# Coordinate time between spatially separated events in Schwarzschild spacetime

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• physlosopher
In summary, the conversation revolves around the physical meaning of coordinate time between two spatially separated events, particularly in the context of a thought experiment involving a black hole. The speaker is seeking clarification on the relationship between coordinate time and proper time, and how they correspond to each other in this scenario. They also question the validity of using coordinate time to measure physical time intervals in this case. The expert summarizer suggests that a difference in coordinate time has no physical significance and that further discussion may not be necessary.
physlosopher
Edit: I'm leaving the original post as is, but after discussion I'm not confused over coordinate time having a physical meaning. I was confused over a particular use of a coordinate time difference to solve a problem, in which a particular coordinate time interval for a particular choice of coordinates is taken to have a physical meaning that I found confusing and problematic.

I'm reading through Collier's A Most Incomprehensible Thing, and am getting tripped up thinking through the physical meaning of coordinate time between two spatially separated events. Any help would be much appreciated!

In the chapter on black holes, Collier asks what it would be like to watch something fall into a black hole from far away. To do this he uses the Schwarzschild metric to calculate the coordinate time separating two events: a signal emitted at ##r_{1}## and the signal being received by an observer at ##r_{2}##, where ##r## is the radial Schwarzschild coordinate, not a proper distance. He considers a signal along a radial line where ##\theta = \phi = 0##, so that the interval is defined by $$ds^2 = (1 - \frac{R_{s}}{r})c^2dt^2 - \frac{dr^2}{1 - \frac{R_{s}}{r}},$$ ##R_{s}## being the Schwarzschild radius, which corresponds to the event horizon. For a beam of light ##ds = 0##, so this reduces to $$0 = (1 - \frac{R_{s}}{r})c^2dt^2 - \frac{dr^2}{1 - \frac{R_{s}}{r}}.$$ Integrating leads to the relationship $$t_{2} - t_{1} = \Delta t = \frac{r_{2}-r_{1}}{c} + \frac{R_{s}}{c} \ln(\frac{r_{2}-R_{s}}{r_{1}-R_{s}}),$$ which gives the coordinate time separating the event at ##r_{1}## and the event at ##r_{2}##.

Collier let's ##r_{1}## approach ##R_{s}## to show that the coordinate time separating the emission of light at the event horizon of a black hole and the reception of that signal far away is infinite.

I understand that this indicates that a signal sent from the event horizon will never reach a point far away (nor, the math seems to suggest, any point with ##r_{2} > r_{1}##?). But I'm having trouble wrapping my head around the physical meaning of a coordinate time difference between two events that are separated in space. Am I correct in thinking that even though for points far from ##r = 0## the proper time will agree with coordinate time, here the coordinate time difference does not correspond to a proper time difference far away?

My reasoning is that it doesn't seem to make physical sense to think about a proper time difference between an event at ##r_{1}## and an event at ##r_{2}## for ##r_{1} \neq r_{2}## for an observer sitting at ##r_{2}##. If we want to talk about an actual physical period of time between these events, since they're spatially separated, wouldn't we need to talk about the interval of proper time on a world line that connects them? And then isn't the argument that ##\Delta t \to \infty## subtly different than "it takes the faraway observer an infinitely long time to receive the signal"? Collier writes "if we can find the coordinate time taken for a light signal, or for a change in position of the freely falling object, we have automatically found the distant observer's proper time measurement for those two events." But I'm having trouble squaring this with the way I'm thinking about the problem. Any advice?

I'm thinking that one sort of remedy might be to modify the thought experiment a bit so that I'm thinking about events that aren't spatially separated. For example, I could look for the coordinate time difference between an object in free fall emitting a photon near the event horizon and then emitting one at the event horizon. But because coordinate time corresponds to proper time far away, wouldn't this coordinate time difference be the same as the proper time difference between the faraway observer receiving each of those signals?

Thanks in advance for any assistance!

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physlosopher said:
But I'm having trouble wrapping my head around the physical meaning of a coordinate time difference between two events that are separated in space.
That’s easy - a difference in coordinate time has no physical significance, so you can stop trying.

Pencilvester
Nugatory said:
That’s easy - a difference in coordinate time has no physical significance, so you can stop trying.

Surely there are at least cases in which coordinate time can inform us about physical reality, or else it wouldn't be useful to talk about it? Can we think about something like a map between it and proper time?

