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?

 

[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".

 

[pervect]

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)

You seem to be stuck on the idea that "time slows down". One can ask "Slows down relative to what? To me, your words suggest you believe in some sort of "absolute time" in which things can be at "absolute rest", this viewpoint has been presented by you several times and hasn't changed through the thread in spite of our efforts to explain that it's not the viewpoint of special relativity. But I'll repeat for emphasis that this is not the viewpoint of special relativity. This, my take on what you are saying is that when you talk about "time slowing down", you think that the time you measure with your clock slows down relative to some sort of absolute time. While such absolute time is present in Newtonian theory, it is NOT present in special relativity. Therfore, if you want to understand special relativity, you need to abandon this notion of absolute time. This make take some effort.

I'll try and break down the concept of time in special relativity. There are at least two sorts of time in special relativity. One is proper time. THis is the sort of time measured with a clock. A clock can be regarded as "travelling through spacetime", and, a bit like a wristwatch, if the clock is present and two events at different times, it will give a reading as to the time difference between the two events.

The second notion of time is different from the above. It's the notion of "simultaneous events". When a clock is not physically present at both events, one needs such a notion to measure the time interval between said events.

There is possibly a third notion related to time here, the notion of causality. But I won't be going into that in this post.

Separating and disambiguiting the two notions of time (proper time and simultaneity) are I think the most that it is reasonable to address in one post.

I will describe my view of "time dilation", and, if you are capable of drawing space-time diagrams, I'd encourage you to draw them. If you're not capable, I'd encourage learning how to draw them. You'll need the related concepts of "events" and "worldlines".

To do this, you will need both of the concepts I outlined - the notion of "proper time" and the notion of "simuttaneity" or "now". And I'll be using some concepts which are standard in space-time diagram, namely events (which happen at some specific time and place), and worldlines (which are the set of all events that some observer experiences).

Draw two worldlines on a space-time diagram, A and B. We'll regard one worldline, worldine A, as our reference. You might call it "stationary". And the second worldine B is a worldine of "an observer moving relative to A". If you're familir with the distinction, we are talking about "timelike worldlines" here, worldlines that are a one-dimensional set of events that an observer "experiences".

Along worldine A, using a clock, we mark regular proper time intervals. Because it's a proper time interval, it has a length, which we measure with a clock. We'll call this interval t_a.

For every point on worldline A, we find the unique event on worldline B that occurs "at the same time" as the event on worldline A. To do this, we need a notion of simutaneity, of events that occur "at the same time", a concept of now. There's a formal way we do this in specail relativty, we use what we call Einstein's clock synchronization convention, and to apply this convention proeprly, we need to specify a specific frame or observer in which we will apply this convention.

These events that we mark along worldline B turn out to also occur at regular intervals of proper time on worldine B, but the value of this interval (which we again measure with a clock) is t_b, which is in general different from t_a. The ratio of t_b to t_a is then the time dilation.

Einstein clock synchronization is important here, and if you haven't studied it, it's something you need to study to grasp special realtivity. The easiest way to study is is to draw space-time diagrams. To provide some physical insight for the space-time diagrams, you need to know how to measure time intervals (proper time intervals0 on the diagram, how to measure space intervals on the diagram (distances with a ruler), and how to synchronize clocks (the Einstein clock syncrhonization method). You need to be able to do all th ree.

We are perfectly free to exhcange the roles of A and B. When we do so, though, it turns out t he process of Einstein clock syncrhonization depends on our choice. The correspondence of points on worldline A to worldine B is a map. The map between A to B in A's frame is different from the map between A and B in B's frame, so when we interchange the observers, we use a dfferent map. THis is important.

Your second point involves GR and not SR. The short answer is that you got this one right, but discussing it at length would make thyis post too long - it's already probalby presenting too much material.

 

[Herman Trivilino]

Therfore, if you want to understand special relativity, you need to abandon this notion of absolute time. This make take some effort.

