## Miscellaneous Science Questions

For the discussion of the sciences. Physics problems, chemistry equations, biology weirdness, it all goes here.

Moderators: gmalivuk, Moderators General, Prelates

jewish_scientist
Posts: 1014
Joined: Fri Feb 07, 2014 3:15 pm UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

gmalivuk, can you explain your answer again. I cannot understand it. Thanks.
"You are not running off with Cow-Skull Man Dracula Skeletor!"
-Socrates

doogly
Dr. The Juggernaut of Touching Himself
Posts: 5510
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

If space and time are both infinitely extensible, there is no privileged choice of x and t coordinates.

If space makes a loop (pac man universe, cylinder), then there is a privileged frame in which x has the minimum value for length around the circle, and t is orthogonal to that.

A rocket which leaves earth and comes back later has traveled a spatial distance which is not this minimal circumference.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

Hypnosifl
Posts: 257
Joined: Wed Mar 02, 2011 10:05 am UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

doogly wrote:If space and time are both infinitely extensible, there is no privileged choice of x and t coordinates.

If space makes a loop (pac man universe, cylinder), then there is a privileged frame in which x has the minimum value for length around the circle, and t is orthogonal to that.

A rocket which leaves earth and comes back later has traveled a spatial distance which is not this minimal circumference.

In relativity wouldn't the globally privileged frame be the one where x has a maximum value, not minimal? If you imagine two buoys which are at rest in the privileged frame, then whatever their separation in this frame, inertial observers in motion relative to them should measure a smaller separation due to length contraction.
Last edited by Hypnosifl on Thu Mar 10, 2016 4:58 pm UTC, edited 1 time in total.

doogly
Dr. The Juggernaut of Touching Himself
Posts: 5510
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

Righto.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

Xanthir
My HERO!!!
Posts: 5389
Joined: Tue Feb 20, 2007 12:49 am UTC
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

Tho I suppose minimal value (of zero) would also work? That would correspond to traveling at C in the wrapping direction; length contraction will bring it down to zero radius, right? And that's the only frame that will see this, I think?
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

gmalivuk
GNU Terry Pratchett
Posts: 26634
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

The speed of light is still the same in all frames, though, and presumably still corresponds to zero-length spacetime intervals in all frames.

Interestingly, I believe a moving observer in a sufficiently small such universe would instantly know they were moving, because the copy they saw in front of them would appear to be less far in the past than the copy they saw behind them.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

Yakk
Poster with most posts but no title.
Posts: 11109
Joined: Sat Jan 27, 2007 7:27 pm UTC
Location: E pur si muove

### Re: RELATIVITY QUESTIONS! (and other common queries)

You are at the bottom of a deep gravity well. All constants are arbitrary, feel free to change them if it makes answering the problems easier (just, you know, don't change 99% to 1% or 0%).

Say, deep enough so that photons lose 99% of their energy getting out to infinity.

The gravity well is generated by one solar mass, and the sphere you stand on is uniform non-charged and not rotating. You are of nearly zero height and mass.

There is a space ship hovering 1 light year away, as measured from your position (it takes 2 years for a photon to reach them and come back, as measured by a local clock).

Question 1:
The space ship fires a light pulse at a mirror you are holding. It bounces off your mirror and returns to them. How far away are you in the space ships frame of reference?

Question 2:
The space ship turns off its engine. How long before it hits you by your clock?

Question 3:
The space ship turns off its engine. How long before it hits you by their clock?

Question 2b and 3b:
As measured by photon round-trip times at each end, what does the distance-to-falling-ship and falling-ship-speed look like?

Question 4:
Suppose instead of a space ship, there was a uniform infinitely thin sphere of dust at 1 light year radius. This dust has sufficient mass to turn the planet into a black hole when it touches the planet. From your perspective, is there any difference in the way it falls compared to the space ship?

