Coordinate time between spatially separated events in Schwarzschild spacetime

[jbriggs444]

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.

 

[Jimster41]

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?

 

[Dale]

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.

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

 

[Jimster41]

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.

 

[jbriggs444]

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.

 

[Dale]

I am saying the path and it’s effect on my the odometer including the effect of acceleration on that effect must be physical.

If you are willing to accept the path as being the physical cause of the effect for the odometer then it should be no problem to accept the path as being the physical cause for the effect for a clock too.

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.

Actually, it is the BIPM, not NIST.

 

[Jimster41]

In the case of an odometer and a path in three dimensions, experiment shows that acceleration has no effect.

What causes two paths in spacetime be of different length?

Actually, it is the BIPM, not NIST.

Noted. My mistake.
But we agree a "second" is given physical meaning by observing a quantum mechanical process, right?

If you are willing to accept the path as being the physical cause of the effect for the odometer then it should be no problem to accept the path as being the physical cause for the effect for a clock too.

I accept that what you are calling "path" is the physical cause. I am asking what the "path" is physically and how the odometer knows how much of it is being covered? And if two spacetimes paths can be of different length because a) there was some acceleration that caused them to have different inertial frames or b) they are like, what, separated on a curved spacetime manifold? My question is how does the odometer account for and "know about" that?

You do agree there is currently no quantum mechanical theory of general relativity right?

 

[Dale]

I am asking what the "path" is physically and how the odometer knows how much of it is being covered?

The path is physically the set of all events at which the odometer/clock was located.

I don't like to anthropomorphize devices. Odometers and clocks don't know anything. They are designed to measure the length of the path being covered. For the design of an odometer it is based on things like Hooke's law and Newton's laws. For clocks it can be based on things like Maxwell's equations, Hooke's law, QED, or other mechanisms depending on the design of the specific clock in question.

And if two spacetimes paths can be of different length because a) there was some acceleration that caused them to have different inertial frames or b) they are on like, what, separated on a curved spacetime manifold.

Neither a nor b is correct in my opinion. They are different simply because they cover a different set of events. The acceleration or curvature certainly may be present and may even be necessary in certain circumstances, but it is incidental. The paths are different because they are different sets of events.

Then my question is how does the odometer account for and "know about" that?

The odometer/clock does not need to "know about" any difference between its path and any other path. All it does is measure the length of the path that it covers. The length of other paths that it does not cover is not relevant to its functioning.

You do agree there is currently no quantum mechanical theory of general relativity right?

Yes, certainly. Although what we are talking about here applies to both GR and QFT individually.

 

[cianfa72]

How fast do these imaginary signals travel? What are the rules?
In the absence of rules, no prediction can be made.

Assume those imaginary signals travel at constant speed $s$ other than the speed of light.
In that case the metric and thus the coordinate time difference for a light beam (always having $ds=0$ equation) sent from $r_1$ at coordinate time $t_1$ and received at $r_2$ at coordinate time $t_2$ should be not given by the previous formula
$$t_{2} - t_{1} = \Delta t = \frac{r_{2}-r_{1}}{c} + \frac{R_{s}}{c} \ln(\frac{r_{2}-R_{s}}{r_{1}-R_{s}}),$$

 

[jbriggs444]

Assume those imaginary signals travel at constant speed $s$ other than the speed of light.

As measured in what frame?

 

[cianfa72]

As measured in what frame?

Right question...I believe the same topic applies for the light beams. How can we "manage" it in that case ?

 

[jbriggs444]

Right question...I believe the same topic applies for the light beams. How can we "manage" it in that case ?

It turns out that all inertial frames will measure the same speed for anything moving at c. But in the case at hand, s is not equal to c. So frame choice matters.

 

[Jimster41]

Honestly, Dale. I am good with this.

Neither a nor b is correct in my opinion. They are different simply because they cover a different set of events. The acceleration or curvature certainly may be present and may even be necessary in certain circumstances, but it is incidental. The paths are different because they are different sets of events.

The odometer/clock does not need to "know about" any difference between its path and any other path. All it does is measure the length of the path that it covers. The length of other paths that it does not cover is not relevant to its functioning.

