The Lorentz factor gamma and proper time

[Trysse]

TL;DR Summary

Proper time along a light-like curve isn’t undefined; it’s zero. The infinities arise only from
$\gamma=1/\sqrt{1-(v/c)^2}$ for $𝑣=𝑐$, not from the geometry itself. Using $\Delta\tau=\Delta t\cdot\sqrt{1-(v/c)^2}$ avoids these issues.

I’ve been following this recent discussion on the speed of light and its implications in special relativity, and it got me thinking about proper time along light-like curves and the use of the Lorentz factor, $\gamma$.

One point that stood out to me is the claim that proper time along a light-like curve is undefined because $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ becomes infinite when $v = c$. While this is technically correct for $\gamma$, the apparent “undefined” nature of proper time comes from $\gamma$’s structure, not from any fundamental issue with proper time itself.

Proper time is often expressed as
$$\Delta\tau=\Delta t/\gamma$$
it can also be written as
$$\Delta\tau=\Delta t\cdot\sqrt{1-(v/c)^2}$$

If we focus directly on $\sqrt{1 - v^2/c^2}$ rather than $\gamma$, the situation becomes much simpler. For light-like motion ($v = c$), $\sqrt{1 - v^2/c^2} = 0$, and so the proper time $\Delta \tau$ is exactly zero. This result aligns perfectly with the idea that no proper time elapses along a light-like curve, such as the path of an electromagnetic signal in Minkowski spacetime.

The infinities associated with $\gamma$ as $v \to c$ are thus artifacts of its structure, not a reflection of the spacetime geometry itself. By focusing on $\sqrt{1 - v^2/c^2}$, we avoid these complications entirely and gain a clearer understanding of proper time along light-like paths.

 

[Orodruin]

Proper time along the world line of light is irrelevant as it is not an affine parameter of that world line. Precisely because it is a null curve. There is not much more to it than that and talking about it as an actual proper time does not offer new insights.

You can however introduce an affine parameter to the null geodesic. It does not have the interpretation of a time measured by a clock, but it does increase monotonically along the null path and as a result also with the increase of coordinate time of any inertial frame.

Events along the null path are distinct and it becomes disingenuous and misleading to talk about it as “time not passing for a photon”.

 

[PeroK]

TL;DR Summary: Proper time along a light-like curve isn’t undefined; it’s zero.

Okay, let's accept that. How does that change the physics? You want to study the motion of a light ray and you have a parameter $\tau$ that is zero at every point on the light's trajectory. That's no use. So, you have to look for a different way to parameterise the light's trajectory.

 

[Ibix]

Typo: $\Delta\tau=\Delta t/\gamma$, not $\Delta t\cdot\gamma$.

The problem is that proper time is a measure of arc length (or interval) along a timelike curve. That's what actually makes it useful - that you can use it to label different events along a curve without reference to anything off the curve. That doesn't work for a null curve because the interval is zero and you cannot label points that way - you need to do something else. That is, the interval along the curve is indeed zero, but proper time isn't just the interval - it's the concept of regularly spaced tick marks along the curve, which isn't possible this way with null curves.

A more practical way to look at it is to note that if we have a timelike curve and I specify an event on it and any finite proper time later, you know which event I'm referring to. Not so with a null curve.

And the most physical way to look at it is to ask you to think about how you plan to build a clock that moves at light speed. It simply isn't possible to lay out a device that changes in any way, moves at an average speed of $c$, and doesn't have any part of it exceed $c$. So there's no physical meaning to "time" along the curve because you can't (even in principle) make a device to measure it.

 

[Orodruin]

So there's no physical meaning to "time" along the curve because you can't (even in principle) make a device to measure it.

Any device returning a constant will work. That speaks to the entire point of being unphysical though …

 

[A.T.]

Proper time along a light-like curve isn’t undefined; it’s zero.

Okay, let's accept that. How does that change the physics?

If it doesn't change the physics, then what is the point of arguing against it? Since the limit also approaches zero, why not just accept it is zero? Even if it's just for the sake of continuity and has no practical application.

 

[PeroK]

If it doesn't change the physics, then it what is the point of arguing against it? Since the limit also approaches zero, why not just accept it is zero? Even if it's just for the sake of continuity and has no practical application.

I agree it's a matter of taste. The issue is that in the other thread, the OP went ahead and divided by $\Delta \tau$ for a photon and got an infinite speed. This, as I read that thread, was something that needed an explanation. We can say that division by zero is not defined. And then we are back to my mathematical analogy!

 

[Trysse]

Typo: Δτ=Δt/γ, not Δt⋅γ.

Corrected in the OP.

 

[phinds]

If it doesn't change the physics, then what is the point of arguing against it? Since the limit also approaches zero, why not just accept it is zero?

Because a photon / beam of light has no rest frame so how can it even be meaningful to talk about a time associated with it?

