# Time like spacetime interval, proper time, and time dilation

• B
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Since the time like spacetime interval equals proper time then it seems time dilation is frame invariant
Since the time like spacetime interval is equal to proper time for stationary or traveling observers, then it seems time dilation (proper time) seen with traveling clock is necessarily frame invariant. Then the so called time between ticks of both identical clocks, with stationary and traveling observers, are also equal/invariant. If traveler has v = .6c then both observers agree that ∆t' records .8∆t elapsed time. However some references state "time between ticks is not equal.

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@morrobay The proper time between successive ticks of identically constructed clocks is invariant, but this tells us absolutely nothing about time dilation. Time dilation is what happens when you apply the relativity of simultaneity to the endpoints of these invariant intervals.

Suppose clock A and clock B are identically constructed and moving relative to one another. The proper time between successive ticks of clock A is one second, as is the proper time between ticks of clock B; these are frame-independent invariant statements. However relativity of simultaneity means that if the two clocks tick at the same time once, their next tick will not happen at the same time: A will say that B’s next tick happened after their own next tick so B is running slow, while B will say that A’s next tick happened after B’s next tick so A is running slow.

dextercioby and PeterDonis
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Since the time like spacetime interval is equal to proper time for stationary or traveling observers, then it seems time dilation (proper time) seen with traveling clock is necessarily frame invariant.
This in no way necessarily follows. Time dilation is ##dt/d\tau##. Although ##d\tau## is invariant ##dt## is not. It is not necessary that the ratio of an invariant quantity and a non-invariant quantity must be invariant.

dextercioby, Grasshopper and Orodruin
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As has been stated already, the proper time per increase of the time coordinate is not invariant because the time coordinate is not invariant even if the proper time is.

However, what is invariant is the time elapsed for an observer between to specific events such as "twin A starts the journey" and "twin A comes back". This is why the twin paradox is not really a paradox at all, but a faulty application of time dilation without taking relativity of simultaneity into account.

dextercioby
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Proper time is the "distance" along the worldline of a clock. Coordinate time is the distance along the worldline of a clock at rest and synchronised with respect to me. You can think of coordinate time as being like the vertical coordinate on a piece of graph paper and the proper time as the length along a (possibly sloped and/or curved) line on that paper. If the line isn't parallel to the vertical of the paper its 1s mark won't occur at 1s above the graph paper's t=0 line. The projection of the line's length onto the coordinate time lines is time dilation.

That's a slightly sloppy Euclidean analogy. You can formalise it as a Minkowski diagram.

jjzeidner, vanhees71, PeroK and 1 other person
cianfa72
Proper time is the "distance" along the worldline of a clock. Coordinate time is the distance along the worldline of a clock at rest and synchronised with respect to me. You can think of coordinate time as being like the vertical coordinate on a piece of graph paper and the proper time as the length along a (possibly sloped and/or curved) line on that paper.
Sorry to resume this thread. In your sentence what do you mean with synchronised with respect to me ? I believe that some synchronization "rule/procedure" is actually implied (even if not explicitly stated).

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I meant Coordinate time is the distance along the worldline of a clock at rest with respect to me and synchronised. As you say, some synchronisation convention needs to be invoked to complete the definition - Einstein's convention most likely.

Gold Member
In your sentence what do you mean with synchronised with respect to me ? I believe that some synchronization "rule/procedure" is actually implied (even if not explicitly stated).
Synchronized with respect to me means synchronized in a frame of reference in which I am at rest. The synchronization rule or procedure (called the convention) may indeed be implied, but that is not the issue prompting the "with respect to me" phrase. Instead, it's the fact that in frames of reference that are in motion relative to me those clocks may not be synchronized.

cianfa72 and Ibix
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You can compute the proper time of an object in any coordinate system if you know the world line. It is an invariant.

morrobay
cianfa72
Synchronized with respect to me means synchronized in a frame of reference in which I am at rest. The synchronization rule or procedure (called the convention) may indeed be implied
I think the are actually two logically distinct aspects in the sentence synchronized in a frame of reference in which I am at rest.
1. being at rest each other: physically we can use the following procedure to define it: take a clock attached to a body (local) and send a light beam from it to the remote body; reflect it back towards the local body and take the time shown on that clock when it eventually comes back. If the elapsed time (according to the local body clock) stays constant in time then we define the two bodies as at rest each other (the other way around for the other body)
2. to be synchronized in the rest reference frame: physically we define this concept using a procedure or rule as for instance Einstein synchronization convention that makes sense only for clocks at rest each other (at rest in the sense of 1.)
I believe definition 1. and 2. are actually related because they basically "share" the postulate of invariance of the speed of the light in vacuum.

