Relativity of simultaneity doubt

[Dale]

And maybe it's just a matter of name (you used the term "convention") but to me it's not a convention, in a simple SR model the straight line, plane or R3 containing simultaneity events are precisely defined for the observer.I s here something more to that?

It is definitely a convention. In fact, I think there are at least 3 conventions used here:

First, it is a convention to say that the one-way speed of light is c. This convention has been extensively investigated by Reichenbach and others. Second, it is a convention to use inertial observers. With non-inertial observers the simultaneity defined by using the first convention is not even momentarily the same as for momentarily co-moving inertial observers. Third, it is a convention to pick which inertial observer to use. Even using the first and second conventions, simultaneity is different for different observers.

From your comments you seem to be aware at least of the third convention, but maybe not the first and second.

 

[LBoy]

I'd define the simultaneity of events wrt. an observer in a frame-independent way by: two events with spacetime fourvectors $x$ and $y$ are simultaneous wrt. an observer with four-velcity $u$ if $u \cdot (x-y)=0$.

I think this definition is not fundamentally different from mine. Before PAllen answers I would like to know your view on the issue of simultaneity: convention or physical reality? Because I admit I don't understand this argument.

 

[PeroK]

I think this definition is not fundamentally different from mine. Before PAllen answers I would like to know your view on the issue of simultaneity: convention or physical reality? Because I admit I don't understand this argument.

It's definitely convention. Take the example where there are two aliens on a planet in the Andromeda galaxy, about 2 million light years from Earth. To simplify things let's assume their planet and the Earth are actually at rest relative to each other.

If one of the aliens moves at a speed $v$ away from the direction of the Earth, and uses the Einstein simultaneity convention, then this changes its "now" on Earth by $\frac v c$ times 2 million years. Even at a speed of $30 m/s$ that amounts to several months on Earth.

Every time the alien changes direction, there is a five-month change in the events that are happening "now" on Earth. How can that be physically meaningful?

 

[LBoy]

It is definitely a convention. In fact, I think there are at least 3 conventions used here:

First, it is a convention to say that the one-way speed of light is c. This convention has been extensively investigated by Reichenbach and others. Second, it is a convention to use inertial observers. With non-inertial observers the simultaneity

Oh yes, 1 and 2 are pretty obvious, using non-inertial observers is beyond my question, the question concerns only the third point: defining simultaneity for a choosen observer is not a convention, it is a precise term (mathematical and physical too - imho) for this chosen observer.

PAllen's comment suggested (to my perception) otherwise, hence my question.

 

[vanhees71]

To define simultaneity of spatially distant events you need a clock-synchronization convention. The one used by Einstein in his famous paper of 1905 with the two-speed light to synchronize clocks at rest relative to an inertial frame of reference is an example, and it's in a sense preferred, because it leads to the pseudo-Cartesian coordinates in Minkowski space related to this frame, leading to the most simple form of the invariant special-relativistic equations of motion.

 

[Dale]

defining simultaneity for a choosen observer is not a convention

I disagree. It is indeed a convention as shown most clearly by Reichenbach. It is clearly a completely standard convention, but nonetheless a convention.

In what way can you specify simultaneity for a chosen observer without using all three conventions?

Oh yes, 1 and 2 are pretty obvious

Yes, they are obvious, but they are still conventions. A convention doesn’t cease being a convention just because it is obvious.

 

[LBoy]

Yes, they are obvious, but they are still conventions. A convention doesn’t cease being a convention just because it is obvious.

It is obvious that they are conventions, my bad that I didn't specify that I was thinking about obvious conventions, not about general things being obvious, I will be more precise from now. :)

In what way can you specify simultaneity for a chosen observer without using all three conventions?

In the simplest 2-D case of SR, inertial observers, simultaneity of two events for an observer is defined as there exists a line orhthogonal to the observer's time vector that is joining these two events. Ie these events are in his 3-D "space".