If so, the question I posed is about a specific case of that: the text I'm reading seemed to imply that a particular difference in coordinate time corresponded to a difference in proper time as experienced by a particular observer, and I didn't follow the logic of the text's argument.

physlosopher said:
Surely there are at least cases in which coordinate time can inform us about physical reality, or else it wouldn't be useful to talk about it? Can we think about something like a map between it and proper time?
There are cases where you can use it to compute and deduce things, yes. However, that does not mean that it holds any particular physical relevance. Ultimately, coordinate time is, well, coordinate dependent.

It also strikes me that you do not seem to grasp what proper time is. Proper time is the time elapsed along a particular world-line. It can be different for different world-lines even if they both start and end at proper times ##t_1## and ##t_2## respectively - even if they start and end at the same event.

vanhees71 and physlosopher
physlosopher said:
If we want to talk about an actual physical period of time between these events, since they're spatially separated, wouldn't we need to talk about the interval of proper time on a world line that connects them?
This is true. But I think the point Collier is making is that if there exists a finite coordinate time that the light reaches the distant observer, then depending on what that observer calls ##t=0## then you could find what time he also assigns to receiving the light.

physlosopher
Orodruin said:
It also strikes me that you do not seem to grasp what proper time is. Proper time is the time elapsed along a particular world-line. It can be different for different world-lines even if they both start and end at proper times ##t_1## and ##t_2## respectively - even if they start and end at the same event.

I understand that, my apologies if I wrote something that was unclear. This is why I was confused when the text I'm reading seemed to imply that the coordinate time difference I described, between events that are spatially separated, could be understood to correspond to some proper time experienced by a faraway observer. I mentioned that a proper time between those events would be defined on a world-line connecting the events, rather than being some globally defined concept by which we can talk about the time elapsed for an observer at ##r_{2}## between events that didn't both happen at ##r_{2}##. The fact that proper time is the time elapsed on a particular world-line is the basis of my question and of my confusion over the way I'm understanding what I've read. My apologies if that wasn't clear.

A clock that measures coordinate time in Schwarzschild coordinates in Schwarzschild spacetime is easy enough to make. It needs to have a rocket so that it hovers at constant ##r## and every time it ticks one second it needs to show ##(1-R_s/r)^{-1/2}## seconds elapsed. Also you need to zero clocks at different altitudes - you can do that by exchanging light signals in this case.

You can then measure the coordinate time difference between two spacelike separated events by noting the times on a coordinate clock at each event and taking the difference. However, my first paragraph should show you that the process of "synchronising" the clocks is pretty arbitrary, so the result depends on choices you made and hence doesn't really mean anything. It also shows you that you can't define time coordinates this way at the horizon - primarily because the rocket on the clock would have to be infinitely powerful, but also because the time multiplier is infinite.

Also, you can see that the time multiplier goes to one as ##r## goes to infinity. That means that there is (in the limit) a one-to-one relationship between a distant clock's proper time and the local coordinate time. But there's no relationship between that time and a "coordinate clock" closer to the black hole, except through the synchronisation process that I outlined. Or any other synchronisation process you prefer.

cianfa72, Jimster41, vanhees71 and 1 other person
Pencilvester said:
This is true. But I think the point Collier is making is that if there exists a finite coordinate time that the light reaches the distant observer, then depending on what that observer calls ##t=0## then you could find what time he also assigns to receiving the light.

Right, and that would have nothing to do with that observer assigning physical meaning to the difference in coordinate time between the light being emitted and the light being received, right?

I think this is the crux of the question I'm trying to ask. You could use the coordinate time difference between those events to deduce when the observer should receive the signal, but that difference does not correspond to an actual physical difference in time, such as "the time between the light being emitted and the light being received." Correct? That physical difference (i.e. in some proper time along some corresponding world-line) seems to be what the text implies, which is what threw me for a loop.

To be as clear as possible, the exact phrasing of the text is: "We saw when looking at gravitational time dilation that the coordinate time ##t## of an event in Schwarzschild spacetime is the same as the proper time ##\tau## measured by a stationary distant observer, i.e. ##d\tau_{\infty} = dt##. This means that if we can find the coordinate time taken for a light signal, or for a change in position of the freely falling object, we have automatically found the distant observer's proper time measurement for those two events." Emphasis is mine, to highlight exactly what's causing confusion.

