- #1

Freixas

- 307

- 42

**Summary**

Almost a year ago, I created a post titled “Understanding the phrase 'simultaneity convention'”. The answers included requirements for defining a simultaneity convention. But some simultaneity conventions, while meeting all the requirements, still appear problematic. What am I missing?

**Answers form my earlier post:**

From @Dale:

*A coordinate chart is a mapping between events in spacetime and points in R4. There are very few requirements. The mapping must be smooth (diffeomorphic) and one-to-one (invertible). Other than that you are not restricted.*

The simultaneity convention is then just the convention you used to choose which events share the same t coordinate. The usual implication is that the coordinate basis vector for t is timelike, but I am not certain even that is actually required.

The simultaneity convention is then just the convention you used to choose which events share the same t coordinate. The usual implication is that the coordinate basis vector for t is timelike, but I am not certain even that is actually required.

From @PeterDonis:

*A spacetime is a 4-dimensional set of events. A "simultaneity convention" is a way of breaking up the spacetime into disjoint 3-dimensional subsets, such that all of the events in each subset are defined to happen "at the same time". This requires that, for each subset, all of the events in the subset are spacelike separated from each other (meaning that no two events can be connected by a timelike or null curve).*

Given the above, the possible range of "simultaneity conventions" is obvious: any division of 4-dimensional spacetime into 3-dimensional subsets that meets the above requirements is a valid "simultaneity convention".

Given the above, the possible range of "simultaneity conventions" is obvious: any division of 4-dimensional spacetime into 3-dimensional subsets that meets the above requirements is a valid "simultaneity convention".

**Discussion**

I have tried searching for a good, layman's definition of a “coordinate chart” and failed. Most definitions read, to me, as gobbledygook. Here is my best guess as to what Dale said:

Spacetime as a collection of events. These events can be assigned coordinates using various systems; for example, Cartesian coordinates, polar coordinates, etc. Could a coordinate chart represent a choice of such a system? Even when we choose a system, we cannot assign coordinates to events without selecting an observer and a simultaneity convention.

If this guess is correct, then I can sort of follow Dale's answer, although it sounds as though a simultaneity convention could be any arbitrary method of assigned a time coordinate as long as the assignment is smooth and one-to-one. I'm not at all sure that my guess is correct.

Peter's answer seems similar, but written more for a layman. He has no diffeomorphic requirement, but I'll assume it's there. Peter adds the concept that within the set of all possible simultaneity conventions, some are valid and some are invalid. Dale noted elsewhere that the requirement Peter added is not universally required by all physicists.

**The missing observer**

We know that events that are simultaneous for one observer will not necessarily be simultaneous for another. Therefore, I would have thought that any definition of simultaneity would include “relative to an observer”. As I mentioned above, perhaps this is implied as part of the coordinate chart. It's not clear.

I would think that the if we wanted to preserve Einstein's first postulate, whatever simultaneity convention we chose would be the same relative to

*any*observer—we wouldn't have one convention for one observer and a different one for another.

**Simultaneity conventions and the one-way speed of light**

Einstein establishes simultaneity by calculating how long a light pulse takes to travel a known distance. If a set of observers are at rest relative to each other, their clocks mark time at the same rate. If one observer sends a light pulse to the others at time 0, then the receiving observers can synchronize their clocks by setting it to the pulse's travel time.

If we choose a one-way speed of light in one direction, then we could apply Einstein's method to establish simultaneity, so the choice of a one-way speed defines a simultaneity convention. But the answers I was given approach the problem from the other end: given a convention, we could presumably calculate the one-way speed of light in a specific direction.

In another thread, I was told that the choice of the one-way speed of light was equivalent to choosing the mathematical system used to solve physics problems. Can we say the same if we instead choose a simultaneity convention?

In general, is the one-way speed of light the flip side of choosing a simultaneity convention?

**Valid and invalid conventions**

Dale's answer simply defines a simultaneity convention without qualifying them as “good”, “bad”, or even “useful”.

Peter introduced the idea that there are “valid” and “invalid” conventions. The restriction he mentioned seems to be widely accepted.

What objective criteria, if any, could be used to separate valid from invalid conventions. Or are all such criteria subjective?

**Investigation**

I hope I've clarified that I am not a mathematician. I would like to think that I can still do some analysis of simultaneity conventions.

If we specify a simultaneity conventions by selecting a constant one-way speed of light in one direction, then this can be represented geometrically on a standard Minkowski spacetime diagram by skewing the X axis by a given amount. I used this approach to create a geometrical method for analyzing a subset of all the possible conventions:

I began by picking a function of the form ##f(x, T) = t## which met all the following criteria:

- ##f(0, T) = T##
- The mapping is invertible: Every ##(x, T)## maps to one and only one ##(x, t)##.
- The mapping is diffeomorphic (“smooth”).
- Every pair of coordinates that share a common ##T## can be connected by a spacelike curve.

By drawing a standard Minkowski ##(x, t)## diagram, choosing a specific ##T##, and drawing the curve for ##f(x, T) = t##, I can project a single line of simultaneity from ##(x, T)## space onto ##(x, t)## space. Because ##f(0, T) = T = t##, I know this curve will cross the ##t## axis at the selected ##T## value. Drawing a number of these allows me to project a grid of simultaneity lines onto my Minkowski diagram.

**A problematic mapping**

Consider $$f(x, T) = \frac {(sin(T) sin(x))} {2} + T = t$$

Here is what the grid for this convention looks like when projected on a normal Minkowski spacetime diagram:

With this convention, it isn't clear that we can establish the one-way speed of light. Do we measure the speed along the main observer's lines of simultaneity for both departure and arrival? Or do we synchronize clocks for when the signal is sent and use the receiver's local time of arrival to determine the speed?

A convention which contradicts physics or creates ambiguities seems like a good candidate for an invalid simultaneity convention.

The convention I chose has other problems: it fails to obey Einstein's first postulate: the choice of an arbitrary 0 time coordinate is significant. Two observers, even with overlapping worldlines, may get different measurements for the one-way speed of the same photon if they choose different 0 time. In general, any convention that allows the one-way speed of light to vary creates unique inertial frames.

The argument for making these conventions invalid is weaker: unique inertial frames exist only within the context of the coordinate chart—there is no way to identify a unique inertial frame without choosing a system that makes that inertial frame unique. Making these conventions invalid is more an aesthetic choice—it keeps the physics cleaner by not implying properties that exist only within that convention.

**Summary**

Many physicists add a spacelike curve requirement to simultaneity conventions to maintain causality ordering. It looks to me as though there should be additional requirements, but I haven't heard of any others. Did I misunderstand something or is my analysis flawed?