 

[ergospherical]

It's straightforward to show that @vanhees71's definition #105 follows from the basic convention of Einstein-Poincaré simultaneity [a special case of Reichenbach simultaneity for $\epsilon = \dfrac{1}{2}$]. If light is emitted from a point $p_1 \in \gamma$ at parameter $\lambda_1$ on the worldline $\gamma$ of an observer, and then reflected at some event $q$ back toward another point $p_2 \in \gamma$ at parameter $\lambda_2$ on the worldline, the event $p \in \gamma$ which is EP-simultaneous with $q$ is the one at parameter\begin{align*}
\lambda = \dfrac{1}{2}(\lambda_1 + \lambda_2)
\end{align*}Assume, to begin, that the 4-velocity $\mathbf{u}$ of the observer is constant, then\begin{align*}
\overrightarrow{p_1 p} = (\lambda - \lambda_1)\mathbf{u} \\
\overrightarrow{p p_2} = (\lambda_2 - \lambda) \mathbf{u}
\end{align*}Meanwhile the vectors $\overrightarrow{p_1 q} = \overrightarrow{p_1 p} + \overrightarrow{pq}$ and $\overrightarrow{qp_2} = \overrightarrow{qp} + \overrightarrow{pq_2}$ are null (they are the paths of a light ray) and therefore\begin{align*}
\overrightarrow{p_1 q} \cdot \overrightarrow{p_1 q} &= (\overrightarrow{p_1 p} + \overrightarrow{pq}) \cdot (\overrightarrow{p_1 p} + \overrightarrow{pq}) \\
&= (\lambda - \lambda_1)^2 \mathbf{u} \cdot \mathbf{u} + 2 (\lambda - \lambda_1)\mathbf{u} \cdot \overrightarrow{pq} + \overrightarrow{pq} \cdot \overrightarrow{pq} \overset{!}{=} 0 \\ \\

\overrightarrow{qp_2} \cdot \overrightarrow{qp_2} &= (\overrightarrow{qp} + \overrightarrow{pp_2}) \cdot (\overrightarrow{qp} + \overrightarrow{p p_2}) \\
&= \overrightarrow{qp} \cdot \overrightarrow{qp} + 2 (\lambda_2 - \lambda) \mathbf{u} \cdot \overrightarrow{qp} + (\lambda_2 - \lambda)^2 \mathbf{u} \cdot \mathbf{u} \overset{!}{=} 0
\end{align*}Assuming $\mathbf{u}$ to be normalised as $\mathbf{u} \cdot \mathbf{u} = -1$, and re-writing $\overrightarrow{qp} = - \overrightarrow{pq}$, this becomes\begin{align*}
-(\lambda - \lambda_1)^2 + 2 (\lambda - \lambda_1)\mathbf{u} \cdot \overrightarrow{pq} + \overrightarrow{pq} \cdot \overrightarrow{pq} = 0 \\

\overrightarrow{pq} \cdot \overrightarrow{pq} - 2 (\lambda_2 - \lambda) \mathbf{u} \cdot \overrightarrow{pq} - (\lambda_2 - \lambda)^2 = 0
\end{align*}subtracting:\begin{align*}
(\lambda_2 - \lambda)^2 - (\lambda - \lambda_1)^2 - 2 ( \lambda_1 + \lambda_2)\mathbf{u} \cdot \overrightarrow{pq} &= 0 \\\implies (\lambda_2 + \lambda_1)(\lambda_1 + \lambda_2 - 2\lambda) - 2( \lambda_1 + \lambda_2)\mathbf{u} \cdot \overrightarrow{pq} &= 0
\end{align*}Therefore $\lambda = \dfrac{1}{2}(\lambda_1 + \lambda_2) \iff \mathbf{u} \cdot \overrightarrow{pq} = 0$. This is the equation of a hyperplane $\Pi \subseteq \mathbf{R}^4$ with normal $\mathbf{u}$, and is referred to as a surface of simultaneity.

If the 4-velocity is not constant along the worldline, then the above reasoning still holds providing the points $q$ are sufficiently "close" to the worldline [i.e. small compared to the curvature $1/(\mathbf{a} \cdot \mathbf{a})$, where $\mathbf{a} = \dfrac{d\mathbf{u}}{d\lambda}$], so that the curvature of the worldline can be neglected.

 

[vanhees71]

Another way to understand that simultaneity of spatially distant events is conventional in the sense that you need a clock synchronization is to remember that at the most fundamental level all observable facts must be expressed as space-time coincidences.

E.g., if you say that "Alice arrives at the same time in Chicago as Bob arrives in New York" you can only make sure that this is correct by observation, if you have synchronized clocks in Chicago and New York, because all there objectively is in relativity (this is even more pronounced in GR than in SR) are coincidences of points in spacetime. So the above sentence in fact means that the synchronized clocks in Chicago and New York showed the same time as Alice was at her clock's position as Bob's clock when he arrived at his clock's position. Thus the simultaneity of these two events A and B depends on how you synchronized these clocks.

