# Proof of time dilation

• Aniket1

#### Aniket1

I don't understand the thought experiment where a photon hitting a reflecting surface and traveling back to the observer is viewed from two frames of reference and the concept of time dilation is suggested arguing that the distances traveled by the photon are not the same in both the frames even though the photon has the same constant velocity.
I could as well think of some other particle instead of a photon and there would be no time dilation in this case.

Other particles aren't photons: photons are unique in that all observers measure the same speed for them.

Einstein postulated that the speed of light was constant in all frames. It's just something he decided to accept (with good reason, I might add, he didn't pull it out of thin air) and see where it led. Via the light-clock thought experiment (Einstein used lightning striking a train - the light-clock was developed in the 1920s, IIRC), the assumption leads to time dilation and length contraction, fixes a fundamental issue in electromagnetic theory and explains all those cosmic ray muons.

As you correctly note, you can't repeat the derivation by substituting electrons for the photons. But there is no reason to think that electrons might travel with any special velocity.

But there is no reason to think that electrons might travel with any special velocity.

I understood that physicists thought it was odd that electrons don't travel at c, and was part of the case for Higgs field.

I don't understand the thought experiment where a photon hitting a reflecting surface and traveling back to the observer is viewed from two frames of reference and the concept of time dilation is suggested arguing that the distances traveled by the photon are not the same in both the frames even though the photon has the same constant velocity
Maybe some spacetime diagrams will help.

Here's one showing a photon (thin blue line) being emitted by an observer (thick blue line) and hitting a reflecting surface (thick red line) and traveling back (thin red line) to the observer. This is for the rest frame of the observer and the reflecting surface:

As you can see, the separation between the observer and the reflecting surface is four feet and it takes 8 nsecs for the photon to make the round trip. It takes an equal amount of time for the photon to get to the reflecting surface as it takes for the reflection to get back to the observer.

Now we will transform this diagram using the Lorentz Transformation process to see what it looks like in a frame traveling at 60%c to the right:

At 60%c, gamma is 1.25 which means the distance between the thick blue and red lines gets contracted to 4/1.25=3.2 feet as you can see in the diagram at the Coordinate Time 0 for example. Also the photon takes longer to go through its cycle by the factor of gamma so now it takes 8 times 1.25 or 10 nsecs. But note that the photon gets to the reflecting surface in just 2 nsecs and it takes another 8 nsecs for the reflection to get back to the observer.

Does this make perfect sense to you?
I could as well think of some other particle instead of a photon and there would be no time dilation in this case.
That's not true.

Let's assume we do the same experiment except with an electron traveling at 80%c. Here is a diagram depicting an electron as the thick black line:

As you can see, it takes 10 nsec for the electron to make the round trip since it is going slower than the photon was. At 80%c, it takes 5 nsecs to go 4 feet (4/5=0.8) and another 5 nesecs to get back.

And as before, we will transform this to a frame moving at 60%c to the right:

Again, the separation between the observer and the reflecting surface is contracted to 3.2 feet but this time the electron leaving the observer is traveling slower than before, it looks like it's taking 2.7 nsecs to go one foot which is a speed of 37%c. But on the way back, the electron is taking 8.6 nsecs to go 8 feet for a speed of 93%c. The net result is that it takes a total of 12.5 nsecs which is the same Time Dilation factor that we got for the photon (12.5/10=1.25).

You can use the velocity addition formula to calculate the exact speeds of the electron and the photon in these two experiments. For the case where the speed of light equals 1 the formula is:

s = (v+u)/(1+vu)

For the case of the photon, v=-0.6 and u=±1.

First, let's calculate it for u=+1:

s = (v+u)/(1+vu) = (-0.6+1)/(1+(-0.6)(1)) = (0.4)/(1-0.6) = (0.4)/(0.4) = 1

The photon is traveling at c away from the observer.

Now we do it for u=-1:

s = (v+u)/(1+vu) = (-0.6-1)/(1+(-0.6)(-1)) = (-1.6)/(1+0.6) = (-1.6)/(1.6) = -1

The reflected photon is traveling at -c (towards the observer).