Question 4b: Now the dust gets heavier. It now has enough mass to create a black hole at 1.5 planet radii. From your perspective, is there any difference in the way it falls compared to the space ship or the Question 4 dust?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

doogly
Dr. The Juggernaut of Touching Himself
Posts: 5510
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

Well, the answer to the first one is 1 light year, because it's not moving. So that's nice. Life is easy.

The other ones require you to do integrals and I can't do them at work. I should probably get back to work. But hey, one down!
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

Yakk
Poster with most posts but no title.
Posts: 11109
Joined: Sat Jan 27, 2007 7:27 pm UTC
Location: E pur si muove

### Re: RELATIVITY QUESTIONS! (and other common queries)

Thanks! I figured that one might be easy, but I wanted to make sure I wasn't missing something.

Like, I dunno, you disagreeing with how much distance there was between you due to something analogous to length contraction.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

tomandlu
Posts: 1108
Joined: Fri Sep 21, 2007 10:22 am UTC
Location: London, UK
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

This came up elsewhere, and I've been trying to find a resource that will deal with this specific set-up, but I can't find anything that helps me...

If you don't have FTL travel, but you do have instantaneous transporters, how would you use that technology to break causality? I sort-of get the examples given when you have spaceships and frames of reference, but I can't work out how those examples would apply with transporters (or, for that matter, how they would apply if you have FTL travel but no communication allowed when a ship was in FTL motion).
How can I think my way out of the problem when the problem is the way I think?

elliptic
Posts: 34
Joined: Tue Aug 14, 2012 2:21 pm UTC
Location: UK

### Re: RELATIVITY QUESTIONS! (and other common queries)

Err... an instantaneous transporter *is* FTL travel...

And "instantaneous" transport in one reference frame can always be backwards-in-time in another. Thats the whole point.

tomandlu
Posts: 1108
Joined: Fri Sep 21, 2007 10:22 am UTC
Location: London, UK
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

elliptic wrote:Err... an instantaneous transporter *is* FTL travel...

And "instantaneous" transport in one reference frame can always be backwards-in-time in another. Thats the whole point.

So can I have an example where I can break causality by using transporter technology? I get that it's possible to observe stuff happening out of sequence, using a star-spanning telescope, but I don't get how you could actually break causality.
How can I think my way out of the problem when the problem is the way I think?

doogly
Dr. The Juggernaut of Touching Himself
Posts: 5510
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

teleport to somewhere else "now," boost a little so that "now" changes meaning, and teleport back into the past lightcone of your original action.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

tomandlu
Posts: 1108
Joined: Fri Sep 21, 2007 10:22 am UTC
Location: London, UK
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

doogly wrote:teleport to somewhere else "now," boost a little so that "now" changes meaning, and teleport back into the past lightcone of your original action.

I know I'm being thick, but that really doesn't help me understand.
How can I think my way out of the problem when the problem is the way I think?

doogly
Dr. The Juggernaut of Touching Himself
Posts: 5510
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

So, you say you understand the examples with FTL spaceships. Could you elaborate on your understanding of how these can break causality as a starting point? A transporter is exactly the same, is the issue.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

tomandlu
Posts: 1108
Joined: Fri Sep 21, 2007 10:22 am UTC
Location: London, UK
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

doogly wrote:So, you say you understand the examples with FTL spaceships. Could you elaborate on your understanding of how these can break causality as a starting point? A transporter is exactly the same, is the issue.

Well, 'understand' would be pushing it, but all the examples seem to depend on communication between ships travelling at FTL. Sub-FTL and transporters don't fit in to the examples in an obvious fashion.

Sure, I can understand that you can observe things out of sequence, but I don't get how you affect things out of sequence.
How can I think my way out of the problem when the problem is the way I think?

doogly
Dr. The Juggernaut of Touching Himself
Posts: 5510
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

This is covered right here actually, in flowy discussiony ways:
/viewtopic.php?f=18&t=51782

and there is a succinct write up here:
http://physics.stackexchange.com/a/54242
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

Copper Bezel
Posts: 2426
Joined: Wed Oct 12, 2011 6:35 am UTC
Location: Web exclusive!

