Why is proper time undefined for spacelike/lightlike paths?

[Frank Castle]

As I understand it, the proper time, $\tau$, between to events in spacetime is defined in terms of the spacetime interval $ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$, such that $$d\tau =\sqrt{-ds^{2}}$$ (where we are using the "mostly +" signature with $c=1$).

Now, for time-like intervals, for which $ds^{2}<0$, it is clear that proper time is well-defined since the quantity $\sqrt{-ds^{2}}$ is positive, and furthermore, one can always find a frame in which the two events occur at the same point in space, such that one can construct a worldline connecting the two events, along which an observer can travel, at rest with respect to both events, such that $d\tau =\sqrt{-ds^{2}}=dt$.

However, why is it the case that for space-like, $ds^{2}>0$, and light-like intervals, $ds^{2}=0$, the notion of proper time is undefined (or perhaps ill-defined)?

For the space-like case, I get that heuristically, one cannot construct a path between the two events along which an observer can travel and so in this sense proper time is meaningless, since a worldline connecting the events does not exist and so no clock can pass through both events. However, can this be seen purely by examining the definition of proper time in terms of the spacetime interval? Is it simply that the quantity $\sqrt{-ds^{2}}$ will become imaginary and so clearly cannot be used to represent any physical time interval?

Likewise, for a light-like interval, only a beam of light can pass between both events and since there is no rest frame for light one cannot construct a frame in which a clock is at rest with respect to the beam and passes through both events. However, purely in terms of the spacetime interval, is it simply because the quantity $\sqrt{-ds^{2}}$ equals $0$, and so the notion of proper time is ill-defined since there is no invertible map between reference frames (here I'm thinking in terms of time dilation, $t =\gamma\tau$ and so for a light-like interval, $\gamma\rightarrow\infty$ meaning that the inverse relation $\tau =\frac{t}{\gamma}$ is ill-defined)?!

 

[Maxila]

I'm trying to understand exactly what you are asking because you appear to answer the question when you say:

one cannot construct a path between the two events along which an observer can travel and so in this sense proper time is meaningless

only a beam of light can pass between both events and since there is no rest frame for light one cannot construct a frame in which a clock is at rest with respect to the beam and passes through both events

Since you already know physically a time could not pass for a light like or space like interval, what are you trying to ask?

 

[PeterDonis]

why is it the case that for space-like, $ds^{2}>0$, and light-like intervals, $ds^{2}=0$, the notion of proper time is undefined (or perhaps ill-defined)?

Because it's only defined for timelike intervals.

As Maxila said, I think you need to clarify exactly what you are asking. The answer to your question as you ask it is trivial, as above.

 

[Frank Castle]

Since you already know physically a time could not pass for a light like or space like interval, what are you trying to ask?

As Maxila said, I think you need to clarify exactly what you are asking. The answer to your question as you ask it is trivial, as above.

I was wondering if one can argue directly from the definition of proper time in terms of the metric, i.e. $$d\tau =\sqrt{-ds^{2}}$$ that proper time is undefined for space-like and light-like (null) intervals?

For space-like intervals we have that $ds^{2}>0$ and so the quantity $\sqrt{-ds^{2}}$ becomes imaginary and therefore cannot be used to describe any physical passage of time, hence proper time for space-like intervals is undefined. However, proper distance $dl^{2}=\sqrt{ds^{2}}$ makes sense for a space-like interval, since $\sqrt{ds^{2}}>0$ and therefore can be used to describe a physical distance. Hence one can parametrise space-like paths with proper distance, but not proper time.

For time-like intervals we have that $ds^{2}=0$, however, I can't see how one can argue that proper time for a null interval is undefined directly from this?!

I have seen people claim that "photons experience no proper time", however, this doesn't sound right to me, since a photon does not have a rest frame (which follows directly from the postulate that light travels at the speed of light $c$ for all observers, hence, there cannot exist any frame in which light travels at less that $c$, ergo there cannot exist a rest frame for photons). Consequently, one cannot construct a reference frame in which a clock is at rest with respect to a photon along its worldline and hence proper time simply cannot be defined for null spacetime intervals.

 

[PeterDonis]

I was wondering if one can argue directly from the definition of proper time in terms of the metric, i.e.

$$
d\tau =\sqrt{-ds^{2}}
$$

that proper time is undefined for space-like and light-like (null) intervals?

