I Coordinate time between spatially separated events in Schwarzschild spacetime

[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