### Re: RELATIVITY QUESTIONS! (and other common queries)

I personally found this diagram helpful for visualizing what light cones mean. Traveling instantaneously through space would be shifting along the horizontal axis of that diagram.
So much depends upon a red wheel barrow (>= XXII) but it is not going to be installed.

she / her / her

Flumble
Yes Man
Posts: 2181
Joined: Sun Aug 05, 2012 9:35 pm UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

What if your teleporters for some reason (magic) need to be at rest wrt. each other and they transport you at 2c wrt. to their rest frame?

doogly
Dr. The Juggernaut of Touching Himself
Posts: 5510
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

Whatever imposed that magic requirement would be a seething abomination unto the notion of causality. But maybe a different sort of abomination?
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

gmalivuk
GNU Terry Pratchett
Posts: 26634
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

But if the entire collection of transporters have to be comoving, then that sets up a privileged rest frame for FTL to happen in, which I believe would avoid the causality problem.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

doogly
Dr. The Juggernaut of Touching Himself
Posts: 5510
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Lexington, MA
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

Right, the privileged rest frame is the abomination.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

speising
Posts: 2335
Joined: Mon Sep 03, 2012 4:54 pm UTC
Location: wien

### Re: RELATIVITY QUESTIONS! (and other common queries)

You could assume there are "jump points" near or in suns (some SF likes to use this mechanism). Stars in the local region don't generally have relativistic velocities relative to each other, so you'd have your nice causality obeying FTL network.

Hypnosifl
Posts: 257
Joined: Wed Mar 02, 2011 10:05 am UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

tomandlu wrote:
doogly wrote:So, you say you understand the examples with FTL spaceships. Could you elaborate on your understanding of how these can break causality as a starting point? A transporter is exactly the same, is the issue.

Well, 'understand' would be pushing it, but all the examples seem to depend on communication between ships travelling at FTL. Sub-FTL and transporters don't fit in to the examples in an obvious fashion.

Sure, I can understand that you can observe things out of sequence, but I don't get how you affect things out of sequence.

You could just have two sets of transporters, one with pads A1 and A2 at rest relative to each other, and another with pads B1 and B2 at rest relative to each other, such that the A1/A2 set are moving at some large sub-light velocity relative to the B1/B2 set. Then I get in A1 and transport instantaneously to A2, with "instantaneously" being defined in the rest frame of the two A's. Then I see that B2 is about to pass right near A2, so I hop on an ordinary spaceship and accelerate to match velocities with B2 when it passes. Then I step onto pad B2 and transport instantaneously to B1, but this time "instantaneously" is defined in the rest frame of the two B's. Now because I have transported instantaneously using two different definitions of simultaneity, it can work out that when I appear at B1, I am actually in the past light cone of the event of my stepping onto pad A1 and transporting away from there (the spacetime diagrams here may be helpful).

If this wasn't possible--if two pads at rest relative to each other (and both inertial) didn't always allow you to transport instantaneously in their own rest frame regardless of how the pads were moving relative to other landmarks like the Earth--then this would violate the first of the two postulates of special relativity, since whatever laws of physics govern the operation of these transports wouldn't work the same way in all inertial reference frames.

gmalivuk
GNU Terry Pratchett
Posts: 26634
Joined: Wed Feb 28, 2007 6:02 pm UTC
Location: Here and There
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

speising wrote:You could assume there are "jump points" near or in suns (some SF likes to use this mechanism). Stars in the local region don't generally have relativistic velocities relative to each other, so you'd have your nice causality obeying FTL network.
If transport is instantaneous relative to the origin point, *any* relative motion allows for causality violation.