Of course there are more or less convenient clock-synchronization conventions. It's like more or less convenient choices of coordinates for a given situation. E.g., Einstein's standard synchronization convention in SR as described in his paper of 1905 is way more convenient than Reichenbach's example for an alternative synchronization procedure, which leads to much more complicated descriptions:

http://arxiv.org/abs/1001.2375

 

[LBoy]

To define simultaneity of spatially distant events you need a clock-synchronization convention. The one used by Einstein in his famous pape

I would like to think that simultaneity is something that is independent of a clock synchronization procedure, like a real and physical measure (or maybe rather a feature of a space time for a choosen observer) with a precise definition (mathematical), regardless of the clock-synchronization procedure. It "is there" as a feature regardless if we can measure it or not.

 

[cianfa72]

With non-inertial observers the simultaneity defined by using the first convention is not even momentarily the same as for momentarily co-moving inertial observers.

Can you elaborate this point, please ? Thanks.

 

[PeroK]

I would like to think that simultaneity is something that is independent of a clock synchronization procedure, like a real and physical measure (or maybe rather a feature of a space time for a choosen observer) with a precise definition (mathematical), regardless of the clock-synchronization procedure. It "is there" as a feature regardless if we can measure it or not.

Looking for simultaneity to be physically meaningful is a dead end, in any case, if you wish to study GR.

 

[LBoy]

Looking for simultaneity to be physically meaningful is a dead end, in any case, if you wish to study GR.

Not a doubt here, eventually it can be preciselly defined locally in a tangent space at a point, but physically I think this has no sense (although there is a local "similarity" between a tangent space and a R4 "space-time" with cartesian coordinates, but this makes less and less physical sense anyway as the curvature of spacetime increases...)

 

[Dale]

Can you elaborate this point, please ? Thanks.

Yes. This is radar-coordinates which can be applied for non-inertial observers. Radar coordinates are defined by assuming that the second postulate holds even for a non-inertial observer. So for every event the observer sends a radar signal out and gets a radar echo back. The distance to the event is the difference in time (echo - emission) divided by two and the time of the event is the sum divided by two.

I learned about this method from Dolby and Gull’s paper: https://arxiv.org/abs/gr-qc/0104077

See figures 5 and 9 for how simultaneity differs for accelerated and inertial observers in the twin paradox.

 

[PeroK]

Not a doubt here, eventually it can be preciselly defined locally in a tangent space at a point, but physically I think this has no sense (although there is a local "similarity" between a tangent space and a R4 "space-time" with cartesian coordinates, but this makes less and less physical sense anyway as the curvature of spacetime increases...)

A tangent space is a local construction. The only definition of simultaneity that would make any sense in GR is that two events have the same timelike coordinate. And that is manifestly a coordinate-dependent definition.

 

[ergospherical]

@LBoy more generally in a curved spacetime, the ability to synchronise clocks via the Einstein-Poincaré simultaneity criterion reduces to the condition that $\displaystyle{\oint} \dfrac{g_{0i}}{g_{00}} dx^i = 0$ around any closed curve, where $i$ runs over $1,2,3$.

[You can show it by starting with the same idea in #113 about sending and receiving light signals, except now writing the metric as $g = g_{ij} dx^i dx^j + 2g_{0i} dx^0 dx^i + g_{00} dx^0 dx^0$ and solving for the zeroes of $g$ w.r.t. $dx^0$; these are the times of emission and reception]

 

[cianfa72]

Radar coordinates are defined by assuming that the second postulate holds even for a non-inertial observer.

I read the paper. Aboout your claim above I think the point is just that we are assuming that in the (t,x) inertial frame the radar signal sent from non-inertial (proper) accelerating observer's worldline has the same equation $x=\pm t$.

Btw, I believe in Figure 4 $u$ and $v$ axes are actually reversed.

 

[Dale]

Btw, I believe in Figure 4 $u$ and $v$ axes are actually reversed.

No, I think those are correct. You can get the $u$ axis by setting $v=0$ which gives the line $y=x$ as shown in the figure.

 

[cianfa72]

No, I think those are correct. You can get the $u$ axis by setting $v=0$ which gives the line $y=x$ as shown in the figure.

Sorry you're right I confused them. What about my claim above about radar signals ? Is it actually the same as yours ? Thank you.