"The coordinate time ##t## of an event in Schwarzschild spacetime is the same as the proper time ##\tau## measured by a stationary distant observer" actually seems fine to me as long as you're strictly talking about events happening infinitely far from ##r = 0##. Unless I'm seriously misreading, the text seems to argue, en route to the claim that ##t_{2}- t_{1}## can be understood as a proper time as measured by the observer receiving the signal, that the coordinate time ##t_{1}## at which the signal is emitted corresponds to the proper time at which it is emitted for the observer who is located somewhere else. The text also states "Because coordinate time is the same as a distant observer's proper time, such an observer will never quite see the object reach the event horizon." Maybe he means to imply only that ##t_{2}##, the time at which the observer receives the signal, can be understood as equivalent to a proper time on a clock located infinitely far away? In fact the more I reread what's written, the more I think this must be what he means.

To everyone, sorry about getting hung up on "physical significance." The text I'm reading seems to imply that in the thought experiment coordinate time is identical to a proper time in a circumstance where I don't understand that as being possible. Regardless of how we philosophically understand "physically significant," my problem was that I couldn't square that implication with the definition of proper time.

Ibix said:
That means that there is (in the limit) a one-to-one relationship between a distant clock's proper time and the local coordinate time. But there's no relationship between that time and a "coordinate clock" closer to the black hole, except through the synchronisation process that I outlined.

Fantastic, this is what I was looking for as a sanity check - thank you.

As I suggested in my last post, I may very possibly just be misreading the text, but regardless I'm glad I'm not totally off base about my (possibly mistaken) interpretation of it being problematic.

physlosopher said:
The text I'm reading seems to imply that in the thought experiment coordinate time is identical to a proper time in a circumstance where I don't understand that as being possible.
For a distant observer at rest with respect to the black hole and using Schwarzschild coordinates, the coordinate time between event A, on the observer's worldline and declared to be simultaneous with the event at ##r_1##, and event B, also on the observer's worldline and declared to be simultaneous with the event at ##r_2##, is equal to the observer's elapsed proper time.

This tells you a fact about Schwarzschild coordinates: far from the black hole they are based on the "natural" coordinates of an inertial observer at rest with respect to the black hole. It doesn't really tell you anything about the physics.

cianfa72, Nugatory, Pencilvester and 1 other person
physlosopher said:
Surely there are at least cases in which coordinate time can inform us about physical reality, or else it wouldn't be useful to talk about it? Can we think about something like a map between it and proper time?
Yes, and indeed that is how we use it. Coordinate time has no physical significance because the relationship between coordinate time and proper time (which does have physical significance) is pretty much completely arbitrary; as long as we respect certain mathematical niceties, we can choose just about any rule for associating time coordinate values with events.

This freedom to choose a mapping allows us to choose a mapping that works well in whatever problem we’re working with at the moment; in practice we generally choose a mapping to make it easy to do exactly that.
For example, if I am floating in empty space it is very natural (so natural that we usually don’t stop to consider that I have other options and am making a choice) to consider myself to be at rest in space and choose to assign time coordinates in such a way that if light from an event at a distance DD away from me reaches my eyes at time TT, then the time coordinate of that event is T−(D/c)T−(D/c). Using this convention, I can easily calculate the proper time between two events on a distant spaceship: if the spaceship is at rest relative to me the proper time between the events is equal to the difference in the time coordinates I have assigned to them, and if it is moving relative to me the proper time will be that difference divided by √1−v2/c21−v2/c2. But this doesn’t mean that the coordinate times have any physical significance; it means that I’ve chosen them in a way that makes it easy to calculate the proper time between events on the distant spaceship.

Of course for this problem it would be perverse of me to choose any other rule for assigning time coordinates, but that’s just because this is an unusually simple problem with an obvious convention right under our noses. Your more complicated black hole example isn’t that way.

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physlosopher
physlosopher said:
physical meaning of coordinate time
In general there is no physical meaning of coordinate time.

The only exception is if the meaning is built into a specific coordinate chart. Then that information is contained in the derivation or the description of the coordinates.

physlosopher
How would you all propose interpreting a claim like "we know that in Schwarzschild spacetime, coordinate time ##t## is the same as proper time ##\tau## as measured by a distant stationary observer."?

This is stated shortly after the other quote I mentioned above. It's the sort of claim that's giving me trouble, because it doesn't seem true unless we're talking specifically about an event on the world-line of an observer who is at rest very far from the central mass. Can we choose a convention that makes it "true" generally, similar to what Nugatory suggested earlier?