And with the separations found between stars, even a very small relative motion will do the trick. At 4.3ly, a relative velocity of 1m/s away from each other will mean signals from the other transporter seem to go back in time 445ms in your transporter's frame. Alpha Centauri is actually coming towards us at about 20km/s, but if it were moving away at that same speed and distance, it would be possible to send signals into the past by nearly 2.5 hours.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
---
If this post has math that doesn't work for you, use TeX the World for Firefox or Chrome

(he/him/his)

Copper Bezel
Posts: 2426
Joined: Wed Oct 12, 2011 6:35 am UTC
Location: Web exclusive!

### Re: RELATIVITY QUESTIONS! (and other common queries)

And even if that wasn't the case, the locally fixed jump points and other fixes like them could only make causality really inconvenient to violate; it'd never be theoretically impossible.
So much depends upon a red wheel barrow (>= XXII) but it is not going to be installed.

she / her / her

tomandlu
Posts: 1108
Joined: Fri Sep 21, 2007 10:22 am UTC
Location: London, UK
Contact:

### Re: RELATIVITY QUESTIONS! (and other common queries)

Thanks all - I'll get back to you when (hopefully) I've got my head round this...
How can I think my way out of the problem when the problem is the way I think?

jewish_scientist
Posts: 1014
Joined: Fri Feb 07, 2014 3:15 pm UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

Gravity bends light. Light is made of photons. Photons are the exchange particle of the electromagnetic force. Therefor, gravity bends electromagnetism.

Imagine I am in deep space and put 2 magnets 1 meter apart. I measure the force of attraction between them as 100 Newtons. I do the same experiment near a massive body. Will my result change; if so, how?
"You are not running off with Cow-Skull Man Dracula Skeletor!"
-Socrates

Flumble
Yes Man
Posts: 2181
Joined: Sun Aug 05, 2012 9:35 pm UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

They're still 1 meter apart in their reference frame, right? Then I guess the attraction will also still be 100 N.

Unless the field lines are what makes magnetism and one side of those is bent more towards the massive body, thus decreasing the resulting attraction. I actually have no idea how magnets work.

Eebster the Great
Posts: 3304
Joined: Mon Nov 10, 2008 12:58 am UTC
Location: Cleveland, Ohio

### Re: RELATIVITY QUESTIONS! (and other common queries)

Locally space still looks flat, so on the small scale it should make no difference. But I imagine large asymmetrical magnetic fields in strong gravitational fields would be warped in some way.

Yakk
Poster with most posts but no title.
Posts: 11109
Joined: Sat Jan 27, 2007 7:27 pm UTC
Location: E pur si muove

### Re: RELATIVITY QUESTIONS! (and other common queries)

Near a strong gravitational field, what do you mean by "1 meter apart". A strong enough gravitational field gradient (bent enough space) makes that an interesting question.

If the field is weak enough that it becomes a boring question, then space is locally flat on that scale, and everything behaves much like it would in flat space elsewhere.

If space isn't flat at a 1 meter scale, then "1 meter apart" isn't a very good description of their relative position.

To be clear, space near earth is really flat at a 1 meter scale. Things "stationary" relative the the earth are just being pushed at 9.8 m/s^2 by some outside force (like the ground, or the air if you are using wings), and you can describe their behavior as if they are in locally flat space with some large object accelerating them "away from the Earth".

Things would get more interesting if you had three 5 cm black holes tightly orbiting each other (at a 50 cm distance). "1 meter apart" becomes a much less useful bit of information there.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Eebster the Great
Posts: 3304
Joined: Mon Nov 10, 2008 12:58 am UTC
Location: Cleveland, Ohio

### Re: RELATIVITY QUESTIONS! (and other common queries)

Yakk wrote:Things would get more interesting if you had three 5 cm black holes tightly orbiting each other (at a 50 cm distance). "1 meter apart" becomes a much less useful bit of information there.

wat

Apart from the fact you just said that distance measurement isn't meaningful, there's no way three black holes could remain in a tight stable orbit.

