True static equilibrium and effects on time

[Herman Trivilino]

I think one aspect of @pervect 's view is that from the point of invariants, and also, for GR, it is arguably better to say that simultaneity is a concept that should be (almost completely) banished, rather than considered relative.

The notion that there is a universal time that is the same everywhere can't be banished by ignoring it. It seems built into our worldview. It is one of the most stubborn notions to oust.

 

[pervect]

(Maybe this discussion of how to teach "relativity of simultaneity" can be split off and given an appropriate title, but linked from this thread.)

(Maybe this should be converted to an Insight.)

---

From my recent consecutive posts
www.physicsforums.com/threads/derive-the-lorentz-transformation-in-minkowski-four-dimensional-spacetime-spacetime.1083553/post-7294556
and
www.physicsforums.com/threads/derive-the-lorentz-transformation-in-minkowski-four-dimensional-spacetime-spacetime.1083553/post-7294560

Spacetime geometry suggests
the "relativity of simultaneity"
is related to two vectors being "orthogonal" or "perpendicular", which comes from the metric.
In particular, given an inertial worldline (which has a future-timelike tangent vector) "timeline",
we want a line (with spacelike tangent) that is "orthogonal" to it--call it a "spaceline".
Loosely, we want to encode the idea "space is perpendicular to time".

I would use much simpler (but also less precise) language to convey a similar thought. I would call "causal diamonds", light clocks. It's just a matter of a different name choice, I am thinking that the idea of a "light clock" might be more familiar for the typical PF reader. I am somewhat guessing at this, to be honest - it's hard to tell exactly what the background of a typical PF reader really is, I am guessing from what I can infer from reading their posts.

BTW - I haven't noticed (until now) seen anyone else propose using light clocks (or causal dimaonds) to teach simultaneity, but I find I like the idea, it's been at the back of my mind for some time. I don't recall reading anything in any of Scherr's research about this particular approach. But it's possible and even probabl that I've missed some one implementing this approach.

I also have a feeling that some discussion of how to draw space-time diagrams is needed to reach the target audience of PF readers. I'll refer to the attached diagram below, which I've labelled similarly to yours with H, L, O, and P.

On a space time diagram, vertical lines represent an object (or the worldline of an object" "at rest". In the diagram below, OP is vertical (or intended to be, the drawing isn't great), so that's what makes it a "light clock at rest". OH and OL represent segments of light beams, as do HL and HP. The conventions I am using (which are fairly standard, but need to be explained) is that in the diagrams I draw, light beams always travel at 45 degree angles to the T and X axes. This is related to the concept that we use units where c=1, so that in 1 unit of time, light travels 1 unit of distance. This means that on a diagram with this choice of scale, light will always travel at a 45 degree angle.

Time, of course, runs up the page - I've labelled the appropriate axis "T". Space runs horizontally. Purely spatial intervals are something we can measure with a ruler. The line segment HL represents a snapshot of such a ruler at one particular instant of time. All events on the line segment HL are simultaneous. Note that this line is not a full space-time diagram of a ruler, because it exists only at one intant of time. An actual ruler exists at all times, not just one instant. A better space-time diagram of a ruler would be a pair of vertical lines, possibly with some hatched fill to show what part of the diagram is the "ruler". If I was better at drawing, I would ideally make a diagram of such a ruler as part of the process of reviewing how space time diagrams represent things in an extreme amount of detail. Ah, I know how to phrase this. "Drawing a space-time diagram of a ruler that exists at all instants of time is an exercise for the reader.". I do think the more diagrams the reader draws themselves, the better, but it is probably more laziness on my part.

As far as simultaneity goes, we can say that all events on HL are simultaneous in the chosen frame of reference. It is useful now to point out how they satisfy Einstein's definition of simultaneity. A paraphrase of a quote from Einstein:

Two events are simultaneous if light signals from them arrive at the midpoint of the events at the same time.

Other definitions might be possible and easier to learn, another topic of potential research. The definition of midpoint may be particularly fuzzy. My opinion is that the space-time diagram of a light clock (or causal diamond) provides a disambiguation and more rigorous defintion of the idea of simultaneity, with Einstein's "midpoint" explanation being more historical and having more associated authority. I can't offhand think of other historical defintions of "simultaneous" that might be useful to compare to the one based on light clocks.

The upper half of the space-time diagram shows two light signals emitted from two eventrs reaching a common point at the same time. What we have to demonstrate is this point is the midpoint. The argument that this is the midpoint is that light signals emitted between the two mirrors of the light clock will return to the center at the same time if and only if they are emitted at the center. For instance, if a light pulse occurs left of center, the light beams will reflelct off the mirrors and intersect right of center, and vica-versa. Again, it'd be good to add in the walls of the light clock and make a better diagram than the one I've sketched out, along with a better diagram of a ruler. Again, I suppose that I can say that this is a recommended exercise for the reader.