Nugatory said:
For example, if I am floating in empty space it is very natural (so natural that we usually don’t stop to consider that I have other options and am making a choice) to consider myself to be at rest in space and choose to assign time coordinates in such a way that if light from an event at a distance ##D## away from me reaches my eyes at time ##T##, then the time coordinate of that event is ##T−(D/c)##. Using this convention, I can easily calculate the proper time between two events on a distant spaceship

Can we use a similar conventional tactic to say something meaningful about the time, for the observer, at which an event near the black hole occurs? And then would Collier's claim above be made true by convention?

physlosopher said:
How would you all propose interpreting a claim like "we know that in Schwarzschild spacetime, coordinate time ttt is the same as proper time ττ\tau as measured by a distant stationary observer."?
I would interpret it as wrong. The coordinate time is defined throughout spacetime (except at the horizon). The proper time of a distant stationary observer is only defined along the worldline of the distant stationary observer. Since they are defined over different regions they are clearly not the same, although where both are defined they do agree.

PeterDonis and physlosopher
Dale said:
Since they are defined over different regions they are clearly not the same, although where both are defined they do agree.

Thank you. That's what I thought. It was starting to drive me insane.

kent davidge
Dale said:
In general there is no physical meaning of coordinate time.

The only exception is if the meaning is built into a specific coordinate chart. Then that information is contained in the derivation or the description of the coordinates.
I believe the point to highlight is that Schwarzschild coordinates in Schwarzschild spacetime define themselves, in a sense, the procedure (or the mapping) to assign spacetime coordinates ##r, \theta, \phi, t ## to events. In other words they define a specific coordinate chart in which the metric is the well known Schwarzschild metric.

Dale
cianfa72 said:
In other words they define a specific coordinate chart in which the metric is the well known Schwarzschild metric.
The coordinates do not define the metric, they just choose the way you end up expressing the metric. The thing about Schwarzschild coordinates is that they reflect the symmetries of the spacetime and have a fairly straightforward interpretation for hovering observers.

Ibix said:
The coordinates do not define the metric, they just choose the way you end up expressing the metric. The thing about Schwarzschild coordinates is that they reflect the symmetries of the spacetime and have a fairly straightforward interpretation for hovering observers.
ok sure; my point was that, as far as i can understand, Schwarzschild coordinates - being a coordinate chart - define let me say the 'thought procedure' to use to assign coordinates to events, don't you?

I'd say it's the other way around - you come up with a methodology for assigning labels to events and your coordinate system is its mathematical expression. But it's a bit chicken-and-egg.

The metric is definitely a distinct entity.

cianfa72
Cool discussion. I almost think I understood it.

Is it correct to say difference in metric is what was portrayed in that scene in Interstellar when the guy who'd been up on the mother ship opens the door and is like, "You were gone for a lot of years" and we were all like... "wow! really didn't seem like it" In other words the metric refers to the above mentioned "symmetries of spacetime" that "cause" (for lack of a better word) proper time?

The part I don't get is how proper-time can "vary" (for lack of a better word) due to curvature or acceleration but we don't really feel it, our local clocks don't register it. I mean it's doing something physically and that something is being altered physically but the laws of physics feel totally the same the whole time. I mean how does it manage that?

The metric is the mathematical tool that describes distances and times between nearby events. It does lead to the notion of "elapsed time" for a person, but the reason people can experience different amounts of elapsed time between events is that they can take different routes through spacetime that have different elapsed times.

So no, it's not "difference in the metric". It's just taking different routes through spacetime. It's closely analogous to the fact that if we zero our odometers in Washington and then meet up in New York, our odometers may not agree.

Jimster41
@Ibix I appreciated your explanations above btw.

I get that analogy but comparing an odometer to the the aging of a human being falls a bit short for me in capturing how cool that scene was (and meant to be accurate, at least I thought?)

I get that there is nothing in the entire process of chemistry that you can point to as you head down to the planet nearer to the black hole and say - "that's different see, buddy on the spaceship is getting old so much faster than me now"

though I am puzzled about "frame dragging". In general, if acceleration - the effect of interacting with our inertia (rather than just being inert) is the only way to tell if you are "driving less far in your car" (to extend the analogy of the odometer) then why don't we run around changing directions all the time you know jumping up and down (to maximize our exposure to the fictional forces due to acceleration of course) so we don't get old?