 

[Dale]

TL;DR Summary: Proper time along a light-like curve isn’t undefined; it’s zero. The infinities arise only from
$\gamma=1/\sqrt{1-(v/c)^2}$ for $𝑣=𝑐$, not from the geometry itself. Using $\Delta\tau=\Delta t\cdot\sqrt{1-(v/c)^2}$ avoids these issues.

the apparent “undefined” nature of proper time comes from γ’s structure, not from any fundamental issue with proper time itself

No, the fact that proper time is undefined on a null worldline comes from the definition of proper time:

“For timelike paths we define the proper time (eq 1.97) which will be positive”

See Sean Carroll’s Lecture Notes on General Relativity https://preposterousuniverse.com/wp-content/uploads/grnotes-one.pdf equations 1.95 - 1.97 and the associated text.

The proper time is undefined on a null worldline because we want it to be undefined and we have deliberately constructed our definition in such a way as to make it undefined there.

The reason that we have chosen to define it this way is that this definition will make other proofs simpler. This choice is not forced on us by the “nature” of anything, but is chosen for its usefulness.

 

[Nugatory]

If it doesn't change the physics, then it what is the point of arguing against it? Since the limit also approaches zero, why not just accept it is zero? Even if it's just for the sake of continuity and has no practical application.

As long as we stop there it’s harmless, but when this question comes up it’s always because someone hasn’t stopped there: they conclude “time stops” and end up in a sea of confusion because now the photon cannot move, try to do calculations in the rest frame of the photon, or some other silliness. Then they try to calculate a $\gamma$ factor from it….

The statement that $\gamma(v)\rightarrow\inf$ as $v\rightarrow c$ means that we can make $\gamma$ as large as we want by choosing a value of $v$ that is close enough to but still less than $c$. Getting from there to $\gamma(c)=\inf$ is mathematically problematic because $v=c$ isn’t in the domain of $\gamma(v)$; $v$ is the coordinate velocity of an object at rest in its own inertial frame.

Thus I prefer to interpret the infinities and physically meaningless zeroes that appear at $v=c$ as the math telling us that we’re abusing it by trying to evaluate a function outside of its domain.

 

[Dale]

If it doesn't change the physics, then what is the point of arguing against it? Since the limit also approaches zero, why not just accept it is zero? Even if it's just for the sake of continuity and has no practical application.

Because then in every subsequent proof and use of proper time you have to do a test to see if it is zero or not before you can divide by it. That is inconvenient.

Also, it is not clear that continuity would be desirable here. There are several senses in which a null path and a timelike path are fundamentally different and placing the restriction forces you to confront that possibility. In the cases where continuity is warranted then you can use the affine parameter which is well defined for both timelike and lightlike worldlines.

 

[Orodruin]

If it doesn't change the physics, then what is the point of arguing against it? Since the limit also approaches zero, why not just accept it is zero? Even if it's just for the sake of continuity and has no practical application.

Let me introduce you to the Orodruin constant. It is a constant characterised by being able to take any fixed value, but with the important property of being universally constant. It also does not have any predictable consequences depending on its value.

I don’t understand why the Nobel committee has not called yet …

 

[thomsj4]

Consider a timelike worldline parameterized in flat Minkowski spacetime by coordinates $(ct(\lambda), x(\lambda), y(\lambda), z(\lambda))$ with the metric signature chosen as $(+,-,-,-)$. Proper time is defined by integrating the interval $d\tau$ along the worldline, where $d\tau^2 = \frac{1}{c^2}(c^2 dt^2 - dx^2 - dy^2 - dz^2)$. For a massive particle traveling with speed $v < c$, we can write $dx^2 + dy^2 + dz^2 = v^2 dt^2$ and thus obtain
$$d\tau = \sqrt{1 - \frac{v^2}{c^2}} dt.$$
Integrating from some initial time $t_1$ to a final time $t_2$ gives
$$\Delta \tau = \int_{t_1}^{t_2} \sqrt{1 - \frac{v^2}{c^2}} dt.$$
In the special case of a massless particle, we know the path is light-like, so by definition $dx^2 + dy^2 + dz^2 = c^2 dt^2$, which forces
$$\sqrt{1 - \frac{v^2}{c^2}} = \sqrt{1 - 1} = 0.$$
Hence each infinitesimal interval of proper time $d\tau$ is zero, and the total proper time accumulated along that path is
$$\Delta \tau = \int_{t_1}^{t_2} 0 dt = 0.$$
The perceived “infinity” when using $\gamma = 1/\sqrt{1 - v^2/c^2}$ arises only if one tries to treat the light-like case as a limit of the timelike formula for velocities approaching $c$, but this limit artifact does not reflect any genuine geometric singularity. Instead, the interval’s vanishing factor, $\sqrt{1 - v^2/c^2}$, correctly signals that photons and other massless quanta do not experience the passage of proper time, and the geometry remains perfectly well-defined.

 

[Dale]

Proper time is defined by integrating the interval dτ along the worldline,

Proper time is defined by integrating the interval along a timelike worldline. See the Carroll above.

In the special case of a massless particle, we know the path is light-like, so by definition

So by definition the proper time is undefined for a massless particle.

