Further Understanding Simultaneity Conventions

[Sagittarius A-Star]

Maybe I need a clearer definition of perverse and non-perverse.

As others said, the perverse coordinates describe some events, that are causally connected, to be simultaneous.

This coordinate system is a mixture of a normal Minkowski diagram and a light-cone diagram, where one light-cone axis is interpreted as time-axis.

 

[Freixas]

You left out a third alternative: ask. Asking doesn't make you sound obtuse. It makes you sound like someone who is actually paying attention to what other people post. That's a good thing.

Well, I thought we (everyone in this thread) were having a discussion on the topic. I have been reading all the responses and thinking about them. I appreciate the help from everyone. I think I'm getting there except for the null curve restriction. I'm still listening.

I will post the current state of my thinking once I've had time to review all responses more carefully.

 

[vanhees71]

As others said, the perverse coordinates describe some events, that are causally connected, to be simultaneous.

This coordinate system is a mixture of a normal Minkowski diagram and a light-cone diagram, where one light-cone axis is interpreted as time-axis.

This is misleading. Using light-cone coordinates you have two light-like basis vectors in the Minkowski plane. Neither of those is a time-axis. The transformation from standard Lorentzian coordinates, ##(x^0,x^1)## to light-cone coordinates are
$$x^{\pm}=\frac{1}{\sqrt{2}} (x^0 \pm x^1),$$
and the Minkowski line element reads
$$\mathrm{d} s^2=(\mathrm{d} x^0)^2-(\mathrm{d} x^1)^2=2 \mathrm{d}x^+ \mathrm{d} x^-.$$

 

[Histspec]

The transformation from standard Lorentzian coordinates, ##(x^0,x^1)## to light-cone coordinates are
$$x^{\pm}=\frac{1}{\sqrt{2}} (x^0 \pm x^1),$$
and the Minkowski line element reads
$$\mathrm{d} s^2=(\mathrm{d} x^0)^2-(\mathrm{d} x^1)^2=2 \mathrm{d}x^+ \mathrm{d} x^-.$$

Light-cone coordinates happen to be identical to the asymptotic coordinates of a hyperbola, with the corresponding squeeze mappings (=Lorentz transformations) being essentially known for thousands of years (Apollonius of Perga, ca. 200 BC).

 

[robphy]

Light-cone coordinates happen to be identical to the asymptotic coordinates of a hyperbola, with the corresponding squeeze mappings (=Lorentz transformations) being essentially known for thousands of years (Apollonius of Perga, ca. 200 BC).

Yes, lots of geometrical results have been known for a long time.
Applications to (say) physics require more than just an equation or a geometrical result.
One needs a mathematical model that maps from the appropriate mathematics to the physics.

Sadly, when applications are found,
many don't get used in situations where they can help us understand the physics better (like special relativity).

 

[Freixas]

@Dale provided a clear definition of a simultaneity convention. This definition is included in the OP.

Most physicists seem to accept a restriction to simultaneity conventions that treats a convention as valid only if any pair of simultaneous events can be connected by a spacelike curve. The explanation I’ve received is that this maintain the causal order of events. Otherwise, two events that are cause and effect may occur simultaneously.

The need for this restriction, justified in this way, has not been clear to me. Consider that we assign various values to the one-way speed of light in one direction. Let’s pick constant speeds. I can project lines of simultaneity onto a standard Minkowski diagram using the method shown in the OP. These lines are just skewed versions of the x axis. As we skew the angle more, the one-way speed increases (a counter-clockwise skew increases the speed of light moving from left to right). With a 45 degrees skew, light travels instantaneously in one direction. With larger skews, light travels backwards in time—it arrives before it leaves!

However, all these skews have no effect on the Minkowski diagram on which it is overlaid. We’re just talking about coordinate systems, right? If we can calculate invariants using a skew of 39 degrees, it seems we could calculate them equally well with a skew of 47 degrees.

I’m stating all this just to show why the necessity of the restriction has not been clear to me.

