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

[Ibix]

I'd like to focus on just this one statement. Perhaps these questions will help identify my confusion:

1. For each simultaneity convention, isn't there a unique set of simultaneity planes?
2. For each simultaneity convention, aren't these planes, by definition, orthogonal to the worldlines of clocks at rest in that coordinate system?

If both these statements are true, then the isotropic convention cannot be the only one in which simultaneity planes are orthogonal to the worldlines of clocks at rest.

I would say 1 is more or less true - more precisely, for a given simultaneity convention and choice of origin clock the planes are unique. Hence the Einstein simultaneity convention yields a different set of planes for two clocks in relative motion.

However, 2 is completely wrong. Sticking to 2d, two lines are orthogonal if vectors in those lines have zero inner product. Given a first vector, only vectors in one direction satisfy this, so if you choose a timelike axis then only one choice of simultaneity plane is orthogonal to it.

I suspect that you are assuming that the Euclidean $\vec v\cdot\vec u=v^xu^x+v^yu^y$ or Minkowski $\vec v\cdot\vec u=v^tu^t-v^xu^x$ hold in all coordinate systems. This is not correct - in non-orthogonal systems you get cross-terms as well. That's the root of why the maths is nastier.

The line of simultaneity is orthogonal to the worldlines of the clocks.

The line you drew is orthogonal in the Euclidean sense to the worldlines on the diagram, yes. But the line that represents in reality may or may not be orthogonal to the worldlines.

Assume for a moment that it is a normal Minkowski diagram using Einstein synchronisation. Add the axes of some other frame. They are orthogonal in reality. Are they orthogonal in the Euclidean sense on your diagram?

I agree that a city is a real thing, independent of any diagram. However, simultaneity planes don't appear to be analogous to cities; simultaneity planes seem to be mathematical constructs only.

I can physically realise a Euclidean line by building a street. I can physically realise a simultaneity plane by taing a flock of clocks, sychronising them by some process, and making them flash a light when they read some pre-agreed time. I wouldn't say either is more or less of a mathematical construct.

Ironically, it was a similar statement that resolved one of my questions: the isotropic convention is the only convention where the loci of the concentric hyperbolas needed to meet the constraints of the problem reside on the same simultaneity plane.

What do you mean by "loci" here? If you mean the locus of the transitions from inertial to accelerated motion then yes. The point is that the plane in which they lie must be orthogonal to the inertial worldlines. You may choose to use that plane as a simultaneity plane, but you are not required to do so and nothing changes physically if you do or do not do so.

 

[Freixas]

I suspect that you are assuming that the Euclidean v→⋅u→=vxux+vyuy or Minkowski v→⋅u→=vtut−vxux hold in all coordinate systems. This is not correct - in non-orthogonal systems you get cross-terms as well. That's the root of why the maths is nastier.

I think you've found the source of my confusion. Thanks!

What do you mean by "loci" here?

Oops, sorry, I realize I both misread this and then drifted down some nonsense path. Ignore my reference to loci. I understand what you wrote now.

This particular problem requires the spacelike line between the acceleration starts to be orthogonal to the hyperbolae so that the hyperbolae can share radii (in the same sense that concentric circles share radii).

 

[PeterDonis]

I think you've found the source of my confusion.

Whenever you make a coordinate change, the form of the metric changes as well. A change of simultaneity convention is a coordinate change.

 

[cianfa72]

I can physically realise a simultaneity plane by taking a flock of clocks, sychronising them by some process, and making them flash a light when they read some pre-agreed time. I wouldn't say either is more or less of a mathematical construct.

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?

 

[Nugatory]

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 long as all the flash events are spacelike-separated from all the others and they completely fill space, they form a spacelike hypersurface. Devise a coordinate system in which they all have the same $t$ coordinate and we can also say that that hypersurface is a simultaneity surface in that coordinate system.

 

[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 simultaneous. But again, people do 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".

 

[cianfa72]

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.

You mean that flash events belonging to the same set must be spacelike separated from each other to define a spacelike hypersurface (the same applies to any set of such flash events). For events in "between" we need an interpolation process to assign them to a spacelike hypersurface.

 

[Ibix]

You mean that flash events belonging to the same set must be spacelike separated from each other to define a spacelike hypersurface (the same applies to any set of such flash events). For events in "between" we need an interpolation process to assign them to a spacelike hypersurface.

Yes.

 

[cianfa72]

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.

