True static equilibrium and effects on time

[atnu8]

If an object is truly still in space, how would it perceive time?

If light is moving so fast that time has completely stopped, then something that is completely still should see the end of the universe no? is there a limit on the speed of time?

—-

MENTOR NOTE: this thread was for discussion of Static Equilibrium. It is now closed and discussion Relativistic Simultaneity continues in the new thread below:

https://www.physicsforums.com/threads/relativistic-simultaneity-and-effects-on-time.1083894/

Thank you all for contributing here.

 

[Ibix]

If an object is truly still in space, how would it perceive time?

There is no such thing. Speed is always a relative concept and you can always consider yourself "at rest" or "moving". As physicists, usually we choose the option that makes the maths easier.

If light is moving so fast that time has completely stopped

It isn't. Proper time is not defined along null curves (the technical term for the paths light follows in vacuum). "Time stops at the speed of light" is nonsense promulgated by popsci trying to use maths that's only valid for speeds below that of light to describe something moving at the speed of light.

s there a limit on the speed of time?

It always passes at one second per second for everybody. It's always other people's clocks that tick slowly.

There is no absolute concept of speed. Right now you may consider yourself at rest, or moving at 0.999999...9c as measured by a passing neutrino. It has no effect on your perception of time.

 

[Herman Trivilino]

If an object is truly still in space, how would it perceive time?

There's no way distinguish between being at rest and moving in a straight line at a steady speed. The two states are equivalent. It's hard to answer your question because I don't know how an object would perceive time. People do, and so the way you would perceive time is the same way you do so now.

If light is moving so fast that time has completely stopped,

That's a common but erroneous notion. At light speed the passage of time is not defined. Which is different from the passage of time being zero.

 

[cianfa72]

There's no way distinguish between being at rest and moving in a straight line at a steady speed. The two states are equivalent.

The notion of traveling along a straight line at constant velocity has different meaning depending on the "physics framework" used. The best/invariant one is the notion of proper acceleration, i.e. the acceleration measured by an accelerometer attached to the body under investigation. Motion in straight line at constant velocity is interpretated as zero proper acceleration.

 

[Dale]

If an object is truly still in space, how would it perceive time?

You are currently truly still in your reference frame. How do you currently perceive time?

 

[atnu8]

Thank you all for the replies, I feel like I may be getting the idea..

The perception of time never changes for the observer, time differences can only be perceived if an observer has a change in acceleration in relation to another observer?

You are currently truly still in your reference frame. How do you currently perceive time?

Just trying to wrap my head around the whole 'reference frame' thing. According to Planck Collaboration et al. (2018) the earth is hurtling through space at 369.82±0.11kms−1.
Does this speed have no influence on our perception of time?

If the earth is speeding up in this direction, would time be slowing down for us or can the difference only be noted if there is something to reference against? (like a copy of the earth that wasn't speeding up)

 

[jbriggs444]

Thank you all for the replies, I feel like I may be getting the idea..

The perception of time never changes for the observer, time differences can only be perceived if an observer has a change in acceleration in relation to another observer?

Experiment shows that an accelerating clock continues to keep accurate time. This is the "clock hypothesis". It is tested by measuring the lifetime of particles traversing an accelerator ring at high speeds and correspondingly high accelerations.

Just trying to wrap my head around the whole 'reference frame' thing. According to Planck Collaboration et al. (2018) the earth is hurtling through space at 369.82±0.11kms−1.

This is the speed of the Earth relative to a frame in which the cosmic microwave background radiation would appear to be isotropic.

Does this speed have no influence on our perception of time?

No. Our clocks still advance at one second per second.

If the earth is speeding up in this direction, would time be slowing down for us or can the difference only be noted if there is something to reference against? (like a copy of the earth that wasn't speeding up)

Yes, the only way you can measure a clock's rate is by comparing it to another clock. The details of how that comparison is done will affect the measured result.

Without specifying the measurement details, a broad statement like "our clocks are running slow" would not be specific enough to be meaningful.