In my experience this definitely takes a great deal of time and effort. Reading something like Spacetime Physics by Taylor and Wheeler and working through all the examples and chapter-end exercises requires time and effort. But there's no substitute for it. Reading popsci explanations is more entertaining and requires less mental effort and time. Either way you're left with misconceptions, but the former gives you the tools to resolve them whereas the latter does not.

 

[pervect]

In my experience this definitely takes a great deal of time and effort. Reading something like Spacetime Physics by Taylor and Wheeler and working through all the examples and chapter-end exercises requires time and effort. But there's no substitute for it. Reading popsci explanations is more entertaining and requires less mental effort and time. Either way you're left with misconceptions, but the former gives you the tools to resolve them whereas the latter does not.

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. Some of the pedagogical literature I've seen suggests that the relativity of simultaneity is difficult, even with all this added support. See for instance Scherr et al, "
Student understanding of time in special relativity: simultaneity and reference frames"
https://web.archive.org/web/20180726114406id_/http://cds.cern.ch/record/573634/files/0207109.pdf.

Scherr has other very similar papers on this topic.

A quote about the observations of the difficulty from the abstract:

The results indicate that after standard instruction students at all academic
levels have serious difficulties with the relativity of simultaneity and with the role
of observers in inertial reference frames.

I try to remain positive, to some extent I feel that scaring people by telling them the problem is hard (in the sense that even after instruction, people get it wrong) isn't productive to getting people to learn. But it might be useful for people to know that it is recognized as hard. To my mind, the difficulty is not so much in the learning, as in forgetting previous ideas that turn out not to work in special relativity.

 

[robphy]

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. Some of the pedagogical literature I've seen suggests that the relativity of simultaneity is difficult, even with all this added support. See for instance Scherr et al, "
Student understanding of time in special relativity: simultaneity and reference frames"
https://web.archive.org/web/20180726114406id_/http://cds.cern.ch/record/573634/files/0207109.pdf.

Here is the arxiv version
Student understanding of time in special relativity: simultaneity and reference frames
Rachel E. Scherr, Peter S. Shaffer, Stamatis Vokos
https://arxiv.org/abs/physics/0207109
https://doi.org/10.1119/1.1371254 Am. J. Phys. 69, S24–S35 (2001)

Scherr has other very similar papers on this topic.

AN INVESTIGATION OF STUDENT UNDERSTANDING OF
BASIC CONCEPTS IN SPECIAL RELATIVITY - Rachel Ellen Scherr
https://www.physics.umd.edu/perg/papers/scherr/dissertation/ScherrDisstn.pdf

A quote about the observations of the difficulty from the abstract:

The results indicate that after standard instruction students at all academic
levels have serious difficulties with the relativity of simultaneity and with the role
of observers in inertial reference frames.

I try to remain positive, to some extent I feel that scaring people by telling them the problem is hard (in the sense that even after instruction, people get it wrong) isn't productive to getting people to learn. But it might be useful for people to know that it is recognized as hard. To my mind, the difficulty is not so much in the learning, as in forgetting previous ideas that turn out not to work in special relativity.

From the Scherr's dissertation, there are 178 instances of "event diagram" and 24 instances of "spacetime diagram".

p. 2

The dissertation includes several appendices. The first of these describes a particular
representation (the “event diagram”) that is used in the curriculum developed in this study.

p. 189 is appendix A: Event Diagrams

APPENDIX A:
EVENT DIAGRAMS
In this dissertation (specifically, in Chapters Four and Five), we introduced a new
representation called an event diagram. Event diagrams are comparable to Minkowski
(spacetime) diagrams in that they are a graphical representation of spacetime, providing a visual
means for organizing events, objects, locations, and so on. An event diagram consists of a series
of pictures, each of which represents events and objects at a single instant. Each picture in an
event diagram is therefore a “snapshot” of the objects, with their relative sizes and distances from
one another indicated. Events that occur at the instant of each picture are indicated in the
appropriate picture at the location of the event. Successive instants appear one below the other.
Event diagrams for different reference frames are drawn separately.