 

[Dale]

Sorry you're right I confused them. What about my claim above about radar signals ? Is it actually the same as yours ? Thank you.

I don’t know. I didn’t really understand the point you were trying to make there. Could you rephrase, perhaps?

 

[cianfa72]

Could you rephrase, perhaps?

Surely. You said that radar coordinates definition assumes the second postulate holds even for non-inertial observers. Second postulate is about the invariant one-way speed of light, right ?

 

[Dale]

Surely. You said that radar coordinates definition assumes the second postulate holds even for non-inertial observers. Second postulate is about the invariant one-way speed of light, right ?

Yes, that is correct.

 

[cianfa72]

Yes, that is correct.

So the assumption is that the one-way coordinate speed of light even for non-inertial observers is always the same c.

 

[Dale]

So the assumption is that the one-way coordinate speed of light even for non-inertial observers is always the same c.

Yes. I think that the only time a speed is not a coordinate speed is when it is the relative speed in a collision. But in any case, that is what I had in mind here.

 

[PAllen]

The Minkowski diagram (Italicus post 30) shows that the two signal reception events by OT and OE are not "colocated at the same event", we are dealing with two different events: OE receives a lightning strike signal, and OT receives two signals. These events are separated both temporally and spatially.

My point was simply that in the initial post where I raised this, you were apparently disputing the claim that two observers colocated at the same event but with relative motion must either both receive two signals or neither. They cannot disagree about such things. The post you were criticizing made no claim about non-colocated observations.

The second issue I have is the existence of simultaneity of events (what you call, I believe, modeling simultaneity). Here I use a mathematical definition: events are simultaneous to an observer if they are orthogonal to its time vector in Minkowski space. And maybe it's just a matter of name (you used the term "convention") but to me it's not a convention, in a simple SR model the straight line, plane or R3 containing simultaneity events are precisely defined for the observer.I s here something more to that?

A mathematical definition is not physical unless it can be observed. By causality in SR, nothing about a spacelike separated event can be observed, in any way. Thus simultaneity is inherently not physical in SR. Further, SR in no way requires you to use standard Minkowski coordinates (let alone a particular choice of these), and any choice of coordinates is convention not physics. The world does not care how you label things. The only physical relation between distant events in SR is that either one is in the causal future of the other, or not. In SR (but not GR, in general), a non-causal relation means the events can be connected by a unique spacelike geodesic.
[edit: as an example, consider the following book:

https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20

standard Minkowski coordinates and the Lorentz transform are not even introduced until all the essential physics of SR is developed in a coordinate free manner. And simultaneity is introduced as convention.]

 

[italicus]

@Dale
As far as I know, the problem of the one-way speed of light has not been solved yet. The constancy of c in vacuum, in any inertial reference frame, is a “postulate “ assumed by Einstein, and is valid in SR only, where you can imagine inertial reference frames extended without limits. His sincronisation of clocks is based on forth and back light signals , and has been criticized by Reichenbach and others.
But in GR the situation is different. It is impossibile to have r.f. extended to the infinite. One of the formulations of the equivalence principle is this : “ In every point of ST you can put an inertial rf. , that is a local frame in “free fall”, abandoned to gravity, in which the laws of physics are those of SR”.
In GR you cannot compare a vector “here “ to a very far vector “there “ , unless you don’t carry the first near the second.
Wolfgang Pauli said very clearly: we cannot say nothing about the speed of light very far from us.
The same J.H. Wheeler used to say: Physics is easy only locally.
There are a lot of works about the one way speed of light problem , it seems that even recently , by means of GPS clocks signals, an anysotropy in c value has been found.
Are you sure that time flow is the same here and in a galaxy 100 millions yl far? I am not so sure.
but in SR a uniform lattice of sticks and synchronized watches is supposed all over ST, have a look at paragraph 2.6of the book “ Spacetime physics “ by Taylor and Wheeler.

But I am not an expert verified scientist, feel free to correct what I have written, where necessary.

 

[cianfa72]

@Dale
As far as I know, the problem of the one-way speed of light has not been solved yet. The constancy of c in vacuum, in any inertial reference frame, is a “postulate “ assumed by Einstein, and is valid in SR only, where you can imagine inertial reference frames extended without limits. His syncronization of clocks is based on forth and back light signals , and has been criticized by Reichenbach and others.