As stated in comment #17, it occurred to me that I could approach the restriction a different way. Because the one-way speed of light is indeterminable, I cannot say that any two events are simultaneous in any absolute way, but I can impose a restriction on “reasonable” simultaneity ranges by looking at events in a causal chain.

This diagram shows a light beam traveling from event A to C and reflected back to B. In any reasonable simultaneity convention, we would expect C to be simultaneous with one of the events from A to B. This may be what people have been referring to as a “non-perverse” coordinate system, but it makes more sense to me when I state it my way.

Note that I haven’t said whether the segment AB is inclusive or exclusive of the endpoints. At this time, I can’t see any reason to make them exclusive, but I’m open to arguments.

I’d like to prove that all simultaneity conventions (as defined by @Dale) that satisfy my premise (that one event between A and B is simultaneous with C) do not contain any simultaneous events connected by a timelike curve. I will try to do this using my method of projecting lines of simultaneity from a given convention onto a Minkowski spacetime diagram.

Let’s say I pick a point D that is simultaneous with A but a bit later in time with respect to the standard Minkowski diagram. Any line from A to D (representing a projection of a line of simultaneity) is clearly timelike. Any curve from A to D will have to have some portion which is timelike.

Given @Dale’s invertible requirement, no other event on the AB segment can be simultaneous with C since it would have to cross the AD curve.

We can use a similar argument by moving D below B and making it simultaneous with B.

We’re not quite done. A line from somewhere between A and B to C will be spacelike or lightlike, but a curve might contain some timelike segments.

All this requires is creating a new light reflection positioned such that, in the new A’, B’, C’ system, the simultaneity convention makes no point on the A’B’ segment simultaneous with C’ and so also violates my initial requirement.

Therefore, the only possibilities left are simultaneity conventions in which no simultaneous events are connected by timelike curves.

Most of you may consider this the long-winded way around something straightforward, but this approach is comprehensible to me. Saying that I can't assign the same time coordinate to two causally-connected events just left me asking "Why?" Now I have a reason.

I haven’t been able to rule out lightlike curves because that requires that I consider an instantaneous one-way speed of light to be “unreasonable” (or maybe “perverse”?). The argument that it makes the emission and reception of a photon simultaneous is equivalent to saying that infinite one-way speed is "unreasonable", so it sounds like a circular argument.

Perhaps someone could make an equivalent causal ordering argument for lightlike connected events? I can’t picture it. Bonus points if you can diagram your answer.

 

[PeterDonis]

Note that I haven’t said whether the segment AB is inclusive or exclusive of the endpoints. At this time, I can’t see any reason to make them exclusive, but I’m open to arguments.

You've already been given the argument. The events A and C, and C and B, are causally connected. That means any "line of simultaneity" that includes both A and C, or both C and B, has causally connected events being simultaneous. But that violates the causal ordering: A is causally before C, and C is causally before B, but having A and C simultaneous, or C and B simultaneous, means there is no ordering. A non-perverse coordinate chart shouldn't do that.

For events between A and B on the blue line, there is no issue, because all of them are spacelike separated from C and so aren't causally connected to C. That is why non-perverse coordinate systems require that simultaneous events be spacelike separated.

I’d like to prove that all simultaneity conventions (as defined by @Dale) that satisfy my premise (that one event between A and B is simultaneous with C) do not contain any simultaneous events connected by a timelike curve.

Sure, that's easy, but it still allows simultaneous events to be connected by null curves. Which is perverse, by the above argument.

Perhaps someone could make an equivalent causal ordering argument for lightlike connected events?

It's already been given. Multiple times now in this thread, including above in this post. If you're just going to stick your fingers in your ears and say you don't believe it, of course we can't stop you, but it makes any further discussion in this thread pointless.

 

[PeterDonis]

The argument that it makes the emission and reception of a photon simultaneous is equivalent to saying that infinite one-way speed is "unreasonable"

The argument that lightlike separated events should not be simultaneous in a non-perverse coordinate chart has nothing whatever to do with the one-way speed of light. See the multiple posts in which the argument has been made in this thread.

 

[Dale]

With larger skews, light travels backwards in time—it arrives before it leaves!

