What does the Lorentz factor actually mean?

[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?

 

[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.

 

[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.

 

[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.

 

[harrylin]

The Lorentz factor is used ubiquitously in relativity for transformation between frames and in describing various relationships.
[..]
The Lorentz factor is defined as:
γ = 1/√(1-v²/c²) = 1/√(1-β²) = dt/dτ
[..] (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?

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

 

[Vanadium 50]

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

 

[Mentz114]

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.

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

 

[Dale]

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?

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"?

 

[robphy]

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

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.

 

[Naty1]

Arind:

I mean, is there any intuitive, physical way in which it can be explained?

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.

The Lorentz transform is a way to bring observers of different velocities at different places together so they can 'compare notes'...see things from a common frame. It is a way to take into account the fact that the speed of light is constant, and finite, but distance and time are not constant!

As a simplistic example, if you are right alongside a house and a friend is a mile away how do you compare observations about the size of the house?? You need some sort of a transform, an adjustment,to reflect your different positions. In relativity this gets more complicated since neither distance nor time are constant between inertial observers.

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

Lorentz transforms are the correct ones to use with all physical laws as suggested by Poincare and proposed by Einstein. These transforms change our concept of space and time. Space contracts in the direction of motion and time slows in a moving frame both measured from another frame. “failure of simultaneity at a distance” ensues as two observers disagree about the timing of an event. (why is this??..because they measure different elapsed times??)

A Lorentz transformation is analogous to a rotation between space and time. We now represent space and time as four vectors, the fourth associated with time. If applied to momentum, the three space parts are like ordinary momentum; the fourth time part like energy. Starting with the Lorentz transform we can notice two important things:

1) time and space are not entirely separate entities but one frame's time gets split into another frame's space and vice versa.
2) there is a notion of "distance" called the spacetime interval which also mixes space and time together and is agreed upon by all reference frames (i.e. is invariant under the Lorentz transform).

 

[Naty1]

Here is a discussion on how you can derive the Lorentz transform:

https://www.physicsforums.com/showthread.php?t=649496&highlight=lorentz

 

[Bill_K]

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"?

Shhhh! You promised not to mention that...

 

[Naty1]

if you know the secret handshake they will let you in and give you a different definition, the "real meaning"?

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

 

[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.

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.

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.

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...

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.

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.

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

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

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 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"?

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...

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"?

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:

I put this together for a previous post... ...agreed upon by all reference frames (i.e. is invariant under the Lorentz transform).

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

Shhhh! You promised not to mention that...

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

 

[ghwellsjr]

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.
...

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.

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

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?

 

[harrylin]

[..] I understand the mathematical explanation [..]

That is incompatible with what you say next:

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...

If you understand it mathematically, then you know that a difference of time rates has qualitatively no effect on the Michelson-Morley experiment.

 

[Dale]

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

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.

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?

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".

 

[ghwellsjr]

Arindamsinha is well aware of the many situations where the Lorentz factor is used. He started his first post with:

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

Nevertheless, I don't see anything wrong with his question:

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?

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]

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?

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?

Obviously, this is not the only physical meaning of the Lorentz factor...

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]

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?

No, that is still not right.

It's between one observer and one reference frame.

Please go back and read my posts.

 

[Dale]

Nevertheless, I don't see anything wrong with his question:
...
Obviously, this is not the only physical meaning of the Lorentz factor

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]

No, that is still not right.

It's between one observer and one reference frame.

Please go back and read my posts.

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]

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.

In arindamsinha's first post, he quoted a definition of Lorentz factor from wikipedia:

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 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]

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.

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]

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.

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]

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.

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]

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?

OK, thanks. I got what you mean.

 

[phyti]

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


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?

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.

https://www.physicsforums.com/attachments/52787

 

[ghwellsjr]

Your diagram and explanation are very difficult to follow.

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.

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.

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.

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.

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.

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]

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?

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.

 

[Dale]

What, no picture?

 

[bobc2]

What, no picture?

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.

https://www.physicsforums.com/attachments/52882

 

[ghwellsjr]

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.

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".

 

[phyti]

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.

The hyperbola is the inverse function of the quarter circle, a fact of geometry.

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".

The rod is only a visual aid to indicate a rigid device moving with A, and represents the photon path as imagined by A. It helps with those new to SR. The photon doesn't hit the rod. The successful photon has to have the correct angle to intercept the A mirror. How would it 'know' this? It doesn't. As stated, the successful photon has a horizontal vt component that equals the speed of the clock. The vertical st component moves toward the mirror but at less than c. The intersection of the arc of photons and the path/(rod) to the mirror is determined by the speed of A. The angle of the ct photon relative to vertical is a function of v/c. A simple examination shows, a lesser speed, a smaller angle and a greater s,
a greater speed, a greater angle and a lesser s.

The drawing is intended to show why 1 tick for the A clock is longer than 1 tick for the U clock for the general case, which it does. The reason is the constant and independent speed of light. The degree of dilation is a function of v/c. If the path is extended, it will not show anything new. Gamma is still the ratio c/s or (A time)/(U time).