For the case of the electron, v=-0.6 and u=±0.8.

First, let's calculate it for u=+0.8:

s = (v+u)/(1+vu) = (-0.6+0.8)/(1+(-0.6)(0.8)) = (0.2)/(1-0.48) = (0.2)/(0.52) = 0.385

The electron is traveling at 38.5%c away from the observer.

Now we do it for u=-0.8:

s = (v+u)/(1+vu) = (-0.6-0.8)/(1+(-0.6)(-0.8)) = (-1.4)/(1+0.48) = (-1.4)/(1.48) = -0.946

The electron is traveling at -94.6%c (towards the observer)

To summarize, these two cases, for the photon and the electron, both show Time Dilation and they both use the same equations, either the Lorentz Transformation or the Velocity Addition Formula.

Does this all make perfect sense to you? Any questions?

Aniket1
Maybe some spacetime diagrams will help.

Here's one showing a photon (thin blue line) being emitted by an observer (thick blue line) and hitting a reflecting surface (thick red line) and traveling back (thin red line) to the observer. This is for the rest frame of the observer and the reflecting surface:

As you can see, the separation between the observer and the reflecting surface is four feet and it takes 8 nsecs for the photon to make the round trip. It takes an equal amount of time for the photon to get to the reflecting surface as it takes for the reflection to get back to the observer.

Now we will transform this diagram using the Lorentz Transformation process to see what it looks like in a frame traveling at 60%c to the right:

At 60%c, gamma is 1.25 which means the distance between the thick blue and red lines gets contracted to 4/1.25=3.2 feet as you can see in the diagram at the Coordinate Time 0 for example. Also the photon takes longer to go through its cycle by the factor of gamma so now it takes 8 times 1.25 or 10 nsecs. But note that the photon gets to the reflecting surface in just 2 nsecs and it takes another 8 nsecs for the reflection to get back to the observer.

Does this make perfect sense to you?

That's not true.

Let's assume we do the same experiment except with an electron traveling at 80%c. Here is a diagram depicting an electron as the thick black line:

As you can see, it takes 10 nsec for the electron to make the round trip since it is going slower than the photon was. At 80%c, it takes 5 nsecs to go 4 feet (4/5=0.8) and another 5 nesecs to get back.

And as before, we will transform this to a frame moving at 60%c to the right:

Again, the separation between the observer and the reflecting surface is contracted to 3.2 feet but this time the electron leaving the observer is traveling slower than before, it looks like it's taking 2.7 nsecs to go one foot which is a speed of 37%c. But on the way back, the electron is taking 8.6 nsecs to go 8 feet for a speed of 93%c. The net result is that it takes a total of 12.5 nsecs which is the same Time Dilation factor that we got for the photon (12.5/10=1.25).

You can use the velocity addition formula to calculate the exact speeds of the electron and the photon in these two experiments. For the case where the speed of light equals 1 the formula is:

s = (v+u)/(1+vu)

For the case of the photon, v=-0.6 and u=±1.

First, let's calculate it for u=+1:

s = (v+u)/(1+vu) = (-0.6+1)/(1+(-0.6)(1)) = (0.4)/(1-0.6) = (0.4)/(0.4) = 1

The photon is traveling at c away from the observer.

Now we do it for u=-1:

s = (v+u)/(1+vu) = (-0.6-1)/(1+(-0.6)(-1)) = (-1.6)/(1+0.6) = (-1.6)/(1.6) = -1

The reflected photon is traveling at -c (towards the observer).

For the case of the electron, v=-0.6 and u=±0.8.

First, let's calculate it for u=+0.8:

s = (v+u)/(1+vu) = (-0.6+0.8)/(1+(-0.6)(0.8)) = (0.2)/(1-0.48) = (0.2)/(0.52) = 0.385

The electron is traveling at 38.5%c away from the observer.