Presumably,
"standard instruction" refers to standard textbook presentations (presumably, words and a maybe a spatial figure or two).
"Event diagrams" appear to be a sequence of aligned spatial diagrams at selected instances.
In my opinion,
both "tools" fall short of what Minkowski spacetime diagrams encode.
However, I feel that "standard presentations of spacetime diagrams" also fall short in developing "spacetime intuition".
So, I think the "relativity of simultaneity" will continue to be difficult
because students are not being taught to use better "tools" (the Minkowski spacetime diagram).

 

[robphy]

Here is the arxiv version
Student understanding of time in special relativity: simultaneity and reference frames
Rachel E. Scherr, Peter S. Shaffer, Stamatis Vokos
https://arxiv.org/abs/physics/0207109
https://doi.org/10.1119/1.1371254 Am. J. Phys. 69, S24–S35 (2001)

Here are some slides from Vokos' talk at https://www.aapt.org/doorway/TGRU/
https://www.aapt.org/doorway/tgrutalks/vokos/Vokos_AAPT_TGRU.pdf

Sadly, there are no spacetime diagrams above.

(I maintain my position.... standard-textbook presentations and event-diagrams are not good enough.)

Students need to learn about spacetime diagrams (position-vs-time graphs)
and aspects of its Minkowski geometry and their physical interpretations.
First: Geometry and trigonometry in spacetime and their physical interpretations,
Later: standard-textbook-formulas... and then, maybe, tensors for physics majors.

 

[pervect]

I'm not sure what the best tools are to treat the relativity of simultaneity. I find space-time diagrams (Minkowskii diagrams) convenient, but I think much of the value lies in learning to draw them, and less in looking at diagrams others have drawn. Looking at diagrams others have drawn can be helpful in learning how to draw them, of course. But, me doing the work to draw a space-time diagram for someone else doesn't seem like it will necessarily teach them anything. Also, it is a bit of a pain. And I'm a bit lazy, I suppose.

I suppose my thoughts on the relativity of simultaneity are that - I'd rather defver talking about it as long as possible, just because I know it confuses people. It is important, unfortunately, for explaining how it is possible for time dilation to occur due to relative motion without there being a state of "absolute rest". This seems like the main question of the OP, so perhaps it can't be avoided.

But using the principle of "avoid the relativity of simultaneity as long as one can", I have realized that I can provide some insight as to why time dilates for a moving observer, filling in the mathematical details of the rather lengthly scenario I outlined, putting in the numbers, and invoking the principle of invariants I espoused in my previous post, the Lorentz interval. This will accomplish the goal of showing why time moves slower for the moving observer, even if it won't necessarily quash the inconsistent-with-special-relativity idea that there is some sort of "absolute rest".

Rather than draw out a space-time diagram, I'm just going to list out some coordinates. It certainly might be easier to digest in a diagram form, but I'm going to encourage the reader who wants to see such a diagram to draw it themselves.

So to recap. We want to construct events occuring at regular time intervals for a stationary observer. With the Minkowskii construction, we can regard these events as lying on a worldline. Until we introduce the space dimension, we could just as well consider these events as lying on a timeline. Timelines are a common pedagogical tool for organizing events in contexts such as history and are not unique to physics. So, there shouldn't be any big mystery or glamour about just imagining these events as lying on a line. But it's convenient to give this line a name, we call it the worldline. And when we talk about moving observers, we'll need both time and space on one diagram. The standard diagram here is the Minkowskii diagram - I've read about a few other approaches, but I haven't used them.

Next, we want to construct the events on a moving worldline that occur "at the same time" as the events on the stationary worldline. We note that because of the relativity of simultaneity we've been talking about, "at the same time" depends on the frame of reference. But we won't stress the point much, we just want to make a note of it. The results we get will be valid and achieve our goal. We won't resolve here why there is no absolute rest, but hopefully even for someone who just can't accept that there is no such thing, we will show how utilizing the principles of invariants gives the correct result that the moving observer has lest time pass than the stationary observer.

Finally, we want to compute the ratio of the elapsed times between the events we've marked out for the moving observer and the same time interrvals for stationary observer. But how do we compute the elapsed time for the moving observer? We are aruging that time depends on motion - what is our guiding principle here to compute the time? Well, the guiding principle here is the one I mentioned earlier, the princple that the Lorentz interval is invariant for all observers. By computing the lorentz intervals, we can compute the time intervals.