I for one do the opposite. I drag my frame more... literally.

Not really trying to be goofy. I just don't get how it is something as physical as Proper time can be changed without some mechanism that we can point to that is changing it... though I guess Einstein's ten equations would describe it, right?

Jimster41 said:
I get that analogy but comparing an odometer to the the aging of a human being falls a bit short for me in capturing how cool that scene was
I think @Ibix was going more for simplicity and clarity in answering your question over coolness.

why don't we run around changing directions all the time you know jumping up and down ... so we don't get old?
You will still get old, it’s just that everyone else will be aged by a couple nanoseconds more than you at the end of your life.

Dale
Jimster41 said:
I get that analogy but comparing an odometer to the the aging of a human being falls a bit short for me in capturing how cool that scene was (and meant to be accurate, at least I thought?)
Can't help you with drama, but the point is that aging is just a rather crude clock. And clocks turn out to measure "distance" through spacetime in much the same sense odometers measure distance through space. I think that's fascinating - but it does reduce all aging "paradoxes" to the rather prosaic "they took routes of different 'lengths'".
Jimster41 said:
though I am puzzled about "frame dragging". In general, if acceleration, the effect of interacting with our intertia (rather than just being inert), is the only way to tell if you are "driving less far in your car" to extend the analogy of the odometer then why don't we run around in changing directions all the time jumping up and down (to maximize our exposure to the fictional forces due to
I don't think you mean frame dragging. You seem to be talking about the twin paradox. You do, in fact, experience less time if you jump up and down compared to staying sat in one place - but the ratio of experienced times is ##\sqrt{1-v^2/c^2}## which, for the kind of speeds you can generate, is about one part in a million billion different from 1. So you gain a milluonth of a nanosecond for each second you jump up and down - more than offset by the difficulties of doing anything while bouncing up and down.

Jimster41 said:
I just don't get how it is something as physical as Proper time can be changed without some mechanism that we can point to that is changing it
In the @Ibix analogy what would be the mechanism you can point to that is changing the odometer readings?

Ibix said:
So you gain a milluonth of a nanosecond for each second you jump up and down - more than offset by the difficulties of doing anything while bouncing up and down.

Yes that probably explains why I make the choices I do. And I love your clarity.

I'm not sure I mean frame-dragging I was trying to see if it was obviously relevant to folks who know and yes I am talking about the twin paradox.

I get the "cars" are rolling down geodesics of different length, measured only by differences in proper (elapsed) time. Other than that abstraction I don't know what you can point to. Is there something you would say? I was imaging there is nothing we know of. I am looking to understand what the state of the theory is.

I am struggling to understand what the physical line (the description of causal connection) is between acceleration or exposure to gravitational curvature and the effect on the twin (biochemical aging)? It's just bugging me.

My understanding (probably pretty wrong) is that Yang-Mills symmetry groups (the standard model) work because you re-gauge the physics as you accelerate

So, my somewhat cheeky guess (since you asked!) is that the ghost field monkeys with the structure constants of the Lie algebra through the curvature or field strength form, the $F$ terms in the Yang-Mills summary expression below... somehow. Maybe it torques spin (like all spin)... as a Geometrical Berry Phase?

$$\mathcal L _\mathrm{gf} = -\frac{1}{2}\operatorname{Tr}(F^2)=- \frac{1}{4}F^{a\mu \nu} F_{\mu \nu}^a$$

https://en.wikipedia.org/wiki/Yang–Mills_theory

I'm dying to understand this every since I read "Deep Down Things" a book that if it did anything good (I loved it) at least gave me an appreciation of Yang-Mills genius and some sense of what the symmetry groups do - and that Yang-Mills and other's insights were significantly related to trying to reconcile QM with GR.

Now maybe it seems bizarre to suggest there is some causal path up from GR-QM to biochemical aging but doesn't there have to be? It can't just be magic - "space-time geodesic changes to shorter and all things follow".

To the OP's original question about some notion of coordinates (not really time at all though time is everywhere on it) that span a Schwarzschild manifold (I think it's not wrong to call it a manifold) as well as ibix' nice image of a stacked set of clocks on orbiting spaceships- what if the two Schwarzschild observers or the clocks on spaceships were one living giant... we would have some expectation that Giant would suffer even though the physics are the same for his feet and his head don't we? If the giant is too dramatic an analogy a long rod sort of suffices - though to my mind it avoids the question that the movie made clear - how does reality, including the whole bucket of electro-megnetism, the periodic table itself get...re-gauged, in-toto, somehow continuously deformed, to support variation in proper time over curvature. My understanding is it's an open question..?