I wasn't thinking of an "awareness" of differential path length in some global sense. I am asking what is different about the events of the accelerating twin? How do those events manifest to the odometer the length of the path... How do they provide information that will later be observed as evidence of difference in path length?

I'm not advocating something mysterious here. I'm admiring the mountain of Yang-Mills looking up at it. I am curious as heck about how re-normalization works as Yang-Mills re-gauges or whatever across non-inertial frames. Obviously the fictional forces are kind of real... we feel acceleration. How does that happen from the QM up? I wish there were more books that got into conceptual depth - but were more about explaining the derivation and purpose of the math involved rather than teach one to do it. "Deep Down Things" was a good one in that regard re the standard model.

[Edit] I can imagine you would say there is nothing different about those events and mean that in the sense the theory of the Standard Model proves that the same physics apply continuously to all the events. I am just trying to understand how the Standard Model navigates that claim. It has to have some mechanism to keep the physics smooth. The mechanism is germane to events that have no acceleration or curvature involved as you say. But when those things are involved my understanding is that "gauge in-variance" is the accepted understanding of the machinery at work. I'm hoping to have a better understanding of what the gauge does and how.

 

[cianfa72]

It turns out that all inertial frames will measure the same speed for anything moving at c. But in the case at hand, s is not equal to c. So frame choice matters.

ok sure you're right. Thus if just light has the same speed $c$ in any and all inertial frame, are we still able to exploit those imaginary signals to define the synchronization among coordinate clocks spatially separated ?

 

[jbriggs444]

ok sure you're right. Thus if just light has the same speed $c$ in any and all inertial frame, are we still able to exploit those imaginary signals to define the synchronization among coordinate clocks spatially separated ?

Wait a bit. You've used a coordinate system to define a process and then used the process to define a coordinate system? Is that not a bit circular?

Edit: To be clear, you are talking about doing clock synchronization using imaginary signals that are defined to move at speed s relative to a particular coordinate system. Then you are talking about using this synchronization mechanism to define the time coordinate of the clocks making up your coordinate system, right?

 

[Nugatory]

What causes two paths in spacetime be of different length?

The same thing that causes two paths in space to have different length. If I drive from Miami to Denver and then to Boston while you drive from Miami to Washington DC and then on to Boston we will find that your odometer has counted fewer miles than mine. We have no problem saying that the odometer readings are different because we took different paths between Miami and Boston. If asked what caused the path through Denver to be longer, there may be no better answer than that different paths have different lengths. There's nothing different about the miles on the Miami-Denver-Boston path, there are just more of them than on the Miami-Washington-Boston path.

And if two spacetimes paths can be of different length because a) there was some acceleration that caused them to have different inertial frames or b) they are like, what, separated on a curved spacetime manifold? Then my question is how does the odometer account for and "know about" that?

The odometer doesn't "know" anything. It just counts the miles/seconds as they pass by, and there are more of them to count along one path than along the other.

[ from one of your later posts]

I am asking what is different about the events of the accelerating twin? How do those events manifest to the odometer the length of the path... How do they provide information that will later be observed as evidence of difference in path length?

These events are just statements of the form "the twin was at this point in spacetime", just as the paths of the two cars are defined by statements of the form "the car passed by this point". The events themselves don't provide any information about path length; the path has a length whether a car drives along or not. We can assign meaning to the statement "the Miami-Denver-Boston path is longer than the "Miami-Washington-Boston path" without involving any evnts at all.

 

[Jimster41]

That is because there is literally more of a thing, road... what is the road in the case of the traveling twin?

I thought Dale's way of getting out of the idea of saying there is any kind of measurable stuff, any sense of "road" (or land on which it is laid) which exists despite me driving over it and which would support the kind of measure that would apply globally to roads on the surface of the planet was to invoke the idea of sequence or network of events, a road that only exists because I drove down it (I had no choice but to drive down some road though). This is consistent with models of QMGR that I have read about and makes sense the way those do. Having said that my understanding of some Bohmian Models is that there is more of a sense of objective Road. But they are more Machian or Gallilean (I sort of lump those two together) than Einstein's theory allows.