Again, Taylor & Wheeler , "Space-time physics" covers this in a lot more depth than I will.

So: for the stationary observer, we get the following list of coordinates (t,x) where t is time and x is position (space).

$$(0,0) \quad (t,0) \quad (2t, 0) \quad (3t,0) ....$$

The list is easy - time advances in constant increments, and the position is always zero, at the origin.

And for the moving observer, we get a different list of coordinates (t,x)

$$(0,0) \quad (t,vt) \quad (2t,2vt) \quad (3t, 3vt)$$

The space coordinate is no longer constant - instead, because the observer is moving at some velocity v, the space coordiante is v *t.

To compute the Lorentz intervals, which will give us the proper times we are looking form, we need some simple formula. If we set the speed of light, c, equal to 1, by the proper choice of units, this is especially simple. For instance, we can use units of seconds as time, units of light seconds as distance, and measure all velocites as some fractional multiple of the speed of light. With this approach, we write:

$$ds^2 = dt^2 - dx^2$$

And we see that the Lorentz interval, and the elapsed time is the same. Thus we can conclude that if we compute the Lorentz interval for both the stationary and moving observers, we will have also computed the time interval.

If we don't make this simplification, we write instead

$$c^2 ds^2 = c^2 dt^2 - dx^2 \quad => \quad ds^2 = dt^2 - \frac{dx^2}{c^2}$$

It basically just makes things a bit more complicated. Is the extra complicity worth it for being able to use standard units? While I tend to think it isn't, I can see where someone might want to use standard units, so I'll include the calculations for that case as well. The result of this complication is that the Lorentz interval is no longer the same as proper time, instead, the Lorentz interval is a multiple of proper time. But, since our goal is to take the ratio of the moving observer to the stationary observer, this multiplying factor cancels out, and isn't of any significance.

The time interval for the stationary observer is obviously just equal to t from in the first list I gave. The time interval for the moving observer is $\sqrt{t^2 - x^2} = \sqrt{t^2-v^2 t^2}$ for the case of the moving observer, and the ratio we are looking for is just $\sqrt{1-v^2}$, where v is expressed in factions of the speed of light.

If we go back to making c something other than 1, we can say that the Lorentz interval in the first case is c*t, and in the second case is $\sqrt{c^2 - v^2t^2} = c t\sqrt{1 - v^2/c^2}$. So we get the ratio is $\sqrt{1-v^2/c^2}$ - basaically the same as before. In the first case, v was a dimensionless fraction of the speed of light, in the second case v has dimensions of velocity, so we need to divide it by c to get the correct dimensionless number.

This is just the usual relativistic formula. We can see that as v approaches the speed of light, the time between events is shorter for the moving observer than the stationary observer.

So to recap. While this simple calculation doesn't necessarilyi "put to bed" the incorrect idea of "absolute rest", it does illustrate that the principle of invariance of the Lorentz intervals makes the prediction that the time interval measured by a clock (proper time) between the events we marked out on a moving observers worldline is lower than the time interval masured byh a clock between the events we marked out on the stationary worldine.

To operationally be able to do special relativity, one simply needs to accept that the calculation works regardless of the state of motion, as opposed to only working in some "special frame".

 

[robphy]

(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".

Physically, as we will motivate,
the events on these "spacelines" are "simultaneous according to the inertial timeline".