Take a flock of clocks filling the entire space. Their worldlines form a timelike congruence. Pick one clock as the master clock. Suppose any clock determinates that the round-trip for light to the master clock doesn't change. The master clock send out a zeroing pulse and any clock, upon receiving it, adds half the round-trip time measured for it. This process defines a coordinate time $t$ (one can build a coordinate system where the flock of clocks are "at rest" -- i.e. their worldlines have fixed spatial coordinates and varying $t$).

Does the above synchronization process define a spacelike hypersurface for events that share a given value of the coordinate time $t$ ?

 

[cianfa72]

Another point related to this: Born rigid timelike congruence means that the congruence has zero expansion and shear. Is it the same as constant proper distance between any two worldlines in the congruence (i.e. constant round-trip light travel time as measured by any observer along each member of the congruence) ?

 

[PeterDonis]

Born rigid timelike congruence means that the congruence has zero expansion and shear. Is it the same as constant proper distance between any two worldlines in the congruence (i.e. constant round-trip light travel time as measured by any observer along each member of the congruence) ?

Yes.

 

[cianfa72]

Ok, so in the global chart where the Born rigid congruence is at rest (constant spatial coordinates and varying timelike coordinate $t$), the metric tensor components are such that the solution of differential equation given from $ds^2=0$ for null paths representing the bouncing light pulses sent and received back from any observer in the congruence, gives constant elapsed proper time between the sending and receiving of the same light pulse.

 

[PeterDonis]

Ok, so in the global chart where the Born rigid congruence is at rest (constant spatial coordinates and varying timelike coordinate $t$), the metric tensor components are such that the solution of differential equation given from $ds^2=0$ for null paths representing the bouncing light pulses sent and received back from any observer in the congruence, gives constant elapsed proper time between the sending and receiving of the same light pulse.

You're just repeating the same thing in different words.

 

[pervect]

I'd like to venture a few comments about orthogonality.

To try and put it more plainly, but less precisely (i.e. using popular langauage rather than mathematical language), we need a way to treat time and space as part of a unified whole to say that one is orthogonal to the other. If we regard time as being totally separate from space, I don't think it makes much sense to talk about orthogonality.

To put it more precisely, I would say that in order to talk about time being orthogonal to space, one needs the underlying mathematical structure of special relativity as a 4-dimensional vector space, with an associated vector product rule, usually called an "inner product". Then this vector product defines orthogonal vectors, orthogonality being the condition when the vector product is zero.

This inner product rule of special relativity has many uses, one of the more fundamental ones is to define the "Lorentz Interval" between two events in space-time. Mathematically, this is done by taking the separation vector between the two events (consisting of a spatial separation and a time difference), finding the vector product of this separation with itself, which defines single number.

The important characteristic property of this Lorentz interval is that it is an invariant - it is the same for all observers. The spatial separation and the time separation between two events varies as one chooses different frames of reference (length contraction, time dilation, the relativity of simultaneity), but the Lorentz interval remains the same in all inertial frames of reference. This is discussed in the reference I gave before, Taylor & Wheeler's "Space-Time Physics".

The Lorentz interval of two events connected by a light beam is zero in the theory of special relativity.

The OP may lack and/or not care about this mathematical structure, but this structure is still a part of the thery of special relativity.

There are some physical justifications for using the Einstein clock synchronization system, though, that we can talk about without introducing the underlying mathematical structure of space-time. This is the observation that the Einstein clock synchronization is the unique clock synchronization that makes Newton's laws work in the low velocity limit, as I discussed in a previous post on this thread, post #20, https://www.physicsforums.com/threa...the-1-way-speed-of-light.1064028/post-7101240

Using the usual two-clock notions of velocity, there is only one way of synrhonizing clocks, usually called a "fair" or "isotorpic" synchronization, that makes Newton's laws works.

We could go further afield if we were willing to use the notion of "proper velocity", where there is only one clock, a clock on the moving object, rather than two clocks. This can lead to a formulation of physics where we use "proper" velocities measured by the distance in some frame divided by the time delta of the clock on the moving object, in which case we can write p = m * v_proper.

 

[Nugatory]

If we regard time as being totally separate from space, I don't think it makes much sense to talk about orthogonality.

If we're thinking about time as being totally separate from space we are - without realizing it - thinking in terms of coordinates in which the t-axis is everywhere orthogonal to the three spatial axes. So yes, it makes no sense to talk about orthogonality - we've already assumed the topic out of the discussion.

 

[PeterDonis]

Moderator's note: A subthread about properties of Born rigid congruences that is a topic change from this thread has been moved to a new thread:

https://www.physicsforums.com/threads/properties-of-born-rigid-congruence.1064188/