There are scenarios where accelerating clocks see time dilation effects. For instance, gravitational time dilation. Which also applies in an accelerating elevator. In these situations, there is a natural way to compare clocks. The "bottom" clock will run slow while the "top" clock will run fast.

 

[Ibix]

According to Planck Collaboration et al. (2018) the earth is hurtling through space at 369.82±0.11kms−1.
Does this speed have no influence on our perception of time?

That's a speed relative to a particularly useful sense of "at rest", one so prevalent in cosmology that people talk about speed relative to it just as casually as we talk about "doing 30mph" when (if we were pedantic physicists all the time) we ought to say "30mph with respect to the Earth's surface". In either case, it's perfectly ok to say you are at rest and the thing we usually call "stationary" is moving. There's no physical test that will prove you wrong or right.

If you are wondering how it works that we can all say that everybody else's clock is running slow, it turns out to be due to different assumptions about what "now" means. It gets messy quite quickly in curved spacetime, but is fairly easy to explain in flat spacetime if you are interested.

 

[cianfa72]

In either case, it's perfectly ok to say you are at rest and the thing we usually call "stationary" is moving. There's no physical test that will prove you wrong or right.

Yes, however the person who says "I'm stationary" must measure for themself zero proper acceleration.

 

[PeterDonis]

the person who says "I'm stationary" must measure for themself zero proper acceleration.

No, this is not a requirement. For example, you're perfectly justified in saying you're stationary as you type your posts here. But you don't have zero proper acceleration (unless you're posting from the International Space Station, which I assume is not the case).

 

[cianfa72]

No, this is not a requirement. For example, you're perfectly justified in saying you're stationary as you type your posts here. But you don't have zero proper acceleration (unless you're posting from the International Space Station, which I assume is not the case).

Ah ok, so here we are taking just a kinematic viewpoint/description, no dynamic is involved.

 

[Herman Trivilino]

Just trying to wrap my head around the whole 'reference frame' thing. According to Planck Collaboration et al. (2018) the earth is hurtling through space at 369.82±0.11kms−1.
Does this speed have no influence on our perception of time?

Is this a rehash of your original question? Look again at the answer I gave you.

The issue you are having trouble wrapping your head around is the fact that motion is relative. Relative to your desk you're motionless. The motion you quote above is relative to the sun. But we're also in motion relative to other things, like the center of our rotating galaxy. Or the center of Earth.

Do a google search for "Galileo in a ship hold". He does a good job of explaining it. He was attempting to dispel the notion that Earth must be at rest because we cannot detect its motion. For example, Earth rotates once per (sidereal) day. Consequently, at my latitude I move a distance of about 24 000 miles in 24 hours. That's 1000 mi/h. A wall just west of me is moving towards me at that speed. If I drop my pen why does the wall not slam into it before it can hit the floor? Galileo's contemporaries regarded the fact that this doesn't happen as proof that Earth is not moving. Galileo dispels that argument.

This idea is embedded in the 1st Postulate of Einstein's relativity. Since Galileo's day it's place in the hierarchy of physics has increased, and is now called the Principle of Relativity.

This idea forms part of the foundation for the issues you are struggling with. You'll do yourself a favor by spending time absorbing it.

 

[pervect]

Thank you all for the replies, I feel like I may be getting the idea..

The perception of time never changes for the observer, time differences can only be perceived if an observer has a change in acceleration in relation to another observer?

Just trying to wrap my head around the whole 'reference frame' thing. According to Planck Collaboration et al. (2018) the earth is hurtling through space at 369.82±0.11kms−1.
Does this speed have no influence on our perception of time?

If the earth is speeding up in this direction, would time be slowing down for us or can the difference only be noted if there is something to reference against? (like a copy of the earth that wasn't speeding up)

I would suggest learning about quantities, called invariants, that are independent of the frame of reference. In special relatiavity, his would be, for instance, the Lorentz interval. This would be discussed in textbooks such as "Space time physics" by E.F. Taylor. An older edition is available for free on his website - this is a standard textbook with a rather informal and chatty style.

See for instance https://www.eftaylor.com/spacetimephysics/

The common definition of distance intervals and time intervals do change with one's reference frame, but there is a way of combining them that does not.