So far, nothing very surprising is happened. We've been setting up familiar territory - how a light clock is related to simultaneity - in the simple case where the light clock is at rest. But now - we want to consider the case of a moving light clock.

Since light travels at a constant velocity for all frames of reference, up to a scale factor we can draw the the space-time diagram of a moving light clock as below:

Note: I should have added prime marks to all the letters on this diagram- but - I didn't.

This is how a light clock must look on a space-time diagram if the light clock is moving. By "the light clock is moving", I mean that O'P' is not vertical on the diagram. Recall that we call observers with a constant position in our chosen frame of reference "stationary", and they are represented by vertical lines on the space-time diagram. "Moving" observers do not have a constant position, and hence they are sloped lines on the space-time diagram.

We have previously argued that H and L were simultaneous according to Einstein's defintion. Nothing about this argument has changed. So we can argue that H' and L' are simultaneous according to the observer represented by O'P' on the diagram. What's confusing here is that someone who is not familiar with relativity and the relativity of simultaneity may be confused about the idea that events that are simultaneous for the observer OP are different from the events that are simutaneous for observer O'P'. This is, however, what we are trying to teach. How successful the approach will be, remains to be seen. I think 10 percent "getting it" might be of the right order of magnitude - I hope it's higher than 1%, and I'd say it's deftinely lower than 100%. But the only statistics I have would be from Scherr, and I don't recollect him trying this approach (which is a pity), so I'm taking a wild guess here.

Not again that I am using "stationary" and "moving" as a convenient notion to distinguish the two observers, but they are just labels. This is a bit confusing to people who want to find out how to tell if one is moving or not - the answer, according to relativity, is that there is no way to tell who is moving and who is not. Historically, Michelson and Morely did an experiment in the context of ether theory, where observers could be regarded as stationary relative to the ehter or moving relative to the ether. But they famously had a null result. Relativity solves the problem by saying that any observer you like can be considered to be stationary - it's just a labael that you can apply however you like, as long as you are consistent.

 

[pervect]

I think one aspect of @pervect 's view is that from the point of invariants, and also, for GR, it is arguably better to say that simultaneity is a concept that should be (almost completely) banished, rather than considered relative. Of, if you insist: any two events such that neither are in the causal future or past of the other, may be treated as simultaneous.

I have espoused that idea, and met a lot of resistance in certain instances. When I have the energy, I still do sometimes like to advocate for the idea that the fundamental idea of physics "should be" presented independent of conventions. But that is more from idealism than practicality.

I think my actual feelings are more that it is a worthwhile pursuit to look at physics that is independent on convention, and think about how to elimante as many conventions that must be follwed to the extent that's possible.

I also think this is probably not the best way to first learn physics. It's definitely not the way we teach it at an introductory level.

If I were to describe how to do special relativity at an advanced level, I would say - just label all the events with "smooth" coordinates for time and position, and then write the quadratic form that represents the invariant interval. The trick of how to interpret this invariant iterval is a necessary step, of course, that I'm skipping over because it's long. Why a quadratic form? Well, a linear form doesn't have a high enough order to work, and the cubic terms are negligible when/if you have some taylor series representation of a smooth function. So quadratic forms are what you need. This is the same way I'd describe how to do Euclidean geometry with arbitrary coordinates, the space-time version just has a sign difference. D istance, and the squared distance given by the quadratic form, is always positive, while the quadratic form that gives the Lorentz interval can be negative - the Euclidean version has positive definite quadratic forms, the space-time version removes this restriction. In the space-time version, postive and negative signs represent time and space, or vica-versa, depending on some sign conventions.

And I gloss over all the details regarding what I actually mean by smooth, leaving that to the mathematicians :).

I wouldn't expect someone just starting out to appreciate or be able to use that point of view though. It's just not a high school level presentation.

When I put on my pragmatic hat, I look instead at what's easiest to teach and learn and use, even if it requires one to adopt and follow certain "rules". In space-time, I like to use the idea of "clocks" and "rulers".

At the moment, for this thread, I have my "pragmatic" hat on, rather than my "idealistic" hat on.

One thing that concerns me a bit is how to teach what conventions must be followed and why. We get a lot of questions that bsically involve breaking the standard convetions, especially from young people. In general there aren't any easy answers - sometimes you get away with it, sometimes it fails horribly.

There's an even more basic, and harder to present, question. How do you know when something has failed horribly, vs being a brilliant idea? This gets into questions of trust (especially trust of our institutions and the previous work that has been done), standards, politics, what-do-we-mean-by-crackpots, media, among others. It's too general to usefully talk much about, though.