To me you if you are saying only the geometry is different and that's not road, or event or anything it feels like you are invoking the idea that there is this abstract thing called "geometry" that underpins physical reality but which itself is not physical. Even stronger it feels like at times the argument from authority is that "it can't be thought of as physical". Maybe that's just my perception as you all strive to be clear when teaching SR and GR (I understand the bias). But I am pretty sure other serious physics folks are still hopeful that there is a way to describe the "beables" of space-time geometry more physically. "Causal networks" of the kind Dale seemed to be alluding to are something I have heard described as one of the candidate models of how events and space time geometry work.

 

[cianfa72]

Wait a bit. You've used a coordinate system to define a process and then used the process to define a coordinate system? Is that not a bit circular?

Edit: To be clear, you are talking about doing clock synchronization using imaginary signals that are defined to move at speed s relative to a particular coordinate system. Then you are talking about using this synchronization mechanism to define the time coordinate of the clocks making up your coordinate system, right?

No, that was not my point.

Starting from the beginning...the idea is to exploit the physical process propagation of an imaginary signal to assign coordinate time to coordinate clocks sitting on rockets hovering at radial coordinate $r$ and at rest each other. Starting from a far away standard clock (here coordinate time is one-to-one with proper time) we send such signals towards remote clocks assigning half the value of the two-way trip upon such signals reaching them. This way we defined a synchronization procedure to adjust the "zero" of each coordinate clocks.

Then we assume, as you highlighted in post #7, that the rate of each coordinate clock at $r$ is adjusted to tick at $(1-R_s/r)^{-1/2}$ of the local proper time (as insted measured locally by a standard clock).

This way we define a procedure to assign globally the coordinate time to each event (or in other words a global coordinate chart when including also the $r, \theta, \phi$ coordinates).

Now the point is: has the metric in this coordinate chart the same expression as in the well known Schwarzschild form ?

 

[Pencilvester]

That is because there is literally more of a thing, road.

You don’t need anything physical, including a road, in order to define the length of a curve in space—just the space itself and a metric. So it is with paths through spacetime.

it feels like you are invoking the idea that there is this abstract thing called "geometry" that underpins physical reality

In GR, the math that describes the physical reality is geometry.

 

[Dale]

I'm admiring the mountain of Yang-Mills looking up at it.

That is fine, but it has nothing to do with the topic at hand.

How do those events manifest to the odometer the length of the path... How do they provide information that will later be observed as evidence of difference in path length?

The odometer is attached to a tire whose circumference is known to within experimental precision. The odometer counts the number of times that the tire rotates and multiplies by the circumference of the tire to obtain the length of the path.

Did you really not know how an odometer works? This is strange, on the one hand you are happy bringing in irrelevant and complicated topics like Yang-Mills and quantum gravity, but on the other hand you profess to not know how an odometer works. I am not sure how to take that.

That is because there is literally more of a thing, road... what is the road in the case of the traveling twin?

There is more distance. The road is just there to help the odometer measure the distance. In the absence of a road we would measure the distance another way, but the geometry remains.

Time is also geometry. For the traveling twin the distance (spacetime interval) is larger. That distance is measured with a clock.

 

[Jimster41]

You can insult me. It’s true.

But I thought we had established that the gold standard of “clock” is an observatory watching quantum mechanical events. I guess you have a better clock?

So how does that odometer work when you accelerate it. Why/how does the “geometry” change.

I apologize for being QM to the GR forum. I will take my leave.

 

[Dale]

You can insult me. It’s true.

Sorry about that. I did not mean to be insulting, but just wanted to make you aware that the direction that this conversation is heading makes me uncomfortable. This topic is simpler than you are making it out to be and you seem highly resistant to reasonable efforts to simplify and clarify.

But I thought we had established that the gold standard of “clock” is an observatory watching quantum mechanical events. I guess you have a better clock?

Sure, but you don't need Yang-Mills for describing the hyperfine transition, QED will suffice. You also do not need a quantum gravity theory, the geometric aspects here are already built into QED.

Further, you don't even need QM at all for this topic. As with the odometer and the road, the key thing is the distance (spacetime interval). How you measure that distance can vary according to the need. We don't need to use the gold standard for this discussion, a classical pocket-watch or even a steady heart-beat is fine. The key point is that there are different paths and the paths are different lengths and the measurement device (clock or odometer) measures that physical length by whatever appropriate means.