That is exactly the problem. The effect occurs before the cause under this simultaneity convention.

There is nothing that forbids this type of coordinate chart, and indeed some authors accept it. But it is also a reasonable objection say we want to use the word “simultaneity” to refer to a subset of coordinate charts that exclude this behavior.

 

[PeterDonis]

That is exactly the problem. The effect occurs before the cause under this simultaneity convention.

And if the simultaneity convention allowed lightlike separated events to be simultaneous, the effect would occur at the same time as the cause, but spatially separated from it. Which is open to the same reasonable objection you give.

 

[Ibix]

Saying that I can't assign the same time coordinate to two causally-connected events just left me asking "Why?" Now I have a reason.

I guess my question is why you would even consider calling a convention that connected two events on one worldline by the name 'simultaneity convention'. That's fundamentally why our arguments appear tautological: they are. Two events that happen one after another (and unambiguously so, since they can happen to the same clock when it has different readings) are not simultaneous by definition, so it would be odd to say that they happen at the same time.

Also note that in 2+1 dimensions you can build planes that connect one event to two on the same worldline via spacelike paths. Surfaces that do not do this are called "achronal".

 

[PeterDonis]

Also note that in 2+1 dimensions you can build planes that connect one event to two on the same worldline via spacelike paths. Surfaces that do not do this are called "achronal".

Yes, and "achronal" (or more precisely "acausal" since we want to rule out null separated events as well) is the more precise technical term for the kind of surface we have been describing as "spacelike". It's not enough for tangent vectors to the surface to be everywhere spacelike: every single pair of events in the surface must be spacelike separated (not just connectible by a spacelike path but spacelike separated in the invariant sense that each is outside the other's light cone).

 

[Freixas]

I guess my question is why you would even consider calling a convention that connected two events on one worldline by the name 'simultaneity convention'.

See @Dale's definition in the OP. By his definition, such simultaneity conventions exist. @Peter (and others) introduced the term "valid" simultaneity convention. I believe @Dale's view is that one can choose to restrict the range of simultaneity conventions one wants to work with. In the earlier thread linked in the OP, he notes that not all physicists restrict themselves to conventions that forbid timelike (or lightlike) conventions.

So, it's not me.

If you introduce some restriction, there should be a good reason for doing so. The one that has been repeated here is that it's nice to keep causally related events in their causal order.

 

[Dale]

If you introduce some restriction, there should be a good reason for doing so. The one that has been repeated here is that it's nice to keep causally related events in their causal order.

That is a good enough reason.

It is a definition of a word. The bar for a “good enough” reason is exceedingly low. But conversely, different people can and do mean different things by the same word. That is just the way language is.

 

[PeterDonis]

not all physicists restrict themselves to conventions that forbid timelike (or lightlike) conventions

A better way of saying this would be that there are coordinate charts used in physics that do not have one timelike and three spacelike coordinates, as does a standard inertial frame in SR. For example, consider the following coordinates on 1+1 Minkowski spacetime: ##u = t - x##, ##v = t + x##. Lines of constant ##u## and of constant ##v## are both null. But nobody calls such lines "lines of simultaneity".

 

[Ibix]

In the earlier thread linked in the OP, he notes that not all physicists restrict themselves to conventions that forbid timelike (or lightlike) conventions.

You can, of course, use coordinate systems that do not have spacelike coordinate planes. Lightcone coordinates have been mentioned by at least two of us. But you would not call any of their coordinate planes "planes of simultaneity". (Not deliberately anyway - people have been known to make mistakes.)

As I said above, a simultaneity condition implies a coordinate system, or at least part of one. But a coordinate system does not necessarily imply a simultaneity condition.

 

[vanhees71]

Light-cone coordinates happen to be identical to the asymptotic coordinates of a hyperbola, with the corresponding squeeze mappings (=Lorentz transformations) being essentially known for thousands of years (Apollonius of Perga, ca. 200 BC).

Yes, and this makes the "hyperbolic" motion such a difficult issue concerning the question "does a hyperbolically moving charged particle radiate" ;-)).