I On the physical meaning of Minkowski's spacetime model

[cianfa72]

TL;DR Summary

On the physical meaning of Minkowski's spacetime model from the point of view of clocks and rulers

Hi, I was thinking about the following.

Suppose we have a geometric mathematical model of spacetime such that there exists a global map $(t,x_1,x_2,x_3)$ in which the metric tensor is in the form $$ds^2 = c^2dt^2 - (dx_1)^2 + (dx_2)^2 + (dx_3)^2$$ i.e. the metric is in Minkowski form $(+,-,-,-)$.

What does it mean from a physical experimental point of view ? I believe the following is true:

We can single out congruences of clocks (i.e. a families of clocks each filling the spacetime such that their worldlines do not cross) such that:

• for each congruence of such clocks we can synchronize them, adjust their rates and assign spatial labels to each of them in a way such that light propagation process happens to be isotropic and occurs with the same fixed invariant speed $c$ (note that such light propagation properties are time invariant i.e. do not change in time).

What do you think about ? Thanks.

 

[Ibix]

I think that's just a restatement of Einstein's two postulates.

 

[cianfa72]

I think that's just a restatement of Einstein's two postulates.

I take this as the experimental fact that we can single out congruences of clocks such that we can Einstein synchronize them in a consistent way (namely if we Einstein synchronize them for the first time then they will stay synchronized into the "future" - as we can check experimentally).

 

[PeroK]

I take this as the experimental fact that we can single out congruences of clocks such that we can Einstein synchronize them in a consistent way (namely if we Einstein synchronize them for the first time then they will stay synchronized into the "future" - as we can check experimentally).

"Singling out congruences of clocks" sounds more like theoretical than experimental physics to me.

 

[cianfa72]

"Singling out congruences of clocks" sounds more like theoretical than experimental physics to me.

Why ? Experimentally you look for physical clocks filling the space that have those properties w.r.t. the light propagation process.

 

[PeroK]

Why ? Experimentally you look for physical clocks filling the space that have those properties w.r.t. the light propagation process.

Not an easy experiment to carry out!

 

[Dale]

Not an easy experiment to carry out!

It is a little easier than you might think because the model tells you how you can construct such a congruence. It tells you that each clock in the congruence is in free fall, and that they are initially at rest with respect to each other. It also tells you that they will naturally remain at rest. It also tells you that you can use rulers to measure the distances and those will agree with radar distances.

So it isn’t just a matter of finding such congruences. We can construct them. If spacetime is flat.

 

[PeroK]

It is a little easier than you might think because the model tells you how you can construct such a congruence. It tells you that each clock in the congruence is in free fall, and that they are initially at rest with respect to each other. It also tells you that they will naturally remain at rest. It also tells you that you can use rulers to measure the distances and those will agree with radar distances.

So it isn’t just a matter of finding such congruences. We can construct them. If spacetime is flat.

That sounds a bit far fetched to me!

 

[cianfa72]

It is a little easier than you might think because the model tells you how you can construct such a congruence. It tells you that each clock in the congruence is in free fall, and that they are initially at rest with respect to each other. It also tells you that they will naturally remain at rest. It also tells you that you can use rulers to measure the distances and those will agree with radar distances.

So it isn’t just a matter of finding such congruences. We can construct them. If spacetime is flat.

Actually, in the context of special relativity (no gravity) I don't think the mathematical model says each clock in the congruence we're looking for has to be in free fall. It simply claims such a congruence of clocks with those properties does exist ! Our experimental task is just single out a such congruence.

And then we will find out that such clocks will actually be in free fall with all the other properties @Dale pointed out.

 

[Dale]

That sounds a bit far fetched to me!

Fair enough. This is very much in the “much easier said than done” category.

I don't think the mathematical model says each clock in the congruence we're looking for has to be in free fall.

It does say that. In particular it says that accelerometers attached to such clocks will read 0, which is free fall. We can use that fact to construct the congruence.

 

[cianfa72]

It does say that. In particular it says that accelerometers attached to such clocks will read 0, which is free fall. We can use that fact to construct the congruence.

Ah ok, your claim is grounded on calculation of the proper acceleration in the model -- i.e. $$\left( \frac {dt} {d\tau}, \frac {dx_1} {d\tau}, \frac {dx_2} {d\tau}, \frac {dx_3} {d\tau} \right )$$ that vanishes for such a congruence of clocks.