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I think the are actually two logically distinct aspects in the sentence synchronized in a frame of reference in which I am at rest.
Yes. The worldlines of clocks at rest with respect to me define the vertical lines on my Minkowski diagram. How I choose to synchronise the clocks defines whether my planes of simultaneity are orthogonal to those worldlines or not.

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Restated question: Is it possible for the stationary observer to calculate the proper time of the travelers clock ? See post # 24 https://www.physicsforums.com/threads/fewer-seconds-or-shorter-seconds.1000854/
Yes. The formula in the post you cite is a general formula that works for any worldline in any coordinate system on any spacetime.

On a slightly more philosophical level, I must be able to calculate the outcome of any experiment you do in any coordinate system I like - otherwise my choice of coordinates would have physical significance, which seems implausible. In fact, it was that your choice of (Galilean) coordinates did seem to affect Maxwell's equations that kicked off all the mess that led to the discovery of relativity.

morrobay
cianfa72
Yes. The worldlines of clocks at rest with respect to me define the vertical lines on my Minkowski diagram. How I choose to synchronise the clocks defines whether my planes of simultaneity are orthogonal to those worldlines or not.
ok, about the first part however I believe we need a physical definition of "being at rest each other", do you ?

About the second one: could you please give an example of clocks synchronization other than Einstein synch convention ? Thanks

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ok, about the first part however I believe we need a physical definition of "being at rest each other", do you ?
I'd say someone at rest with respect to you is someone who returns zero Doppler when I bounce a radar pulse off them. Or you can use rulers.

Whether you need that as a definition or whether that's a derived fact, I think, depends on what assumptions you are starting from. You do always need to link observations to quantities in the mathematical model, though, yes.

cianfa72
cianfa72
I'd say someone at rest with respect to you is someone who returns zero Doppler when I bounce a radar pulse off them. Or you can use rulers.
If you use rulers in principle you should keep placing rulers side by side one after the other all the time. If you use radar pulse, perhaps it may be easier to define when remote objects are at rest w.r.t you - all the time.

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Staff Emeritus
ok, about the first part however I believe we need a physical definition of "being at rest each other", do you ?

About the second one: could you pleasegive an example of clocks synchronization other than Einstein synch convention ? Thanks

Having no doppler shift is the criterion I use for "being at rest" relative to each other in special relativity. This means for instance an emitted signal of <X> Hz is received as <X> Hz. It's a somewhat informal approach.

As for an example of clock synchronization other than Einstein syncrhonization, consider wall clocks on the Earth's surface keeping UTC time (or the TAI time on which it is based). This is perhaps an unfortunate example in that it can involve some aspects of General relativity, though if one considers only clocks "on the geoid" (informally, at "sea level"), the GR aspects can be minimized.

cianfa72
cianfa72
As for an example of clock synchronization other than Einstein synchronization, consider wall clocks on the Earth's surface keeping UTC time (or the TAI time on which it is based). This is perhaps an unfortunate example in that it can involve some aspects of General relativity, though if one considers only clocks "on the geoid" (informally, at "sea level"), the GR aspects can be minimized.
ok, wall clocks on the geoid keeping UTC time are at rest each other because of "no doppler shift" criterion, for instance (minimizing GR aspects).

Which is the rule to be used as "synchronization rule or procedure" to synchronize them (or in other words to define their common time) ?

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Mentor
You can compute the proper time of an object in any coordinate system if you know the world line. It is an invariant.
To be clear, the proper time is invariant not time dilation. They are different quantities that the OP seems to have trouble distinguishing.

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being at rest each other: physically we can use the following procedure to define it: take a clock attached to a body (local) and send a light beam from it to the remote body; reflect it back towards the local body and take the time shown on that clock when it eventually comes back. If the elapsed time (according to the local body clock) stays constant in time then we define the two bodies as at rest each other (the other way around for the other body)
By this definition, a clock circling a local clock is "at rest" relative to the local clock.

PeroK and Motore
cianfa72
By this definition, a clock circling a local clock is "at rest" relative to the local clock.
ok, so what is actually our favorite notion of "being at rest" ?

Homework Helper
ok, so what is actually our favorite notion of "being at rest" ?
Set up a coordinate system in which Newton's laws hold good. An object is "at rest" if its spatial coordinates are constant in such a system.

Staff Emeritus
ok, wall clocks on the geoid keeping UTC time are at rest each other because of "no doppler shift" criterion, for instance (minimizing GR aspects).

Which is the rule to be used as "synchronization rule or procedure" to synchronize them (or in other words to define their common time) ?

