Relativistic simultaneity and symmetry problem

[Ibix]

Locally, your relative velocity will never exceed c. That's a physical fact. Light will always win a fair race.

When you get into remote measurements, it's possible to have coordinate velocities that exceed c. This, however, is a cheat from the same school as "I'm 180cm tall and you're 2m - 180>2 therefore I'm taller than you". In a non-inertial coordinate system the (coordinate) speed of light is not a constant, so naively comparing non-inertial coordinate speeds to the invariant speed of light is wrong in the same sense that my 180>2 argument is wrong. Apples and oranges.

 

[DrGreg]

That relative-velocity has a magnitude strictly less than c.

You can certainly have v>c in non inertial frames. There is nothing wrong with that.

?

Dale is talking about coordinate velocity $\text{d}x/\text{d}t$, which can, in some non-inertial coordinate systems, can take all values "up to infinity".

Robphy is talking about what some people call "relative speed", a coordinate-independent value that can be defined in terms of 4-velocities $\textbf{U}, \textbf{V}$ by $$
\frac{c^2}{\sqrt{1-v^2/c^2}} = | U_a V^a |
$$Defined this way, $v < c$.

 

[Dale]

?

This is one problem discussing non inertial frames. All of the comments that you have received are correct. But they can be talking about different things.

I was talking about the coordinate speed (since that is what you were describing). Coordinate speed is not physically meaningful and in non inertial frames it can easily exceed c (and light doesn't need to go at c in such coordinates either)

 

[Dale]

What is the meaning of "v" in this context?
Regardless of the nature of the frame of reference, an observer's 4-velocity is restricted to pointing into the interior of the future light-cone [and thus its spatial component is always smaller than that of a light-signal].

This v is just the coordinate speed i.e. $dx/dt$, which is what he described earlier

 

[PAllen]

Locally, your relative velocity will never exceed c. That's a physical fact. Light will always win a fair race.

When you get into remote measurements, it's possible to have coordinate velocities that exceed c. This, however, is a cheat from the same school as "I'm 180cm tall and you're 2m - 180>2 therefore I'm taller than you". In a non-inertial coordinate system the (coordinate) speed of light is not a constant, so naively comparing non-inertial coordinate speeds to the invariant speed of light is wrong in the same sense that my 180>2 argument is wrong. Apples and oranges.

coordinate speed exceeding c happens quite naturally in non-inertial coordinates, and not just the trivial case of rotating coordinates. Two examples:

1) In Fermi-Normal coordinates for the typical world line of the non-inertial twin in the twin 'paradox', with turnaround smoothed, the coordinate speed of the inertial twin may exceed c at some point.

2) For the Milne congruence of the Milne (flat spacetime) cosmology model, if you set up coordinates based on one flow line, using radial proper distances from it measured on constant cosmological time slices, you will find superliminal coordinate speeds for distant flow lines. Note, even though each flow line of the congruence is inertial, these coordinates are not, because the constant cosmological time slices are hyperbolic, not Euclidean

 

[Chris Miller]

Really appreciate all the explanation and clarification. The problem for me is trying to translate an ideographic language like math into a phonetic language like English (and pretty much every other). How would A expect to see B's clock perform during her (A's) turnaround second, in which B "jumps" from halfway there to halfway back?

 

[Dale]

How would A expect to see B's clock perform during her (A's) turnaround second, in which B "jumps" from halfway there to halfway back?

Did you read the arXiv paper I linked to? There is no single answer to your question, but that paper gives one good approach.

 

[Ibix]

coordinate speed exceeding c happens quite naturally in non-inertial coordinates, and not just the trivial case of rotating coordinates.

Agreed. But the coordinate speed of light exceeds c in those cases too. Which makes the comparison of coordinate speeds to c somewhat irrelevant, no? Or at least, only a graphic illustration of the importance of the "inertial" bit of "invariant in all inertial frames".