The above makes sense in the mathematical model. So the next step is to associate the mathematical proper acceleration "object" with the measurement performed by a physical accelerometer device. This last step, I believe, is the actual correspondence between the mathematical model and the physics.

 

[Dale]

So the next step is to associate the mathematical proper acceleration "object" with the measurement performed by a physical accelerometer device. This last step, I believe, is the actual correspondence between the mathematical model and the physics.

Yes. That is what makes a scientific theory science and not math. It is called the minimal interpretation. Without a minimal interpretation you are not doing science.

The specific mapping of the mathematical curvature of a worldline to the physical measurement of proper acceleration with an accelerometer is the same in GR and SR. This is why the Minkowski metric does say what I said it says.

 

[cianfa72]

Without a minimal interpretation you are not doing science.

The specific mapping of the mathematical curvature of a worldline to the physical measurement of proper acceleration with an accelerometer is the same in GR and SR. This is why the Minkowski metric does say what I said it says.

Yes, furthermore the fact that the spacetime lenght of a timelike curve in the mathematical model (proper time) corresponds to the clock's elapsed time following that curve in spacetime is another element of that minimal interpretation.

Edit: the same, as you pointed out, holds for the ruler distances that will agree with radar distances (based on round-trip travel delay).

 

[strangerep]

[...] the metric tensor is in the form $$ds^2 = c^2dt^2 - (dx_1)^2 + (dx_2)^2 + (dx_3)^2$$ i.e. the metric is in Minkowski form $(+,-,-,-)$.

Huh? Your line element suggests a metric with signature $(+,-,+,+)$.

 

[cianfa72]

Huh? Your line element suggests a metric with signature $(+,-,+,+)$.

Yup, my fault. It was actually $$ds^2 = c^2dt^2 - (dx_1)^2 - (dx_2)^2 - (dx_3)^2$$
Another point: does the Minkowski spacetime structure force us to assume Einstein's synchronization procedure to synchronize clocks in an inertial frame?

 

[PeterDonis]

does the Minkowski spacetime structure force us to assume Einstein's synchronization procedure to synchronize clocks in an inertial frame?

You're looking at it backwards. An inertial frame is defined as one in which Einstein synchronization is used on free-falling clocks that are at rest relative to each other. You're not forced to use such a frame.

 

[cianfa72]

You're looking at it backwards. An inertial frame is defined as one in which Einstein synchronization is used on free-falling clocks that are at rest relative to each other. You're not forced to use such a frame.

In other words the global chart in which the Minkowski metric is in its standard form is an inertial chart/frame (i.e. Einstein synchronization procedure is used on free-falling clocks that are at rest each other as measured by exchanging light pulses between them -- their round-trip travel times do not change).

 

[PeterDonis]

In other words the global chart in which the Minkowski metric is in its standard form is an inertial chart/frame (i.e. Einstein synchronization procedure is used on free-falling clocks that are at rest each other as measured by exchanging light pulses between them -- their round-trip travel times do not change).

Yes.

 

[cianfa72]

Sorry to be pedantic. Basically the minimal interpretation of spacetime mathematical model includes the following:

1. timelike curves in the spacetime model correspond to worldlines of physical objects (clocks)
2. the spacetime lenght of a timelike curve in the model (i.e. its proper time) corresponds to the clock's elapsed time
3. null paths in the spacetime model correspond to light propagation worldlines
4. timelike paths with zero proper acceleration in the spacetime model correspond to worldlines of physical objects such that accelerometers attached to them read zero (this is the physical operational definition of free-falling)

From Minkowski spacetime mathematical model in standard coordinates + the above minimal interpretation it follows that:

• there are free-falling physical clocks that stay at rest each other according round-trip radar measurements
• we can Einstein synchronize them and assign spatial coordinates (i.e. labels) to each of them such that the radar distance between them is consistent with the "difference of spatial label values" in the context of Euclidean geometry
• the radar distances "at the same time" (i.e. proper distances) obey the Pythagorean theorem (i.e. the geometry "at the same time" is Euclidean)

 

[PeterDonis]

the minimal interpretation of spacetime mathematical model includes the following:

1. timelike curves in the spacetime model correspond to worldlines of physical objects (clocks)
2. the spacetime lenght of a timelike curve in the model (i.e. its proper time) corresponds to the clock's elapsed time
3. null paths in the spacetime model correspond to light propagation worldlines
4. timelike paths with zero proper acceleration in the spacetime model correspond to worldlines of physical objects such that accelerometers attached to them read zero (this is the physical operational definition of free-falling)

Yes.