Now we do it for u=-0.8:

s = (v+u)/(1+vu) = (-0.6-0.8)/(1+(-0.6)(-0.8)) = (-1.4)/(1+0.48) = (-1.4)/(1.48) = -0.946

The electron is traveling at -94.6%c (towards the observer)

To summarize, these two cases, for the photon and the electron, both show Time Dilation and they both use the same equations, either the Lorentz Transformation or the Velocity Addition Formula.

Does this all make perfect sense to you? Any questions?

Thank you so much for your elaborate answer. You couldn't have explained it better :)

I understood that physicists thought it was odd that electrons don't travel at c, and was part of the case for Higgs field.

If you heard that somewhere, you misunderstood.

I understood that physicists thought it was odd that electrons don't travel at c, and was part of the case for Higgs field.

I don't think electrons interact with the Higgs field.

I gather that trying to put a straightforward E=mc2 term for the mass of the massive gauge bosons (W and Z) into the Standard Model Lagrangian causes infinities to pop up everywhere in the solution. There was some head-scratching about how to fix this, which was widely reported (fifty years after the first solution was proposed, when we began to check it) as "scientists say particles shouldn't have mass", with the implication that "everything should travel at the speed of light!", but that was just journalist nonsense.

I don't think electrons interact with the Higgs field.

In the Standard Model, every particle with nonzero rest mass acquires that rest mass through its interaction with the Higgs field. That includes the electron. The detailed mechanism for fermions, like the electron, is different than for the gauge bosons (the W and Z particles), but it's there. See further comments below.

I gather that trying to put a straightforward E=mc2 term for the mass of the massive gauge bosons (W and Z) into the Standard Model Lagrangian causes infinities to pop up everywhere in the solution.

It's not really an ##E = mc^2## term; that's not what the mass term in a QFT Lagrangian looks like. The issue with adding a mass term directly to the Lagrangian for the gauge bosons is that it breaks gauge invariance. The issue with doing it for fermions is that it breaks chiral symmetry. The Higgs mechanism offers a way around both of these difficulties.

If you want to discuss this further, you should probably start a separate thread in either the quantum physics or particle physics forum, since it's getting off topic for this thread and isn't primarily a relativity issue.

If you want to discuss this further, you should probably start a separate thread in either the quantum physics or particle physics forum, since it's getting off topic for this thread and isn't primarily a relativity issue.
There's a few (!) there already, one of which I'd read and thought I understood. Thanks for the correction.

I understood that physicists thought it was odd that electrons don't travel at c, and was part of the case for Higgs field.

If you heard that somewhere, you misunderstood.

In the Standard Model, every particle with nonzero rest mass acquires that rest mass through its interaction with the Higgs field. That includes the electron.

Googling it I still read it the same, my confusion maybe assuming that particles with zero rest mass must travel at c think I've read that on pf and other places many times; with respect to the electron having a rest mass due to higgs field.

Googling it I still read it the same, my confusion maybe assuming that particles with zero rest mass must travel at c think I've read that on pf and other places many times; with respect to the electron having a rest mass due to higgs field.
Yes, you correctly understand that all massless particles travel at c and you correctly understand that electrons have mass due to the Higgs field. So what's the problem?

Googling it I still read it the same

Links, please? It's hard to help disentangle confusion if we can't see the original sources that are causing the confusion.

Links, please? It's hard to help disentangle confusion if we can't see the original sources that are causing the confusion.

I don't have links to where a I can quote a scientist as saying "The electron should move at c, but doesn't because of the higgs mechanism."

Subsequent work showed that the Brout-Englert-Higgs mechanism (or “Higgs mechanism,” for short) could give mass not only to weak particles, but also to electrons, quarks, and other fundamental particles. The more strongly a particle interacts with the Higgs field, the more massive it is. It’s important to note, however, that most of the mass in composite particles, like protons, nuclei, and atoms, does not come from the Higgs mechanism, but from the binding energy that holds these particles together. LINK

I imagine during the course of these studies/theory development or even at the time of determining an electron as a fundamental particle (and all other fundamental particles) or the unification into a new more fundamental electroweak force it became curious why the electron had a mass at all, or the reciprocal or why they don't move at c. idk, this goes well beyond what I can learn from googling stuff.