One can imagine the clocks being synchronized from an emission point at the center of the Earth, for instance by using barycentric coordinates. This works in theory and is easy to communicate, but isn't a very practical scheme. There are physically implementable standards in place about how one syncrhonizes clocks according to the TAI standard that use satellites (usually GPS satellites), see for instance https://en.wikipedia.org/wiki/Two-way_satellite_time_and_frequency_transfer (which is an extremely short "stub" article).

Note that Einstein sychronization of all clocks on the geoid is not possible because of the Earth's rotation - see for instance the Hafele-Keating (HK) experiment. As synchronizing clocks via "slow clock transport" is theoretically equivalent to Einstein clock synchronization the HK experiment demonstrates it's impossible to slowly transport a clock around the Earth and maintain synchronization, hence it's also impossible to Einstein-synchronize clocks all around the geoid.

cianfa72
Set up a coordinate system in which Newton's laws hold good. An object is "at rest" if its spatial coordinates are constant in such a system.
That's good fo SR I believe. What about in the context of GR ?

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That's good fo SR I believe. What about in the context of GR ?
"Rest" isn't well-defined in general unless you are co-located, at which point the SR definition is fine.

jbriggs444
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"Rest" isn't well-defined in general unless you are co-located, at which point the SR definition is fine.
But in that case, it’s going to be when you are in free fall, right?

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But in that case, it’s going to be when you are in free fall, right?
Two people in free fall can pass each other at arbitrarily high velocity - consider two objects in counter-rotating orbits. "Free fall" just means you aren't accelerating in any physically meaningful sense, not that you aren't moving.

Gold Member
Two people in free fall can pass each other at arbitrarily high velocity - consider two objects in counter-rotating orbits. "Free fall" just means you aren't accelerating in any physically meaningful sense, not that you aren't moving.
But what does “moving” mean? Surely if you’re in free fall you are justified in saying you aren’t moving, and in that special case I imagine Newton’s laws hold over small distances.

But then again you’re always at rest in your own reference frame.

I think that describes a local inertial frame, rather than “at rest.” Seem to have mixed the terms.

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But what does “moving” mean?
On it's own it doesn't mean anything. You need to say moving with respect to something. So...
Surely if you’re in free fall you are justified in saying you aren’t moving, and in that special case I imagine Newton’s laws hold over small distances.
...you are always entitled to consider yourself as "at rest" if that's convenient. And Newton's laws, or SR more generally, will always hold over small regions of spacetime, whether you treat yourself as stationary or moving.
I think that describes a local inertial frame, rather than “at rest.”
A local inertial frame is a concept in curved spacetime rather like the fact that you can draw a Cartesian grid on a small region of the Earth. Similarly, you can always draw a Lorentzian grid on a small region of spacetime. You may draw your little grid with its timelike axis parallel to your worldline (and then you are at rest in this little frame) or not (in which case you are moving in this little frame).

Dale
Staff Emeritus
ok, so what is actually our favorite notion of "being at rest" ?

Being "at rest" usually is an informal way of saying that one's spatial coordinates are constant. It's a coordinate dependent statement which has no meaning until one defines the specific coordinates being used. Sometimes the coordinates are spelled out, other times they are only implied. This is somewhat sloppy, but no harm is done if both parties know and agree on which coordinates are being used. Confusion arises if both parties use different coordinates. An object "at rest" on the Earth's surface in earth-fixed coordinates is moving in barycentric coordinates, due to the rotation of the Earth, for instance.

Compare and contrast the notion of "at rest" to the different notion of "zero proper acceleration", which can be defined in an observer independent manner. An object "at rest" on the Earth's surface has a non-zero proper acceleration. An example of an object on the Earth that has zero proper acceleration would be a thrown baseball (ignoring air resistance).

cianfa72 and Dale
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Possibly you need to take a step back. Maybe the notion of being "at rest" is not what you really want to talk about.

You had been suggesting that a remote object is "at rest" if it remains at a constant distance (constant round trip light time) from a selected local object. There is a term used to characterize a set of objects that maintain a fixed distance from one another: Born Rigidity

cianfa72
Mentor
That's good fo SR I believe. What about in the context of GR ?
In GR you don’t need to worry about Newton’s laws. Otherwise the same definition works. Perhaps with the caveat that the coordinate system should have three spacelike and one timelike coordinates.

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Somewhat related question: would a sufficient definition of “inertial reference frame” in special relativity (assuming isotropy and homogeneity) simply be a reference frame in which Newtons law of inertia holds good? If not, what more would be needed?

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I do not read the previous part of the thread but the definition of IFR is as you said.

cianfa72
Set up a coordinate system in which Newton's laws hold good. An object is "at rest" if its spatial coordinates are constant in such a system.
So this notion of "at rest" is actually w.r.t. a coordinate system. What about the notion of "being at rest" w.r.t. another body ?