So how does that odometer work when you accelerate it. Why/how does the “geometry” change.

When you accelerate then your worldline is not straight. The odometer simply measures the distance on that non-straight worldline the same way that it measured distance on a straight worldline: it counts the number of revolutions of the wheel on the non-straight path and multiplies by the circumference.

it feels like you are invoking the idea that there is this abstract thing called "geometry" that underpins physical reality but which itself is not physical

Why wouldn’t geometry be physical? I have a table here, it is about as physical a thing as there is. The top is flat and rectangular, the legs are equal lengths, all perpendicular to the top, and all parallel to each other. The geometry is an inherent part of what makes my physical table a table. How can you say its geometry isn’t physical?

Certainly you can have abstract geometry that isn’t physical, but that doesn’t imply that the geometry of physics is not physical. Spacetime’s geometry isn’t material, but it is still physical.

 

[PeterDonis]

am asking what is different about the events of the accelerating twin?

Acceleration does not affect clock rates. Spacetime path length is proper time along an accelerating worldline just as it is along a geodesic worldline. Acceleration is just path curvature: it means the path is not a geodesic, it's curved instead of straight.

In the odometer analogy, suppose one car exactly follows a great circle path while the other doesn't. The first car's path is a geodesic--it's straight; this corresponds to inertial, free-fall motion in spacetime. The second car's path is not geodesic--it's curved; this corresponds to accelerated motion in spacetime. But both odometers register mileage along their respective paths just fine; the curvature of the path does not affect the odometer's ability to do that.

 

[PeterDonis]

So how does that odometer work when you accelerate it. Why/how does the “geometry” change.

You undergoing accelerated motion instead of free-fall motion doesn't change the geometry of spacetime at all. It just makes you follow a curved path through spacetime instead of a straight one.

 

[cianfa72]

No, that was not my point.

Starting from the beginning...the idea is to exploit the physical process propagation of an imaginary signal to assign coordinate time to coordinate clocks sitting on rockets hovering at radial coordinate $r$ and at rest each other. Starting from a far away standard clock (here coordinate time is one-to-one with proper time) we send such signals towards remote clocks assigning half the value of the two-way trip upon such signals reaching them. This way we defined a synchronization procedure to adjust the "zero" of each coordinate clocks.

Then we assume, as you highlighted in post #7, that the rate of each coordinate clock at $r$ is adjusted to tick at $(1-R_s/r)^{-1/2}$ of the local proper time (as insted measured locally by a standard clock).

This way we define a procedure to assign globally the coordinate time to each event (or in other words a global coordinate chart when including also the $r, \theta, \phi$ coordinates).

Now the point is: has the metric in this coordinate chart the same expression as in the well known Schwarzschild form ?

Any comments about this point ? Thanks

 

[PeterDonis]

has the metric in this coordinate chart the same expression as in the well known Schwarzschild form ?

This coordinate chart is standard Schwarzschild coordinates, so yes.

 

[cianfa72]

This coordinate chart is standard Schwarzschild coordinates, so yes.

Thus even using different signals other than light beams to synchronize spatially separated coordinate clocks (or even other procedures), provided that coordinate clocks rates are "adjusted" properly (see post #7 about it), does it result in a standard Schwarzschild coordinate chart (obviously including the spatial coordinates) ?

 

[PeterDonis]

even using different signals other than light beams to synchronize spatially separated coordinate clocks (or even other procedures), provided that coordinate clocks rates are "adjusted" properly (see post #7 about it), does it result in a standard Schwarzschild coordinate chart (obviously including the spatial coordinates) ?

Unless you tell me specifically what other signals you are going to use, and what other clock synchronization procedure you are going to use, I have no idea.

 

[cianfa72]

Unless you tell me specifically what other signals you are going to use, and what other clock synchronization procedure you are going to use, I have no idea.

Actually that was my point around all my posts: to define a coordinate chart for spacetime we need a procedure (possibly thought). Here for instance the nature of the signals involved and the clock synchronization procedure has to be specified in order to define the coordinate time. Then in that just defined coordinate chart the spacetime metric will be described accordingly