"The Parable of the Surveyor", the first chapter of the reference I just gave, is perhaps a bit longwinded but describes how north-south and east-west are part of a larger concept in surveying. It is worth thinking about why we regard north-south and east-west as being combined into a larger structure, rather than two separate entities.

This sounds simple enough - and it is - but there is one obstacle that people stumble over all the time, and something that Einstein had to struggle with to formulate the special theory of relativity. This is the notion that the idea of "at the same time", or "now", depends on the frame of reference.

Many, many people find this very hard to grasp, and it is a notorious obstacle to understanding special relativity. It's unclear if bringing this up now is the best approach, but it just doesn't make sense that switching from distance and time intervals (which vary with the reference frame) suddenly become frame independent if they don't, somehow, interact with each other. It turns out that while space and time intervals do vary with the reference frame, a fairly simple combination of them does not.

The actual math formula is actually simple - if dx is a distance interval, and dt is a time interval c^2 dt^2 - dx^2, and it's inverse dx^2 - c^2 dt^2, are invariant and independent of the frame of reference, while dt and dx are not independent.

This speciflally means that if dt equals zero in one frame of reference it may not be (and usually is not) zero in a different frame of reference. Hence, the phenomenon that is called "the relativity of simultaneity".

This is a simple example with only one dimension of space, but that's more than good enough to get started. Note that for any two points connected by a beam of light, this formula gives Lorentz interval of zero. That' where the remarks about "null intervals" come from.

It's a LOT easier to keep tract of things that don't change with the frame of reference than it is to keep tract of things that do deped on the frame of reference. For one thing, one winds up tediously haveing to give the exact details of what frame of rerference one is using. This is doable, and sometimes can't be avoided, but it is in general much easier to talk about things that are the same in all frames of references - things that are invariant.

 

[PeterDonis]

so here we are taking just a kinematic viewpoint/description, no dynamic is involved.

"Stationary" is a matter of "kinematics" as you are using the term here. It's just a choice of reference frame. You can do it with any "dynamics" you like.

 

[cianfa72]

"Stationary" is a matter of "kinematics" as you are using the term here. It's just a choice of reference frame. You can do it with any "dynamics" you like.

Ok, so in the context of Newtonian physics let's pick the frame in which I am at rest. What if this isn't inertial? To do dynamic w.r.t. it, one is forced to add inertial forces appearing to act on all objects (including me) in this frame.

I'm not sure whether from a dynamic perspective I've the "right" to say "I'm stationary" though.

 

[PeterDonis]

so in the context of Newtonian physics let's pick the frame in which I am at rest. What if this isn't inertial?

The frame in which you are at rest right now is not inertial. You are stationary in this frame--because you are at rest in the frame. Why is that a problem? We all use such a frame in our daily lives all the time.

To do dynamic w.r.t. it, one is forced to add inertial forces appearing to act on all objects (including me) in this frame.

Well, of course. That's how a non-inertial frame works. Why does that make it a problem to use the word "stationary" for an observer at rest in the frame?

I'm not sure whether from a dynamic perspective I've the "right" to say "I'm stationary" though.

Why wouldn't you?

I simply don't see why any of this is an issue.

 

[atnu8]

I would suggest learning about quantities, called invariants, that are independent of the frame of reference. In special relatiavity, his would be, for instance, the Lorentz interval. This would be discussed in textbooks such as "Space time physics" by E.F. Taylor. An older edition is available for free on his website - this is a standard textbook with a rather informal and chatty style.

See for instance https://www.eftaylor.com/spacetimephysics/

This book is great! I love how it's written. I just read chapter 4: Trip to Canopus and found that extremely enlightening, thank you for sharing.

I appreciate all of you for being patient with me!

How does this sound:

All measurements of time are dilated when moving at high speed relative to an inertial observer.
(Time slows down for moving observers compared to non-moving observers)

Because the speed of light is finite, moving through space causes any physical mechanical processes to take longer.

If a rocket was to counteract the velocity of the earth's orbit around the sun, staying still in relation to the sun but moving parallel with the sun through space, when the earth met with the rocket again in space and completing a full orbit of the sun, the clocks on earth would measure less time elapsed than the clocks in the rocket.