The definition you give only applies to timelike intervals, by definition. The interval $ds^2$ is obviously well-defined for all three cases (timelike, null, and spacelike), but the term "proper time" is only used to refer to it, by definition, for the timelike case.

You need to take a step back and think carefully about what you are asking. So far you have only asked about terminology, not physics. "Proper time" is an ordinary language term, and what kinds of intervals it applies to is a question about words, not physics. What question about physics are you asking?

 

[Nugatory]

You are making an easy question harder than it needs to be.

The convention is that we use the phrase "proper time" to describe timelike spacetime intervals, so the answer to the question in the title is "because that's what the convention is". You could as usefully ask why male horses aren't mares.

 

[Maxila]

However, proper distance dl2=√ds2dl2=ds2dl^{2}=\sqrt{ds^{2}} makes sense for a space-like interval, since √ds2>0ds2>0\sqrt{ds^{2}}>0 and therefore can be used to describe a physical distance. Hence one can parametrise space-like paths with proper distance, but not proper time.

Again I think you answer your own question, possibly without realizing it? Yes, we can extrapolate physics for a space like interval; however those physics cannot affect two events with such an interval since nothing can travel faster than light, including gravity. In other words, how could you attribute a time between two events where no observation or physical interaction is possible. It would be infinite which is nonsensical in terms of a physical time.

 

[Frank Castle]

The definition you give only applies to timelike intervals, by definition. The interval $ds^2$ is obviously well-defined for all three cases (timelike, null, and spacelike), but the term "proper time" is only used to refer to it, by definition, for the timelike case.

You need to take a step back and think carefully about what you are asking. So far you have only asked about terminology, not physics. "Proper time" is an ordinary language term, and what kinds of intervals it applies to is a question about words, not physics. What question about physics are you asking?

You are making an easy question harder than it needs to be.

The convention is that we use the phrase "proper time" to describe timelike spacetime intervals, so the answer to the question in the title is "because that's what the convention is". You could as usefully ask why male horses aren't mares.

Again I think you answer your own question, possibly without realizing it? Yes, we can extrapolate physics for a space like interval; however those physics cannot affect two events with such an interval since nothing can travel faster than light, including gravity. In other words, how could you attribute a time between two events where no observation or physical interaction is possible. It would be infinite which is nonsensical in terms of a physical time.

I think I may be over thinking things. Apologies for going round in circles a bit.

Is there an intuitive reason for why proper-time is defined in terms of the space-time interval as $d\tau =\sqrt{-ds^{2}}$? I get that it is in part from the requirement that it is a Lorentz invariant quantity, but does the definition also follow from the fact that the proper-time of an object is the time as measured in the rest frame of that object, and so the corresponding space-time interval in this frame is given by $ds^{2}=-d\tau^{2}$.

Why can't one define a notion of proper time for null space-time intervals? Is it simply because there is no rest frame for particles traveling at the speed of light?

 

[jbriggs444]

Why can't one define a notion of proper time for null space-time intervals?

You can define whatever you like in any way you like. No meaning is obliged to flow from the resulting definition. Others have asked whether there is a physical question here rather than a question about terminology and definitions.

 

[Frank Castle]

You can define whatever you like in any way you like. No meaning is obliged to flow from the resulting definition. Others have asked whether there is a physical question here rather than a question about terminology and definitions.

It's just that I've had people tell me before that "time stands still" for photons since the space-time interval is null in this case. To make this statement they are obviously assuming the definition of proper time in terms of the space-time interval (since then $d\tau =0$), which then conflicts with the definition being in terms of timelike intervals. Physically, is it even meaningful to quantify a "proper-time" (a photons "own time") for null intervals?

 

[PeterDonis]

I think I may be over thinking things

It's not so much that you're over-thinking as that you're thinking about words instead of physics. Again: what physics are you asking about? If you think the term "proper time" implies some particular physics, what?

 

[PeterDonis]

It's just that I've had people tell me before that "time stands still" for photons

Yes, and they are wrong.

Physically, is it even meaningful to quantify a "proper-time" (a photons "own time") for null intervals?

This is still a question about words, not physics; but I think I see at least one question about physics that you are groping towards. Reading this forum FAQ might help:

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

 

[Frank Castle]

It's not so much that you're over-thinking as that you're thinking about words instead of physics. Again: what physics are you asking about? If you think the term "proper time" implies some particular physics, what?

Up until now I've thought of proper time of an object as a physical quantity, measuring the elapsed time in the objects rest frame, or in other words, it is the the time measured by a clock that travels along the worldline of the object.