That said, without the higgs field interaction, would electrons move at c?

without the higgs field interaction, would electrons move at c?

According to our current Standard Model, yes, they would. So would quarks. Protons, nuclei, and atoms might not; see below.

It’s important to note, however, that most of the mass in composite particles, like protons, nuclei, and atoms, does not come from the Higgs mechanism, but from the binding energy that holds these particles together.

Note the key term, "composite particles". According to our current Standard Model, composite particles are "particles whose fields do not appear in the Standard Model Lagrangian". Charts of the "Standard Model particles", such as the one on this Wikipedia page, only include the "elementary" particles (which appear in the Lagrangian), so any particle not on this list (such as protons, nuclei, and atoms, which are all composed of multiple particles from the list) is a "composite" particle.

The point about composite particles is that they are held together by various interactions between the elementary particles that compose them. Protons are held together by the strong interaction between their quarks. Atoms are held together by the electromagnetic interaction between the nucleus and the electrons. These interactions have energy associated with them, and this energy contributes to the mass of the composite particle. Also, the individual elementary particles that make up the composite particle may be moving (quarks move inside protons and neutrons, and electrons move inside atoms--actually the nuclei move too, in the center of mass frame of the atom--by "move" here I mean "move in the CoM frame"--but so slowly that they can usually be considered stationary); and the kinetic energy associated with this motion also contributes to the mass of the composite particle.

The quote you give is actually misleading, because it implies that most of the mass of atoms comes from binding energy instead of the masses of its constituents. That's not true. For an atom, the binding energy between the nucleus and electrons is actually negative--the mass of the atom is less than the mass of the nucleus + electrons. (The difference is very small compared to the total mass of the atom but it's there.) The kinetic energy of the electrons adds some positive mass back, but the overall mass of the atom is still less than the sum of the masses of its constituents.

The statement you quoted is usually only made about strongly interacting particles, like protons. For them, it is true that most of the mass we measure is not due to the Higgs mechanism acting on the quarks; most of it is due to strong interaction energy (which here is positive--it adds to the mass from the Higgs mechanism) and kinetic energy of the quarks. (Technically, this means that most of the mass of an atom does not ultimately come from the Higgs mechanism either, since almost all of an atom's mass comes from the nucleus, so the quote you gave is not, strictly speaking, incorrect; it's just misleading.)

If the Higgs mechanism weren't there, individual electrons and quarks would move at c, but composite particles like nucleons and atoms might still exist and move at speeds less than c, because the interaction energy between the quarks might effectively add invariant mass and make the whole bound system timelike instead of lightlike. I don't know if anyone has ever actually tried to construct a test theory along these lines, though, to see if it's still consistent.

A -
The quote you give is actually misleading, because it implies that most of the mass of atoms comes from binding energy instead of the masses of its constituents. That's not true.

B - The statement you quoted is usually only made about strongly interacting particles, like protons. For them, it is true that most of the mass we measure is not due to the Higgs mechanism acting on the quarks; most of it is due to strong interaction energy (which here is positive--it adds to the mass from the Higgs mechanism) and kinetic energy of the quarks. (Technically, this means that most of the mass of an atom does not ultimately come from the Higgs mechanism either, since almost all of an atom's mass comes from the nucleus, so the quote you gave is not, strictly speaking, incorrect; it's just misleading.)

Thanks for the reply PeterDonis, it was informative.

However I am confused by these A/B comments; they seem to be at odds with A saying B is not true. I understand that an atoms mass is mostly from the binding energies within the nucleus and that by comparison all the other masses of the constituents are quite small.

wiki - The rest masses of the quarks [of a proton] are thought to contribute only about 1% of the proton's mass. The remainder of the proton mass is due to the kinetic energy of the quarks and to the energy of the gluon fields that bind the quarks together.