From Minkowski spacetime mathematical model in standard coordinates + the above minimal interpretation it follows that:

• there are free-falling physical clocks that stay at rest each other according round-trip radar measurements
• we can Einstein synchronize them and assign spatial coordinates (i.e. labels) to each of them such that the radar distance between them is consistent with the "difference of spatial label values"
• the radar distances "at the same time" (i.e. proper distances) obey the Pythagorean theorem (i.e. the geometry "at the same time" is Euclidean)

Yes.

 

[cianfa72]

Last but not least, in the conditions and physical setup of post#19, light propagation process is by definition homogeneous and isotropic and occurs with the (one-way) fixed invariant speed $c$.

 

[PeterDonis]

Last but not least, in the conditions and physical setup of post#19, light propagation process is by definition homogeneous and isotropic and occurs with the (one-way) fixed invariant speed $c$.

In the sense that adopting Einstein clock synchronization and standard Minkowski coordinates make these statements true, yes.

 

[Dale]

Sorry to be pedantic. Basically the minimal interpretation of spacetime mathematical model includes the following:

1. timelike curves in the spacetime model correspond to worldlines of physical objects (clocks)
2. the spacetime lenght of a timelike curve in the model (i.e. its proper time) corresponds to the clock's elapsed time
3. null paths in the spacetime model correspond to light propagation worldlines
4. timelike paths with zero proper acceleration in the spacetime model correspond to worldlines of physical objects such that accelerometers attached to them read zero (this is the physical operational definition of free-falling)

From Minkowski spacetime mathematical model in standard coordinates + the above minimal interpretation it follows that:

• there are free-falling physical clocks that stay at rest each other according round-trip radar measurements
• we can Einstein synchronize them and assign spatial coordinates (i.e. labels) to each of them such that the radar distance between them is consistent with the "difference of spatial label values" in the context of Euclidean geometry
• the radar distances "at the same time" (i.e. proper distances) obey the Pythagorean theorem (i.e. the geometry "at the same time" is Euclidean)

Last but not least, in the conditions and physical setup of post#19, light propagation process is by definition homogeneous and isotropic and occurs with the (one-way) fixed invariant speed $c$.

This does not follow from the points 1-4 in your post 19. It does follow in the bullet point list because you added the important condition "in standard coordinates". In particular, the fixed invariant speed $c$ does not hold in non-standard coordinates.

 

[cianfa72]

This does not follow from the points 1-4 in your post 19. It does follow in the bullet point list because you added the important condition "in standard coordinates". In particular, the fixed invariant speed $c$ does not hold in non-standard coordinates.

Just to be clear: the Minkowski mathematical model in standard coordinates + minimal interpretation as in points 1-4 in post #19 tells us that, in the physical world (no gravity at all), there must exist a congruence of clocks with properties in the post#19 bullet points (notably such a congruence of clocks is consistently Einstein synchronizable). Then if we choose to adopt the Einstein synchronization (and therefore we Einstein synchronize them) it follows that the light propagation process is homogeneous and isotropic and occurs with the (one-way) fixed invariant speed $c$.

 

[Dale]

The Minkowski spacetime model + the minimal interpretation gives the existence of the congruence. Then adding the standard coordinates is equivalent to Einstein synchronization with its homogenous and isotropic invariant speed of light, $c$.

 

[cianfa72]

The Minkowski spacetime model + the minimal interpretation gives the existence of the congruence. Then adding the standard coordinates is equivalent to Einstein synchronization with its homogenous and isotropic invariant speed of light, $c$.

The standard coordinates in Minkowski spacetime mathematical model tell us that the associated congruence of physical clocks (that we know exists from Minkowski spacetime model + minimal interpretation as you pointed out) has got furthermore the property to be Einstein synchronized (hence the properties in your quoted post of light propagation in that frame).

 

[Dale]

The standard coordinates in Minkowski spacetime mathematical model tell us that the associated congruence of physical clocks (that we know exists from Minkowski spacetime model + minimal interpretation as you pointed out) has got furthermore the property to be Einstein synchronized (hence the properties in your quoted post of light propagation in that frame).