Yes, and they are wrong.

I get that no rest frame exists for a particle traveling along a null path, so given the physical interpretation I have phrased in the sentence above for proper time, is this the reason why it is not meaningful to define such a quantity, since it is impossible for a clock to measure time along the worldline of a photon?!

 

[PeterDonis]

Up until now I've thought of proper time of an object as a physical quantity, measuring the elapsed time in the objects rest frame, or in other words, it is the the time measured by a clock that travels along the worldline of the object.

Yes, and this is the same as the definition you gave earlier, since in the mathematical model the arc length along a timelike curve is what represents this physical quantity.

is this the reason why it is not meaningful to define such a quantity

What quantity? It's certainly meaningful to define the arc length along a null curve; it's just that the definition (the interval) gives a result of zero, regardless of which points on the null curve we pick. This fact makes "proper time" an unsuitable term for describing such an arc length, but that's still a matter of choice of words, not physics.

it is impossible for a clock to make time along the worldline of a photon?!

This is getting closer to the physics. A more precise statement would be that, in order to build a clock, you need more than just one light beam; in the simplest kind of clock, a light clock, you also need a pair of mirrors, traveling on timelike worldlines. (There are more complicated ways to construct a series of timelike separated points by looking at the intersections of light rays traveling in different directions, but we'll leave that aside here.)

But the physical difference between timelike intervals and null intervals involves more than just clocks. Objects traveling on timelike worldlines can have different relative speeds in different frames. Objects traveling on null worldlines have the same speed in all frames; but that doesn't mean they're unaffected--light rays, for example, change their frequency/wavelength (relativistic Doppler shift). So changing frames (Lorentz transformation) has fundamentally different effects on the two kinds of objects. (And spacelike intervals are different from both: a Lorentz transformation can change a spacelike interval from "future-directed" to "past-directed"--or even make it neither, make the two events it connects simultaneous--but it can't do that to timelike or null intervals.)

 

[Frank Castle]

Is this why one can't physically measure time along a null geodesic since even the most basic of clocks requires time to be measured along a timelike geodesic?!

Would what be the correct argument for why the statement that a photon "experiences no time" is incorrect?

 

[PeterDonis]

Is this why one can't physically measure time along a null geodesic since even the most basic of clocks requires time to be measured along a timelike geodesic?!

Would what be the correct argument for why the statement that a photon "experiences no time" is incorrect?

I think these questions have already been answered. I have described the physics.

 

[Mister T]

Would what be the correct argument for why the statement that a photon "experiences no time" is incorrect?

You already answered this when you said that a clock can't co-move with a photon.

The notion that photons experience no time is poorly stated, but it does serve an explanatory purpose. For example, before neutrino oscillations were confirmed experimentally it was thought that perhaps neutrinos are massless and hence travel at speed c. One great mystery was that of the missing solar neutrinos. Detectors pick up only a fraction of what should be there, given the proposed reactions that produce the known temperature and energy output of the sun. In other words, the nuclear reactions needed to account for the sun's temperature and energy output should produce more electron neutrinos than we detect. If neutrinos are massless and travel at speed c then there's no way for them to change, on their journey from the sun to Earth, from one type to another because they can't, loosely speaking, experience time.

That is at best a loose way to put it, and at worst a misleading and erroneous way to put it. I've heard it stated that way, though, by a Nobel laureate speaking to a group of physicists, with of course the added caveat that it's "loose". I suppose a correct way to state it is that if neutrinos are massless they travel on null geodesics and hence can't oscillate. We now know that they are massive, they do oscillate, and their speed is less than c. But they don't have a mass eigenstate.

 

[SiennaTheGr8]

You could just as well ask why we don't define the proper distance in terms of the spacetime interval.

For entirely practical reasons, we define proper time as the time measured by a co-moving inertial observer. Then we find that $ds = \sqrt{|(c \, d \tau)^2|}$ for timelike intervals.

Likewise, we define the proper distance $\sigma$ between two events as their spatial separation as measured by an inertial observer for whom they occur simultaneously. That leads to $ds = \sqrt{|(d \sigma)^2|}$ for spacelike intervals.

What if we dropped the "for [timelike/spacelike] intervals" caveats? Well, then we'd basically have three different names for the same thing. What's the point? Proper time and proper distance are useful concepts only insofar as they aren't redundant with each other or with the spacetime interval.