Time Dilation for Photons: Explained

[NODARman]

TL;DR Summary

.

So, I have a question.

The time dilation formula is:
t = t₀ • 1 / √(1 - v²/c²)

Let's take a photon that travels at c. In my opinion, for a photon "clock doesn't tick" and its life is just a moment.
But when we calculate time dilation by this formula, then c over c is 1 and the root of 1 minus 1 is 0. So we get 1 over 0. We can't divide by zero. So time dilation for a photon should not be either zero or infinite.

How is it?

 

[Dale]

Time dilation is undefined for a pulse of light.

 

[Sagittarius A-Star]

Light in vacuum moves with $c$ in every inertial reference frame. Therefore, a reference frame, in which light in vacuum is at rest, does not exist.

 

[DaveC426913]

Sag's comment is worth repeating.

If one were to try to see what time behaved like from a photon's POV, one would have to be at rest in the reference frame of the photon.
But photons are observed to travel at c in all reference frames.
Which creates a paradox: You cannot have something at rest wrt you and traveling at c wrt you simultaneously.
The result is that there is no such thing as a photon's reference frame, and therefore no time dilation.

 

[PeroK]

Mathematically speaking $\frac 1 0$ is undefined. It's not "infinity".

 

[Mister T]

The time dilation formula is:
t = t₀ • 1 / √(1 - v²/c²)

Valid only when $v<c$.

 

[Orodruin]

Put slightly differently: A photon does not have a ”clock” as a clock is a device measuring the Minkowski path length along timelike world lines.

 

[PeterDonis]

In my opinion, for a photon "clock doesn't tick"

Your opinion is incorrect. The correct statement is that the concepts of "proper time" and "clock ticking" do not even apply to photons. Those concepts only apply to objects that travel on timelike worldlines. Photons travel on null worldlines.

and its life is just a moment.

This is also incorrect. The worldline of a photon is a continuum of distinct events in spacetime, just like the worldline of a timelike object. The fact that the spacetime interval between any two of those events is zero does not mean they are all the same event.

 

[PeterDonis]

In my opinion

Please note that physics is not based on opinions. It is based on models that make accurate predictions.

 

[PeterDonis]

The result is that there is no such thing as a photon's reference frame

Note that we have a FAQ entry on this:

https://www.physicsforums.com/threads/rest-frame-of-a-photon.511170/

 

[Hornbein]

TL;DR Summary: .

So, I have a question.

The time dilation formula is:
t = t₀ • 1 / √(1 - v²/c²)

Let's take a photon that travels at c. In my opinion, for a photon "clock doesn't tick" and its life is just a moment.
But when we calculate time dilation by this formula, then c over c is 1 and the root of 1 minus 1 is 0. So we get 1 over 0. We can't divide by zero. So time dilation for a photon should not be either zero or infinite.

How is it?

Relativity is hyperbolic. So the way I see it, when length gets shorter then time also grows shorter. And indeed it does : if we look at the clock at a passing by space ship we see that less time is measured. Take the rate of passage of time on your clock and multiply it by √(1 - v²/c²) to get the rate of passage of time on the other ship. And like magic your problem disappears.

My understanding of relativity improved when I started to think in terms of time contraction. The result is exactly the same, just a change in perspective and terminology.

Now let's discuss the result. The passage of time for the photon is zero, from emission to absorption. This is why we are able to observe galaxies from 13 billion years ago. Photons don't change at all in flight except for red shift due to the universe's expansion. I say that during the open interval that is their life span, it is legitimate to say (or at least think in the privacy of one's brain) that time does not pass for a photon.

 

[Hornbein]

Relativity is hyperbolic. So the way I see it, when length gets shorter then time also grows shorter. And indeed it does : if we look at the clock at a passing by space ship we see that less time is measured. Take the rate of passage of time on your clock and multiply it by √(1 - v²/c²) to get the rate of passage of time on the other ship. And like magic your problem disappears.

My understanding of relativity improved when I started to think in terms of time contraction. The result is exactly the same, just a change in perspective and terminology.