The integral that you write does evaluate to 0 but that integral is not the proper time, which is undefined.

correctly signals that photons and other massless quanta do not experience the passage of proper time

That is already captured in the definition. The problem arises when, instead of correctly saying “massless quanta do not experience the passage of proper time”, people incorrectly say “massless quanta experience the passage of 0 proper time”.

 

[A.T.]

The reason that we have chosen to define it this way is that this definition will make other proofs simpler. This choice is not forced on us by the “nature” of anything, but is chosen for its usefulness.

Convenience is a valid reason. But for some, conceptual continuity might be more important.

 

[Dale]

Convenience is a valid reason. But for some, conceptual continuity might be more important.

I don’t think that the concepts that a different definition would make “continuous” should be made “continuous”.

 

[PeroK]

Convenience is a valid reason. But for some, conceptual continuity might be more important.

We can think about spacetime always in some global (inertial) reference frame where light is the limit of subluminal trajectories. However, all subluminal trajectories have a four-velocity of magnitude $-1$. And light, dare I say, has no four-velocity. Also, subluminal particles have energy and momentum that tend to infinity as $v \to c$. Whereas, light has a finite energy and momentum.

Moreover, we must eventually wean ourselves off the security of a global inertial frame, and consider a coordinate-independent view of physics. In this view, the null wordline of light is in no way the limit of time-like worldlines - which are all equivalent geometrically. In other words, there is no such thing as a massive particle that is closer to light-like than another. Whatever one you choose, it has a rest frame where it is as far from light-like as possible. As geometric objects, the four-momentum of any particle is discontinuous from the four-momentum of light.

In terms of conceptual clarity, I would say it's important to see this discontinuity between time-like and light-like worldlines - the latter are invariant and map out the structure of causality throughout spacetime.

In terms of understanding the geometric, coordinate invariant theory of GR, it's important to move away from the concept of a null worldline being the limit of time-like worldlines.

 

[robphy]

If it doesn't change the physics, then what is the point of arguing against it? Since the limit also approaches zero, why not just accept it is zero? Even if it's just for the sake of continuity and has no practical application.

The following quote from Geroch’s Mathematical Physics (p81-82) seems appropriate… in particular, the comment about “irrelevant things”:

“It is intended that the discussion above make the point that all the structure of (V,g) has physical meaning, that anything one says within this structure can be interpreted as a physical statement, that any theorem about this structure is a physical prediction, etc. There are no "irrelevant things" around. We emphasize, however, that a Minkowski vector space is to be nothing more and nothing less than its mathematical definition above. One must prove things within that framework, providing physical interpretations where they are interesting or useful.”

 

[A.T.]

To me, it's all a bit like insisting that the area of a degenerate triangle is not zero, but undefined. Despite the fact that the area formula gives a perfectly normal numerical result for that special case, motivated by inconvenience in some proofs where you need to divide by the area.

 

[PeterDonis]

To me, it's all a bit like insisting that the area of a degenerate triangle is not zero, but undefined.

The issue is not that the arc length along a null worldline is undefined. It is well defined: it's zero.

The issue is interpreting that arc length as a "proper time". That interpretation is not justified for a null worldline, because it depends on the arc length being an affine parameter along the curve, and for a null worldline, arc length is not an affine parameter. To put it another way, you can't use arc length along a null curve as a clock, and being able to use arc length as a clock is the physical basis for interpreting the arc length as a "proper time".

 

[Dale]

To me, it's all a bit like insisting that the area of a degenerate triangle is not zero, but undefined. Despite the fact that the area formula gives a perfectly normal numerical result for that special case, motivated by inconvenience in some proofs where you need to divide by the area.

I don’t think it is at all similar. We are not just doing math we are doing physics. We want our mathematical quantities to map to our physical measurements.

Any of your standard mechanisms for physically measuring areas will give a valid 0 result in the case of the triangle. So calling it 0 area is reasonable.

The proper time is mapped to the measurement of a clock. A clock that measures 0 proper time is only ever measuring a single event, never a pair of null separated events. There is no “conceptual continuity” between a clock measuring 0 time and the interval between distinct events on a null worldline.

If we used the word “proper time” for any spacetime interval, we would still need another word to specifically refer to the interval on a timelike worldline, since that is a physically useful concept that can be measured with a clock. And once we have that word then we are back here again.

We use the definition as it is because it is a useful concept. Changing the word to refer to a broader concept wouldn’t change the importance of the narrower one.

Plus, the broader concept already has a word: the spacetime interval. Why does it need two?

 

[PeroK]

To me, it's all a bit like insisting that the area of a degenerate triangle is not zero, but undefined. Despite the fact that the area formula gives a perfectly normal numerical result for that special case, motivated by inconvenience in some proofs where you need to divide by the area.

This is mathematically untenable. A point or line is not a triangle. You are free to develop your own idiosyncratic mathematics and use a different definition of a triangle than everyone else. But, that's what it is - your own mathematics that will have its own definitions and theorems.

In Euclidean geometry the sum of the internal angles on a triangle is 180 degrees. There's no possibility in standard mathematics for a triangle to have no internal angles.