Maybe my analogy to comparing mumbers without units is a bit of a stretch. Perhaps comparing someone's wage today to the average wage in my grandfather's time is a better analogy. The modern number will almost certainly be bigger, but that doesn't mean we're almost all above average income. It means that it's inappropriate to compare a wage to another 70 years of inflation later. Similarly it's inappropriate to compare a coordinate speed in an inertial frame to the invariant speed of light. The context is wrong.

 

[Chris Miller]

Thanks Dale. Found this googling for a comprehensible definition of radar time: On the radar method in general-relativistic spacetimes
"Summary. If a clock, mathematically modeled by a parametrized timelike curve in a general-relativistic spacetime, is given, the radar method assigns a time and a distance to every event which is sufficiently close to the clock. Several geometric aspects of this method are reviewed and their physical interpretation is discussed."

Whereas your "On Radar Time and the Twin ‘Paradox’" paper seems to say that the twin paradox can be resolved (correctly understood) via radar time without resorting to general relativity. The math is beyond my fluency. But I find their assertion that (prior to this paper) so much was misunderstood interesting.

Perhaps wrongly, I'm more interested in what SR would predict a clock in another frame of reference, however remote or non inertial, would read, and less interested in how it might theoretically be read.

Related to this, and something that still bothers me, is how quantum entanglement ( Einstein's “spooky action at a distance” ) would be effected across A's and B's frames. Their different views of simultaneity (that A's now is not B's now) seems to call into question the concept of entanglement. In other words, how can you have "instantaneous connection over long distances" where instantaneous is not mutually agreed upon?

 

[PAllen]

Perhaps wrongly, I'm more interested in what SR would predict a clock in another frame of reference, however remote or non inertial, would read, and less interested in how it might theoretically be read.

Wrongly, indeed. The crucial understanding is that the answer to this question is purely a matter of convention, not physics. To make it a question of physics, you must posit at least an in principle measurement. Then, no matter which convention you use for 'now at a distance', you will make the same prediction for the physics, i.e. the result of the measurement.

 

[Nugatory]

Perhaps wrongly, I'm more interested in what SR would predict a clock in another frame of reference, however remote or non inertial, would read, and less interested in how it might theoretically be read.

The two questions are inseparable. Asking what a remote clock would read is equivalent to asking what it reads "now"; that requires identifying the point on the worldline of that clock which you're going to call "now", and that requires specifying how you would go about reading the clock - the procedure for reading the clock provides the definition of the "now" point.

 

[Nugatory]

Related to this, and something that still bothers me, is how quantum entanglement ( Einstein's “spooky action at a distance” ) would be effected across A's and B's frames. Their different views of simultaneity (that A's now is not B's now) seems to call into question the concept of entanglement. In other words, how can you have "instantaneous connection over long distances" where instantaneous is not mutually agreed upon?

An easy way to escape this dilemma is to not introduce the word "instantaneous" into the discussion. "Instantaneous" is not part of the mathematical formalism of quantum mechanics; it's something that shows up in non-relativistic collapse interpretations, so of course it is a poor fit with relativistic thought experiments.

Causality and the relative time ordering of events is a more interesting question. But consider a classic entanglement experiment: Bob measures his particle and Alice measures her particle. Bob gets spin-up and Alice gets spin-down. If their measurement events are space-like separated, there will be inertial frames in which Alice's measurement happened first and Bob's result was determined by Alice's measurement; and there will be inertial frames in which Bob's measurement happened first and Alice's result was determined by Bob's result. As with any other frame-dependent situation, both descriptions are equally valid and consistent with any imaginable experiment.

[Further discussion of this interesting digression should happen over in the quantum mechanics subforum - but only after you have reviewed the innumerable threads on the impossibility of using entanglement to communicate]

 

[Dale]

Whereas your "On Radar Time and the Twin ‘Paradox’" paper seems to say that the twin paradox can be resolved (correctly understood) via radar time without resorting to general relativity.

