the elapsed-time of P (and thus of Q) since O.

10. Nov 5, 2012

### Naty1

Arind:
I put this together for a previous post......but ultimately Mentz's post above is the reality....we really don't know why things work as they do. [Maybe they work differently in a different universe.] Einstein came to the conclusion, correctly, that in this universe the speed of light is constant, but but space and time are NOT! So he rejected the idea of an invisible 'ether'. Apparently he used Maxwell's equations, in part, in arriving at this conclusion.

and I saved these related comments for my own notes....I think from Wikipedia:

11. Nov 5, 2012

### Naty1

12. Nov 5, 2012

### Bill_K

Shhhh! You promised not to mention that...

13. Nov 5, 2012

### Naty1

I knew it!!! A secret 'physics society' on how stuff really works....

14. Nov 6, 2012

### arindamsinha

Thanks for all the responses.

I have been trying to come up with a simple yet intuitive explanation, shorn of all mathematics, for this important concept in relativity.

My thoughts have led me to this description
• The Lorentz factor is 'the ratio of time passage rates (or time-speeds) between two observers'.
Is that a correct description of the Lorentz factor? Or too simplistic - i.e. does not cover all situations?

Or, if you consider this description obvious, inane or plain stupid, I would like to know that too.

This is really the reason for my original post - I am trying to validate my thinking about this. I didn't put the above description in my original post, to see if I could get validation or refutation of the same independently.

I understand the mathematical explanation, but am looking for something more intuitive - i.e. something one can describe in simple English without the mathematics. Would like to know your opinion on the description I have given.

No issues with the above statement. Still, among this counter-intuitiveness, I am trying to get an intuitive description of the 'Lorentz factor'. Let me know if you agree with the description I have given earlier in this post.

From the SR definition perspective, completely agree. Is it the same thing as the description I have put?

Understand. However, I am trying to see if we can describe it simply as a ratio of time rates between different observers, rather than being a conversion factor between other dimensions like length etc...

I think the other members who responded understood my question quite well. The description at the top of this post may help clarify what I was looking for...

There is a conspiracy. Unfortunately, it is hatched by the physical laws of the Universe, rather than any human agency. So I am not very hopeful that you will be able to help with the 'secret handshake code'... :tongue:

Thanks for the response. Is this ultimately same as the description I provided at the top of the post?

Last edited by a moderator: May 6, 2017
15. Nov 6, 2012

### ghwellsjr

No, it is not the same. You have stated that the Lorentz factor is the ratio of time passage rates between two observers. That is wrong.

Instead, it is the ratio of time passage rates between the coordinate time of an inertial reference frame and an observer (or a clock) that is moving in that frame. Even if you want to consider a second observer at rest in the frame, he cannot observe the Lorentz factor ratio between his own clock and that of the moving observer.

Your definition should obviously be incorrect to you because it is symmetrical, which only means that the ratio could not ever be anything other than 1. You have to at least make one of the observers different than the other one in order to have a ratio that is greater than 1. That difference is that one of the observers is at rest in an inertial reference frame in which times at distant locations have been synchronized to his clock. We imagine that there are many synchronized coordinate clocks throughout the reference frame at every possible location. Then, the moving observer is comparing the time passage rate of his clock to the time passage rate of whichever clock he is closest to as he is moving past these imaginary coordinate clocks. He finds that those clocks are ticking faster than his own but he's not comparing his one clock to just one other coordinate clock, he's comparing his one clock to many other coordinate clocks as they appear to be flying past him.

Does that help?

Last edited: Nov 6, 2012
16. Nov 6, 2012

### harrylin

That is incompatible with what you say next:
If you understand it mathematically, then you know that a difference of time rates has qualitatively no effect on the Michelson-Morley experiment.

17. Nov 6, 2012

### Staff: Mentor

I think it is a fundamentally flawed effort. The Lorentz factor is $\gamma = (1-v^2/c^2)^{-1/2}$. That is it. That is the definition. There is nothing more nor less than that. It is not possible to come up with a "shorn of mathematics" explanation since it is inherently a mathematical concept.

It doesn't cover all situations. E.g. it doesn't cover length contraction or relativity of simultaneity, both of which also contain Lorentz factors. It also doesn't cover momentum, energy, mass, force, acceleration, current density, charge density, scalar potential, vector potential, or any of the many other places that it crops up where there may not be a pair of clocks that you are interested in.

The Lorentz factor is that number. It is a number that crops up very often, and time dilation is just one of the many places where it appears, not the defining feature. You know the definition. That is the "entire actually really real mostest meaningful meaning".

18. Nov 6, 2012

### 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.

19. Nov 6, 2012

### 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.

20. Nov 6, 2012

### ghwellsjr

No, that is still not right.

It's between one observer and one reference frame.

21. Nov 6, 2012

### Staff: Mentor

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.

Last edited: Nov 6, 2012
22. Nov 7, 2012

### 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.

23. Nov 7, 2012

### 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.

24. Nov 7, 2012

### 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.

25. Nov 7, 2012

### 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.