Btw, it can be proven that the constancy of one-way speed of light follows (logically) from the universal speed of light over closed paths -- https://arxiv.org/pdf/gr-qc/0211091.pdf

 

[vanhees71]

A tangent space is a local construction. The only definition of simultaneity that would make any sense in GR is that two events have the same timelike coordinate. And that is manifestly a coordinate-dependent definition.

In GR all there is are local observations, i.e., spacetime-point coincidences. This was in fact a pretty much debated point in the 8-10 years from Einstein's first approaches till the final version of GR has been found. To discuss "simultaneity" one has to refer to some coordinate time, and it's thus even more conventional than in SR, where you have at least a preferred global simultaneity convention already formulated in Einstein's 1905 paper using the two-way speed of light with the "light clock" to synchronize a continuum of clocks at rest within an inertial reference frame. Already for accelerated observers in you only have a local definition of simultaneity. An illuminating example that you cannot have a global definition of simultaneity or clock synchronization are observers on a rigidly rotating disk:

https://en.wikipedia.org/wiki/Born_coordinates

 

[Dale]

Btw, it can be proven that the constancy of one-way speed of light follows (logically) from the universal speed of light over closed paths -- https://arxiv.org/pdf/gr-qc/0211091.pdf

Be careful. You are overstating the claim made in that paper. The actual claim is that the universal speed of light over closed paths (ie the two way speed of light) implies the possibility of a one-way speed of light synchronization convention. It is an “existence” proof not a “uniqueness” proof. It does not exclude other synchronization conventions.

Also, it is better to cite the arxiv abstract page instead of the pdf: https://arxiv.org/abs/gr-qc/0211091

 

[Dale]

As far as I know, the problem of the one-way speed of light has not been solved yet.

I am not sure what you mean. I don’t know what problem there is. The one way speed of light is a convention to choose, not a problem to solve.

As far as I can tell the only problem is that people think it can be measured. But that is an education problem.

 

[cianfa72]

Be careful. You are overstating the claim made in that paper. The actual claim is that the universal speed of light over closed paths (ie the two way speed of light) implies the possibility of a one-way speed of light synchronization convention. It is an “existence” proof not a “uniqueness” proof. It does not exclude other synchronization conventions.

ok got it. Basically it proves that -- under the hypothesis of constant universal speed of light over closed paths -- we can define a synchronization convention such that all clocks at rest each other can be consistently Einstein synchronized and the one-way speed of light results to be the invariant constant c.

 

[cianfa72]

Just a point about what we said above: the proof in the Minguzzi/Macdonald paper assumes clocks at rest each other.

How can we actually define 'mutually at rest' if not using again light signals over closed paths ? (i.e. defining two clocks as at rest each other if the round-trip time of a light signal exchanged between them does not change).

 

[italicus]

I am not sure what you mean. I don’t know what problem there is. The one way speed of light is a convention to choose, not a problem to solve.

As far as I can tell the only problem is that people think it can be measured. But that is an education problem.

The following article is one of a lot, that can be found on the web:

https://www.intechopen.com/chapters/39778#B14

Furthermore, have a look at paragraph 84 , chapter X , of Landau-Lifshitz “classical theory of fields” , for distances and time intervals in GR.

The problem is that, if the light speed from A to B is different wrt that from B to A, the Einstein sincronization is no longer possible. The cited author concludes that the Lorentz transforms aren’t correct, and should be replaced by others. I hope I have read the article in the right way.

 

[weirdoguy]

The following article is one of a lot, that can be found on the web:

Sorry, but this author is not a physicist. In one of his papers he writes:

Einstein's equivalence principle, the idea underpinning the general theory of relativity (GTR), is examined and shown to be invalid. As a result, GTR collapses and can no longer be considered a viable physical theory.

which is nonsense. Check your sources, please. For physicists there is no problem with one way speed of light. For others - well, not everyone understands relativity. That's their problem

 

[italicus]

Sorry, but this author is not a physicist. In one of his papers he writes:
which is nonsense. Check your sources, please. For physicists there is no problem with one way speed of light. For others - well, not everyone understands relativity. That's their problem

Thank you for the information, I didn’t really check the source, sorry . But I wish to inform you that professor Franco Selleri, cited in the article, was (he died some years ago) a well-known physicist, taught physics and relativity in many universities, wrote a lot of books and articles and made a lot of conferences all over the world. You can check his profile on the Internet.
For him, that understood relativity very well, the one-way speed of light was a problem.

Anyway, this doesn’t mean that I reject relativity , on the contrary! But there are some unsolved problems.