 

[Nugatory]

This book is great! I love how it's written. I just read chapter 4: Trip to Canopus and found that extremely enlightening, thank you for sharing.

You are just cheating yourself out of essential background understanding (which is the really fun part) if you've skipped chapters 1-3.
(Edited to add: I said above that I had learned relativity from that book - actually the first edition, which I still prefer. By far the hardest part was realizing that those first chapters were not "yes of course" obvious stuff that I already knew, but instead a gentle but very firm process of getting me to see that all of my intuitions about time, distance, motion were seriously incomplete).

How does this sound:
All measurements of time are dilated when moving at high speed relative to an inertial observer.
(Time slows down for moving observers compared to non-moving observers)
Because the speed of light is finite, moving through space causes any physical mechanical processes to take longer.

Not right, and a strong hint that you have missed the point.
(You should never say "moving" without also saying what the motion is relative to. This is not a pedantic quibble, the habit is essential to being able to think about spacetime clearly).

Consider: You are sitting on the surface of the earth. Your speed is zero if we use a frame (this is in chapter 2) in which the surface of the earth is not moving - another way of saying that your speed is zero relative to the surface of the earth. Your speed is many kilometers per second if we use a frame in which Mars is at rest. That's two different speeds, so if moving through space changed the speed of physical processes, then all the physical processes in your body would be happening at two different speeds at the same time, which is impossible.

If a rocket was to counteract the velocity of the earth's orbit around the sun, staying still in relation to the sun but moving parallel with the sun through space, when the earth met with the rocket again in space and completing a full orbit of the sun, the clocks on earth would measure less time elapsed than the clocks in the rocket.

Not for the reason you're thinking. It has nothing to do with motion affecting the passage of time, it is because the rocket and the earth have taken different paths through spacetime, the different paths have different lengths, the length of a path through spacetime is measured in seconds by a clock following that path just as the length of a path through space is measured by the odometer of a car following that path. (Correctly calculating the length of that path when the earth is rotating, orbiting, and everyone is affected by gravity is a seriously non-trivial problem... Don't take it on until you understand the simple out and back no-gravity Twin Paradox case).

Time dilation between observers in motion relative to one another is a different thing, the result of relativity of simultaneity, which is in chapter 3.

 

[Herman Trivilino]

(Time slows down for moving observers compared to non-moving observers)

You're still not getting it. If we are moving inertially relative to each other then either of us could validly claim to be at rest and the other in motion. Each of us would observe the other's time to be dilated. A notion that stumped Einstein himself as he was creating his theory. He resolved it when he realized that it could be explained by the fact that simultaneity is relative. He claimed that when the realization came to him it caused him to sit upright when he had been lying in bed.

There is no such thing as "moving through space".

 

[Herman Trivilino]

Yes - I agree. I suspect that a full classroom environment, with homework and grading of the homework to provide feedback to the audience/students studying the material is a huge help in the success rate of understanding

Well, I was referring to a self-learner as I presumed is the case with the OP.

 

[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.

 

[pervect]

In geometry and Riemannian geometry, one uses tangents to curves and normals to curves.
So, in the Minkowskian spacetime geometry of special-relativity and the Lorentz-signauture pseudoriemannian geometry of general-relativity, one also uses tangents to worldlines and spacetime-normals to those worldlines.


The projection-operator spacetime-orthogonal to t^a (namely, h_{ab}=g_{ab}-t_a t_b (+---) ) is used to get the "spatial-according-to-t^a- component" of a 4-vector or a 4-tensor....
So, decomposing a particle's 4-velocity u^a ,
one gets the "spatial component according to t^a" (associated with simultaneity according to t^a,

I'm on board with this - I regard it as the theory of timelike congruences, which I first stumbled across in Wald, and later read more about in Poisson's "A Relativists Toolkit".

Before that, I went with using Fermi Normal coordinates to gain a physical understanding. At that point I couldn't single out space without explicitly writing down the basis vectors, as you allude to later. So when I became aware of it, I preferred the methods of timelike congruences to Fermi Normal coordinates, as it was enlightening to have to specify only one vector field at every point, rather than four.