Jimster41 said:
I am struggling to understand what the physical line (the description of causal connection) is between acceleration or exposure to gravitational curvature and the effect on the twin (biochemical aging)?
There is no direct physical link. The entire effect is due to one of the travelers having followed a longer path through spacetime. Acceleration only comes in because we had to add something to the thought experiment to push them onto different paths. Without it they’d both be following the same path through spacetime so there would be no difference in the amount of time (heartbeats, clock ticks, aging, ...) they experience.

Biochemical aging works just like any other physical process: you age one second for every second that passes.

cianfa72
Edit, I don't know. The Giant may be perfectly fine. It's confusing.

Jimster41 said:
I get the "cars" are rolling down geodesics of different length, measured only by differences in proper (elapsed) time. Other than that abstraction I don't know what you can point to. Is there something you would say?
It is the same with proper time. Why are you willing to accept that something as physical as the odometer reading can be changed without some mechanism that we can point to that is changing it, but not proper time? They are equivalent. The difference in both cases is the path, not a change to the measuring device.

Pencilvester
Ibix said:
It needs to have a rocket so that it hovers at constant ##r## and every time it ticks one second it needs to show ##(1-R_s/r)^{-1/2}## seconds elapsed. Also you need to zero clocks at different altitudes - you can do that by exchanging light signals in this case.
To be as clear as much as possible, suppose it was possible to exchange sound signals in vacuum to synchronize (zeroing out) coordinate clocks attached to rockets each hovering at a different radial Schwarzschild coordinate ##r##. In that case I believe the coordinate time difference for a light beam sent from ##r_1## at coordinate time ##t_1## and received at ##r_2## at coordinate time ##t_2## should be not given by
$$t_{2} - t_{1} = \Delta t = \frac{r_{2}-r_{1}}{c} + \frac{R_{s}}{c} \ln(\frac{r_{2}-R_{s}}{r_{1}-R_{s}}),$$

Thus the definition of Schwarzschild coordinate system (as a global coordinate chart) includes "inside itself" the very procedure to use to synchronize coordinate clocks at different radial coordinate ##r##, I believe...

cianfa72 said:
o be as clear as much as possible, suppose it was possible to exchange sound signals in vacuum
How fast do these imaginary signals travel? What are the rules?

In the absence of rules, no prediction can be made.

Dale said:
It is the same with proper time. Why are you willing to accept that something as physical as the odometer reading can be changed without some mechanism that we can point to that is changing it, but not proper time? They are equivalent. The difference in both cases is the path, not a change to the measuring device.

I’m not “willing to accept” that there are unphysical things such as “paths” that are somehow unphysical vs odometers that are physical.

I know the theoretical limit of the Standard Model says believe there is currently nothing physical we can point to.
Question for you and @Nugatory: what’s a second?

Jimster41 said:
I’m not “willing to accept” that there are unphysical things such “paths” vs odometers that are physical.
I don't understand this statement. Are you saying that paths are unphysical so you don't accept the fact that two cars traveled on different paths as the physical reason that their odometers read differently? If so, then what is the physical reason for the different odometer readings? If you can answer that question then there should be an analogous answer for the proper time question. Personally, the fact that the paths are different lengths seems like a sufficient answer to me.

Believing that paths are not physical seems bizarre, so I hope that I am misunderstanding you.

Jimster41 said:
Question for you: what’s a second?
The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency ##\Delta\nu_{Cs}##, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s^–1.

https://www.bipm.org/en/measurement-units/base-units.html

Pencilvester and weirdoguy
Yes, I think we are misunderstanding each other. I am saying the path and it’s effect on my the odometer including the effect of acceleration on that effect must be physical. I would surprised if you didn’t agree with that.

And yes I was hoping we’d agree that a human outfit called NIST had come up with a scheme to calibrate all our aging to one thing... a quantum mechanical thing.

weirdoguy
Jimster41 said:
I am saying the path and it’s effect on my the odometer including the effect of acceleration on that effect
In the case of an odometer and a path in three dimensions, experiment shows that acceleration has no effect.

weirdoguy

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