For the neutron it's the same deal, being made of quarks held together via the strong force.

wiki - The mass of the three quarks sum to only about 12 MeV/c2 ,[from higgs mechanism] whereas the neutron's mass is about 940 MeV/c2, for example.[14] Like the proton, the quarks of the neutron are held together by the strong force, mediated by gluons.[15]

wiki - In quantum chromodynamics, the modern theory of the nuclear force, most of the mass of the proton and the neutron is explained by special relativity. The mass of the proton is about 80–100 times greater than the sum of the rest masses of the quarks that make it up, while the gluons have zero rest mass. The extra energy of the quarks and gluons in a region within a proton, as compared to the rest energy of the quarks alone in the QCD vacuum, accounts for almost 99% of the mass. The rest mass of the proton is, thus, the invariant mass of the system of moving quarks and gluons that make up the particle, and, in such systems, even the energy of massless particles is still measured as part of the rest mass of the system.

The last bold / underlined part, in what form of energy is a gluon moving at c? what ever it is it must be allot considering the force it mediates.

With the strong force being so ridiculously strong and the mediating force for quarks I imagine that is the majority of an atoms energy/mass, and seems to be what wiki says about an atoms most massive constituents (neutron/proton)

I don't know mass/energy equivalence or potential energy well enough to think through things like if the movement of the quarks to and fro each other and the subsequent increasing / decreasing potential energy of the strong force has any impact on what is the mass of an atom/system. I suppose the separation between quarks must be net against what is pushing them apart (their kinetic energy?), as it's all within the same system so null.

Long and short of it is atoms are made of the elementary particles, quarks & electrons. Those particles have mass due to the higgs mechanism, and that mass in comparison to the binding & kinetic energies is quite small; the mass of a few electrons and a dozen or so quarks is very small in comparison to the binding energy between the quarks (strong force). due to mass/energy equivalence, that energy is most of the mass of an atom, next is the kinetic energies and lastly the higgs field "interaction".

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in what form of energy is a gluon moving at c?

Gluons move at c because they are massless, like photons. The energy stored due to the strong force between quarks is probably not best thought of as being carried by individual gluons; it's better to think of a proton, for example, as being composed of three "valence" quarks (two up quarks and one down quark, the way it's usually listed on tables of particles), plus a cloud of virtual quarks and gluons. The cloud of virtual quarks and gluons collectively contains the energy stored due to the strong force, which is most of the measured mass of the proton. Gluons moving at c can still contribute to that mass because an individual gluon doesn't move at c for very long, since it's constantly interacting with the quarks (both valence and virtual quarks); every time it interacts, it can change direction, so the net motion of the cloud of gluons is timelike, not lightlike, i.e., the whole system (the proton) is an ordinary object with an ordinary rest frame, even though an individual gluon is not.

the mass of a few electrons and a dozen or so quarks is very small in comparison to the binding energy between the quarks (strong force). due to mass/energy equivalence, that energy is most of the mass of an atom, next is the kinetic energies and lastly the higgs field "interaction".

The binding energy between quarks is most of the mass, yes. But the interaction between quarks is not what holds the atom together; it's only what holds the nucleus together. What holds the atom together is the interaction between the electrons and the nucleus, and the binding energy of that interaction is (a) very small in magnitude, and (b) negative--the atom's mass is less than the masses of nucleus + electrons individually. So the binding energy that holds the atom together makes a small, negative contribution to the atom's mass. Sorry if I didn't make that distinction clear before.

nitsuj
^^ Thanks PeterDonis, that does clarify it for me.

To Peter Donis, I understand what you are saying, but in reading some of the replies they are saying similar things to me, they just do it in question form.
Is question form acceptable. By that, I mean that I could express a question, doubt, or alternative to look at the question at hand. Right?
I don't plan to express or suggest my non peer tested opinion, I mean to challenge statements that do not take other things into consideration. Science is supposed to be open to this dialogue.

bligh, did you mean to post in another thread? This is the first time you've posted in this one, so I don't understand what you're referring to.

As a general rule, asking questions is fine. "Challenging statements" might not be. Your question should not assume that mainstream physics is wrong. It should assume that you have not properly understood something.

Dale