Bell's Spaceship Paradox, Born Rigidity, and the 1-Way Speed of Light

[Ibix]

Using whatever physical process to synchronize a such flock of clocks filling the space, does the set of events when they flash a light at a pre-agreed time value always define a spacelike hypersurface in spacetime?

As Nugatory says, the flash events must be spacelike separated from all others, and you need an interpolation process for the spaces between the flashes. It's possible to design a "synchronisation" process maliciously to violate this but most sane processes would be fine.

An example of a valid process is to pick a master clock, have every other clock establish the round trip time for light to the master clock, then have the master clock send out a zeroing pulse, and information on the chosen value of the "fast" speed of light and the direction in which that magnitude is chosen to occur. All clocks can then synchronise themselves to the master clock consistent with the chosen anisotropy.

 

[pervect]

Hyperplanes in spacetime are not just mathematical constructs. Nor is the geometry of spacetime as a whole. That is analogous to the city, and hyperplanes in spacetime are analogous to city streets or blocks--invariant geometric objects.

What is analogous to "simultaneity convention" is drawing map coordinates on the city. Calling a specific set of hyperplanes "simultaneity planes" is defining a simultaneity convention, not defining the hyperplanes themselves; they're there regardless of what you call them.

I'd like to give an enthusiastic thumbs-up to this response.

The overall point is to illustrate the similarities between a map of a 2d surface (assumed for simplicity to be flat), and a space-time diagram. Both are two-dimensional surfaces.

A simultaneity convention on a space-time diagram is like defining the directions "east" and "west" on a map. This would be the "x" axis on a space-time diagram. Selecting the notion of which objects are considered to be "stationary" is like defining the directions "north" and "south" on a map. This would be like the "t" axis on a space-time diagram.

We can do geometry if north-south and east-west are not orthogonal, but - the Pythagorean theorem won't work without modifications.

My point is that we do not HAVE to have north-south and east-west orthogonal to do geometry, but it's a convention, and formulas like the Pythagorean theorem will need to be modified if we make the choice to do something like this. Standardized formulas that are based on the assumption that north-south and east-west were orthogonal would no longer work. But while standard formulas don't necessarily work, it can be done with sufficient care.

Similarly, we do not HAVE to have the time direction, the direction traced out by objects "at rest" on a space-time diagram, orthogonal to the "simultaneity" direction, the direction traced out by what we assume is simutaneous. But again, people don't usually assume orthogonality, and some standardized formula would break if one makes this choice.

Space-time geometry is discussed in Taylor and Wheeler, where they present the space-time equivalent of the Pythagorean theorem to the reader. While I am tempted, I won't go into that here, but I will give some references.

Specifically, I'd like to mention 'The Parable of the Surveyor', part of a larger work, the textbook "Space-time Physics" by Taylor and Wheeler. I will note however that while their discussion is illuminating, it does not specifically address "maps" where time and space are not orthogonal, even though it does illustrate nicely some of the basics and motivations of treating space-time geometrically, drawing useful analogies between maps of space and maps of space-time, aka "space-time diagrams".

Google finds https://phys.libretexts.org/Bookshe...etime_Overview/1.01:_Parable_of_the_Surveyors

 

[pervect]

I have one other point to make, that I missed in my last point. That point is this - geometry exists without coordinates. We do not need to draw a grid on a plane in order to be able to do geometry. Special Relativity also has a specific geometry (it is not Euclidean geometry, though). This geometry is also independent of any "grids" we might assume. Arguing about simultaneity conventions is like arguing about the grid layout - it doesn't address the deeper issues, the geometrical ones. The question arises with the one-way speed of light - are we arguing about how to do the geometry of special relativity with oddball grids, or are we arguing about some new geometry that's different from the geometry of space-time of special relativity, known as "Lorentzian geoemtry".