Without one of these two tools, I find the it hard to get any physical intuition. As you note, with the method of timelike congruences, you don't have to pick out specific spatial vectors - it allows one to focus on only picking out the one that represnts one idea of "time".

It's hard for me to guess what's taught nowadays - I'm guessing that one or the other of these two ideas is presented very briefly, but little time is devoted to it, leaving it up to the more dedicated student to dive into it in detail, with those who just want to pass the course passing it by.

In my opinion, after the "principle of relativity"
(seen in both special-relativity and galilean-relativity, with an analogous* concept in Euclidean geometry),
the first key difference is the "finite maximum-signal speed" (light cone structure),
and
the second key difference is the "relativity-of-simultaneity" (non-parallel tangents to the Minkowski-circle
[analogous to the non-parallel tangents to a circle]).

This made me scratch my head a bit, but I think what you're saying that rotational symmetry in Euclidean geometry is similar to boost symmetry in special relativity, one way of interpreting Taylor's "The Parable of the Surveyor" (from spacetime physics), which is one of the works I read that had a huge impact for me. Possibly I got it worng, though. But I'm not sure I'm quite taking your point here.

So, in my opinion, although not the most-important, the relativity-of-simultaneity can't be neglected.
And, since it has a direct Euclidean-analogue that likely poses no paradox,
I think it can more easily be treated using the "tangent to the circle" idea.
Or, if you want something "more physical" and more-spacetimey,
use the radar-method with light-signals and interpret it in terms of light-cones.

At some point, the relativity of simultaneity can't be avoided, I agree. But I find the idea generates active resistance in PF readers not familiar with it. So I am not in a rush to point it out until I think it's needed. This does have a problem, though. A typical issue is that the reader is confused by some "paradox" that can only really be resolved by understanding the relativity of simultaneity, but they are focused on resolving the "paradox" and not interested in a long - or even a short - lecture on the Relativity of Simultaneity, especially since they don't immediately see how it is relevance to their "paradox", and are are likely on some subconscious level actively resisting the idea anyways. I don't have a great solution for this, but I don't think anyone really does anyway, so I just try different things and watch most of them probably not work. (It's hard to guess when a presentation on PF gets through).

Relativity is often associated with "the geometry of spacetime".
Let's show the geometry and the algebraic-formulas
and not just "algebraic formulas relating invariants".

What are you suggesting here? At some point, it's great to introduce differential geometr, but for trying to get someone on-board with the basics of relativity, I find the geometric ideas are going to be a challenge. I'll push people towards reading "Spacetime physics", sometimes, but I think it takes a certain amount of maturity to really appreciate the geometric methods. I haven't written a lot about it, but for the very basic introductions, I like Bondi's old approach in "Relativity and Common sense", also known as k-calculus (though it's just algebra, not calculus).

(Last gripe:
These days when I see relativistic collision problems done with invariants,
I admire the compactness of the calculation...
but I feel lost as to why it works or what it is telling me.
Yes, it's a system of equations, expressed in terms of neat chunks to substitute.
But when viewed as a polygon analyzed with hyperbolic-trigonometry
(as a geometry problem, analyzed like a free-body-diagram),
the meaning of the algebraic-obtained result becomes clearer to me.)

 

[PAllen]

I think my objection to the utility of simultaneity is its use at large scales, rather than small. Tangents and normals are local objects, and a local frame makes sense. However, globally, giving credence to simultaneity (as an element of reality), even in SR, leads to such nonsense as: for a pacing walker, events in Andromeda as simultaneous to multiple points on their world line, and events hundreds of years apart shift back and forth from future to past and back (relative to 'now'). To me, this robs the concept of any meaning over large scales.

 

[jedishrfu]

MENTOR NOTE: This thread has been split into two parts. This part discusses the Static Equilibrium and is now closed, while the following thread continues to cover Relativistic Simultaneity.

https://www.physicsforums.com/threads/relativistic-simultaneity-and-effects-on-time.1083894/