Yes, that is correct. The twin paradox is in flat spacetime so GR is not needed.

Perhaps wrongly, I'm more interested in what SR would predict a clock in another frame of reference, however remote or non inertial, would read, and less interested in how it might theoretically be read

I don't understand the distinction you are making here.

 

[Chris Miller]

I don't understand the distinction you are making here.

Probably neither did I. Nugatory cleared it up a bit for me with his, the procedure for reading the clock provides the definition of the "now" point.

 

[Chris Miller]

coordinate speed exceeding c happens quite naturally in non-inertial coordinates, and not just the trivial case of rotating coordinates. Two examples:

1) In Fermi-Normal coordinates for the typical world line of the non-inertial twin in the twin 'paradox', with turnaround smoothed, the coordinate speed of the inertial twin may exceed c at some point.

2) For the Milne congruence of the Milne (flat spacetime) cosmology model, if you set up coordinates based on one flow line, using radial proper distances from it measured on constant cosmological time slices, you will find superliminal coordinate speeds for distant flow lines. Note, even though each flow line of the congruence is inertial, these coordinates are not, because the constant cosmological time slices are hyperbolic, not Euclidean

Thanks, PAllen. This helped. I believe 1) is the case I tried to describe in my question (except both twins are non-inertial). I think there's some confusion between c as the speed of light for the "world line [frame of ref?] of the non-inertial twin" and c as 299,792,458 m/s? Like where "the coordinate speed of the inertial twin may exceed c" it's clear you mean 299,792,458 m/s and not the coordinate speed of light (which would always be > the twin's). Maybe c is a constant with caveats...

 

[PAllen]

Thanks, PAllen. This helped. I believe 1) is the case I tried to describe in my question (except both twins are non-inertial). I think there's some confusion between c as the speed of light for the "world line [frame of ref?] of the non-inertial twin" and c as 299,792,458 m/s? Like where "the coordinate speed of the inertial twin may exceed c" it's clear you mean 299,792,458 m/s and not the coordinate speed of light (which would always be > the twin's). Maybe c is a constant with caveats...

That is a very good point! Even when coordinate speed of light may vary from event to event (in those coordinates), and may greatly exceed a standard value, you never have the trajectory of a material body overtaking nearby light.

 

[nitsuj]

Thanks, PAllen. This helped. I believe 1) is the case I tried to describe in my question (except both twins are non-inertial). I think there's some confusion between c as the speed of light for the "world line [frame of ref?] of the non-inertial twin" and c as 299,792,458 m/s? Like where "the coordinate speed of the inertial twin may exceed c" it's clear you mean 299,792,458 m/s and not the coordinate speed of light (which would always be > the twin's). Maybe c is a constant with caveats...

That is a very good point! Even when coordinate speed of light may vary from event to event (in those coordinates), and may greatly exceed a standard value, you never have the trajectory of a material body overtaking nearby light.

agreed it's a great way to illustrate the difference between measurements and calculations & physically fundamental constants. So often people use speed in quotes in the comment the "speed" of light. c less like some traditional or attainable speed / velocity.

 

[Dale]

Thanks, PAllen. This helped. I believe 1) is the case I tried to describe in my question (except both twins are non-inertial). I think there's some confusion between c as the speed of light for the "world line [frame of ref?] of the non-inertial twin" and c as 299,792,458 m/s? Like where "the coordinate speed of the inertial twin may exceed c" it's clear you mean 299,792,458 m/s and not the coordinate speed of light (which would always be > the twin's). Maybe c is a constant with caveats...

Personally, I try to consistently use "c" to mean the defined constant 299792458 m/s. I try to use "speed of light" to refer to the speed at which a pulse of light travels. In inertial frames they are the same. If I wish to refer to the frame invariant geometric concept then I try to say "invariant speed".

I wouldn't say there are caveats, just different concepts that are closely related with inconsistent terminology.