Now let's discuss the result. The passage of time for the photon is zero, from emission to absorption. This is why we are able to observe galaxies from 13 billion years ago. Photons don't change at all in flight except for red shift due to the universe's expansion. I say that during the open interval that is their life span, it is legitimate to say (or at least think in the privacy of one's brain) that time does not pass for a photon.

In the standard terminology you are calculating the dilation factor for the timeline of the passing spaceship. Timelines are entirely imaginary so they may be stretched to infinity.

 

[PeterDonis]

Now let's discuss the result. The passage of time for the photon is zero, from emission to absorption.

This is not correct. Several previous posts in this thread have explained why.

This is why we are able to observe galaxies from 13 billion years ago.

This is also not correct; we can receive things other than photons from distant objects, that have timelike worldlines instead of null worldlines. Cosmic rays and neutrinos are two examples.

Photons don't change at all in flight except for red shift due to the universe's expansion.

There is a mathematical sense in which this is true. Can you give it?

I say that during the open interval that is their life span, it is legitimate to say (or at least think in the privacy of one's brain) that time does not pass for a photon.

No, it isn't. Again, several previous posts in this thread have explained why.

Timelines are entirely imaginary so they may be stretched to infinity.

I have no idea what you mean by this.

 

[Hornbein]

Timelines are entirely imaginary so they may be stretched to infinity.

I have no idea what you mean by this.

It's an unfortunate conflict of terminology. I'm defining a timeline as the number line I learned about in junior high school. This is an imaginary line labeled with sequential integers or real numbers stretching off to infinity. One may think of time as an arrow moving this number line pointing at the time which is the present. I then call it a timeline. One may calculate the rate at which the arrow moves and call it time contraction. Alternatively, one may think of the arrow moving at a constant rate while the timeline stretches and call that stretch factor time dilation. The result is the same.

Maybe you can think of a name that doesn't conflict with the physicist's use of the word.

 

[PeterDonis]

It's an unfortunate conflict of terminology. I'm defining a timeline as the number line I learned about in junior high school. This is an imaginary line labeled with sequential integers or real numbers stretching off to infinity. One may think of time as an arrow moving this number line pointing at the time which is the present. I then call it a timeline. One may calculate the rate at which the arrow moves and call it time contraction. Alternatively, one may think of the arrow moving at a constant rate while the timeline stretches and call that stretch factor time dilation. The result is the same.

Maybe you can think of a name that doesn't conflict with the physicist's use of the word.

I think you should learn the standard terminology and math of relativity.

Any object has a worldline: a curve that is the path the object takes through spacetime.

For objects with nonzero invariant mass, the worldline is timelike, and arc length along it is the object's proper time. "Time dilation" is the ratio between that proper time and coordinate time in some coordinate chart.

For objects with zero invariant mass, such as light, the worldline is null, and there is no such thing as "proper time" or "time dilation" at all. The concepts simply do not apply to null worldlines.

 

[Orodruin]

Photons don't change at all in flight except for red shift due to the universe's expansion.

This is incorrect. There is nothing particular happening to a photon due to the expansion of the universe. It still follows a lightlike geodesic in spacetime. Wavelength is not an intrinsic property of a photon, its 4-frequency is. Cosmological redshift is an effect that occcurs relative to a particular family of observers.

 

[vanhees71]

One should also note that a photon is not a massless point particle but the Fock state of the electromagnetic field. In GR one has to be particularly careful to describe it ("quantum field theory in curved spacetime").

In GR textbooks there's a kind of "pseudo photon" in use, which however has to be interpreted as the eikonal approximation of classical electrodynamics in curved spacetime. In other words, it's ray optics. When calculating a null geodesics in a given spacetime what you actually calculate is the "light ray" of ray optics and you can calculate wave quantities like the cosmological redshift from it. For more details, see

https://itp.uni-frankfurt.de/~hees/pf-faq/gr-edyn.pdf

 

[ergospherical]

https://itp.uni-frankfurt.de/~hees/pf-faq/gr-edyn.pdf

Nice!