Yes.

 

[cianfa72]

I found this Einstein synchronisation on Wikipedia related someway to this topic.

Basically is a theorem (iff) that the consistency of Einstein synchronization procedure and the one-way speed of light constancy over the frame (w.r.t. Einstein's synchronizated clocks) is equivalent to the Laue–Weyl's round-trip condition: the time needed by a light beam to traverse a closed path of length $L$ is $L/c$, where $L$ is the length of the path and $c$ is a constant independent of the path.

As said in that link The importance of Laue–Weyl's condition stands on the fact that the time there mentioned can be measured with only one clock; thus this condition does not rely on synchronisation conventions and can be experimentally checked.

Indeed, it has been experimentally verified that the Laue–Weyl round-trip condition holds throughout an inertial frame.

 

[Dale]

Yes. This is also the reason why Einstein synchronization is not possible (globally) in a rotating reference frame

 

[cianfa72]

This is also the reason why Einstein synchronization is not possible (globally) in a rotating reference frame

Do you mean that for clocks at rest in a rotating frame a light beam traversing a closed path of length $L$ is not always $L/c$ (as measured by clocks at fixed points in rotating frame) ?

 

[Dale]

Do you mean that for clocks at rest in a rotating frame a light beam traversing a closed path of length $L$ is not always $L/c$ (as measured by clocks at fixed points in rotating frame) ?

That is correct. A light beam traversing clockwise or counterclockwise will give different times.

 

[cianfa72]

That is correct. A light beam traversing clockwise or counterclockwise will give different times.

Which is the reason for this ?

 

[Dale]

Which is the reason for this ?

The second postulate.

 

[cianfa72]

The second postulate.

Sorry, not sure to understand. The second postulate is about the invariance of speed of light as measured in inertial frames.

 

[Dale]

Sorry, not sure to understand. The second postulate is about the invariance of speed of light as measured in inertial frame.

Yes, from that you can derive the fact that in the rotating reference frame a light beam traversing clockwise or counterclockwise will give different times.

 

[Ibix]

Sorry, not sure to understand. The second postulate is about the invariance of speed of light as measured in inertial frame.

So analyse what happens to two light pulses going in opposite directions in a rotating closed (circular is easiest) path. Assuming the emitter is attached to the rotating apparatus do simultaneously emitted pulses return to the emitter simultaneously?

 

[cianfa72]

So analyze what happens to two light pulses going in opposite directions in a rotating closed (circular is easiest) path. Assuming the emitter is attached to the rotating apparatus do simultaneously emitted pulses return to the emitter simultaneously?

Ah ok, one can analyze it from the point of view of the inertial frame where the second postulate holds. No, the two emitted pulses do not return to the emitter simultaneously as measured in the inertial frame by Einstein's synchronized clocks in it. This latter fact is frame invariant (i.e. there is no coincidence of the two reception events), hence a light beam traversing clockwise or counterclockwise will give different times.

 

[Ibix]

Yes.

 

[Nugatory]

No, the two emitted pulses do not return to the emitter simultaneously as measured in the inertial frame by Einstein's synchronized clocks in it.

No qualification about clocks or synchronization is needed.

 

[cianfa72]

No qualification about clocks or synchronization is needed.

This because we are looking at the coincidence of events (whether they are the same spacetime point or not).

 

[Sagittarius A-Star]

Ah ok, one can analyze it from the point of view of the inertial frame where the second postulate holds.

You can also analyze it in a rotating frame. The circumference with respect to this frame shall be called $U'$.

You can define an inertial reference frame with only one $x'$ axis in the range $-U'/2 < x' < +U'/2$, curled around the rim of the circular disk and rotating with it, and one $t'$ axis. This happens all in the same potential of the pseudo-gravitation caused by the centrifugal force.

But the standard Lorentz transformation to/from the non-rotating inertial frame is only permitted, if the coordinate time $t'$ is define by an Einstein-synchronization along the rim of the disk. That means you need 2 different clocks at the locations $x'\approx-U'/2$ and $x'\approx+U'/2$, even if the locations almost coincide. The Sagnac-effect uses only one clock, with measures as time-difference 2x the term for "relativity of simultaneity" in the LT (independent of the signal-velocity in an optical fiber).

Source:
http://www.physicsinsights.org/sagnac_1.html