Yakk
Poster with most posts but no title.
Posts: 11109
Joined: Sat Jan 27, 2007 7:27 pm UTC
Location: E pur si muove

### Re: RELATIVITY QUESTIONS! (and other common queries)

I didn't say stable. And yes, that'll be complex. My point is, you can knot space-time, and something like that would do it.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Flumble
Yes Man
Posts: 2181
Joined: Sun Aug 05, 2012 9:35 pm UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

Yakk wrote:Near a strong gravitational field, what do you mean by "1 meter apart". A strong enough gravitational field gradient (bent enough space) makes that an interesting question.

Isn't "1 meter apart" well-defined as the distance at which a light roundtrip takes 2/c?
Although, bend it enough and you get multiple light paths between the objects. But even then, one* is the shortest path, right?

*I'd say 'the original' or 'real' one, but that's unfair to the other paths.

Yakk
Poster with most posts but no title.
Posts: 11109
Joined: Sat Jan 27, 2007 7:27 pm UTC
Location: E pur si muove

### Re: RELATIVITY QUESTIONS! (and other common queries)

Why must there be one? Whose clock are you using?

But more importantly, in flat space with slow objects, we expect objects to be mostly where they are in the time it takes light to go 1 meter.

In a non-flat space, by the time light goes "1 meter" in "1 second" (whose clock?), it might take ridiculous or impossible forces for light to make a return trip of the "1 meter" in "1 second".

Maybe there is a reasonable way to deal with these problems, but I don't know enough physics to be confident of it.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Flumble
Yes Man
Posts: 2181
Joined: Sun Aug 05, 2012 9:35 pm UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

We're using the clock of the rest frame of both objects. I, for one, am assuming that, because you can only have meaningful measurements of force between the objects when they're at rest w.r.t. each other and you're measuring at one of the objects (since force is mass-, distance- and time-dependent).

Hypnosifl
Posts: 257
Joined: Wed Mar 02, 2011 10:05 am UTC

### Re: RELATIVITY QUESTIONS! (and other common queries)

Flumble wrote:Isn't "1 meter apart" well-defined as the distance at which a light roundtrip takes 2/c?

In special relativity, that definition would only make sense if the light emitter and the light reflector are at rest relative to each other, right? And in general relativity I think there's no coordinate-independent notion of things at different locations being at rest relative to each other (see the discussion here).

If you pick a particular spacelike path between points in spacetime, there is a coordinate-independent notion of proper distance along the path, analogous to proper time along timelike worldlines. So if you "foliate" the 4D spacetime into a stack of 3D spacelike surfaces (which is the same as picking a simultaneity convention--there is no single correct way to do this because of the relativity of simultaneity), then in any given surface, you can look exclusively at spacelike paths which are confined to that surface and which connect a given pair of points in that surface, then I guess you could define the length of the shortest path as "the distance" between those points, relative to this simultaneity convention.

Eebster the Great
Posts: 3304
Joined: Mon Nov 10, 2008 12:58 am UTC
Location: Cleveland, Ohio

### Re: RELATIVITY QUESTIONS! (and other common queries)

Obviously you have to pick a reference frame even in classical physics. But the following logic seems pretty solid to me.

In a given reference frame, either one path between two objects is the shortest or multiple paths are the shortest. Regardless, there is some set of paths in your reference frame for which no shorter path exists, and you can measure their length the way you normally measure things.

Right?

Yakk
Poster with most posts but no title.
Posts: 11109
Joined: Sat Jan 27, 2007 7:27 pm UTC
Location: E pur si muove

### Re: RELATIVITY QUESTIONS! (and other common queries)

When do you measure? Is this measurement symmetrical?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Eebster the Great
Posts: 3304
Joined: Mon Nov 10, 2008 12:58 am UTC
Location: Cleveland, Ohio

### Re: RELATIVITY QUESTIONS! (and other common queries)

Not sure I understand the question. The reference frame is a specific event at a specific velocity. That's where and when you make the measurement. You measure the geodesic distance. It has to be symmetrical . . .