---

Method 1: CAUSAL DIAMOND (RADAR METHOD)

At event O, light-flashes are emitted. (Draw the future light-cone of O.)
At event P (along some inertial worldline), the radar-echoes of those flashes are received. (Draw the past light-cone of O.)
The intersection events H and L of those two light-cones are
"simultaneous according to the timeline OP" (and equidistant from OP according to OP)...
but they are clearly not-simultaneous according to the lab frame.
In fact, according to OP, the intersection-events H and L are simultaneous with the midpoint-event of OP (call that midpoint-event M).
The "spaceline of OP through event O" is gotten by drawing a parallel to HL through event O.

based on desmos.com/calculator/tqjkyhzn40 (interactive version below)

• OP would use her clock-readings at O and P to assign (t,x/c)-coordinates to distant-events H and L and the local event M=midpoint(O,P):
  
  the coordinate-time is "half of the sum of the clock-readings" (the mid-time)
  the coordinate-position is "half of the difference of the clock-readings" divided by signal-speed (the round-trip time of the signal, divided by the signal-speed)
  
  in OP's coordinates using the "radar method"
  $$\begin{align}
  t_P=(\tau_P+\tau_P)/2=\tau_P&\qquad x_P/c=(\tau_P-\tau_P)/2=0\\
  t_O=(\tau_O+\tau_O)/2=\tau_O&\qquad x_O/c=(\tau_O-\tau_O)/2=0\\
  t_L=(\tau_P+\tau_O)/2 &\qquad x_L/c=(\tau_P-\tau_O)/2 \\
  t_H=(\tau_P+\tau_O)/2 &\qquad x_H/c=-(\tau_P-\tau_O)/2 \\
  \nonumber \\
  \mbox{since M is the midpoint on OP, } t_M=(\tau_P+\tau_O)/2 &\qquad x_M/c=0
  \end{align}$$
• We can sketch the "causal diamond of OP" (the intersection of the future of O with the past of P),
  which are events which can "be influenced by event-O, then influence-P".
  The timeline OP is along the timelike-diagonal and its spacelines are parallel to the spacelike-diagonal-HL.
  In Minkowski spacetime, OP is Minkowski-orthogonal to HL.
  (It turns out that the area of the diamond is {\cal A}_{OP}=(width)(height)=(\tau_P-\tau_O)^2.
  In fact, the area of the diamond is also (in OP's coordinates)
  \begin{align}{\cal A}_{OP}&= ((t_P-t_O)^2-(x_P-x_O)^2/c^2)\\&=- ((t_L-t_H)^2-(x_L-x_H)^2/c^2) \end{align}. Check it!
  (It turns out, in another observer's (t,x/c)-coordinates, these expressions take the same form. This is the invariant square-interval.)
  
  
• We can draw a grid of diamonds to help us count ticks and calculate.
  Here each grid-diamond is (1/6)(\tau_P-\tau_O)
  in OP's coordinates, followed by a grid in lab coordinates.
  You should be able to count 36 grid diamonds in each case.
  The square-root is the elapsed-time along OP... 6 ticks =6( (1/6)(\tau_P-\tau_O) )=(\tau_P-\tau_O)
• 
  

---

Method 2: The "CIRCLE" in Minkowski-spacetime and the TANGENT-TO-THE-CIRCLE

• Along all inertial timelines through event-O,
  consider all future events from event-O that also have proper-time also equal to (\tau_P-\tau_O).
  
  This is the Minkowski-spacetime analogue of a circle with "radius" (\tau_P-\tau_O):
  (\tau_P-\tau_O)^2=(t - t_O)^2 - (x-x_O)^2/c, which is a hyperbola centered at event-O.
  
  
• For any event (call it P_{arbitrary}) on this hyperbola,
  draw the "RADIUS" vector from event-O to P_{arbitrary},
  and draw the TANGENT line through P_{arbitrary}.
  With this "CIRCLE", the TANGENT is PERPENDICULAR to the RADIUS.
  
  Note that this tangent-line is parallel to the spacelike-diagonal of the associated causal diamond. (Tune v).

Clearly, the tangent-lines for different choices of P are not parallel.
So, simultaneity is relative in special-relativity.

• (Note that the area of the causal-diamonds are the same for arbitrary choices of P on this hyperbola.
  Counting with the lab-frame grid of grid-diamonds.
  In the left-case, the area is (12)(3). In the right-case, the area is (6)(6).)
  
  
• (To see the case in Galilean relativity and Euclidean geometry,
  tune the E-slider below (which is initially +1 for Minkowski) to "0" for Galilean and to "-1" for Euclidean.
  
  Note the shape of the "circle of equal proper time" for the Galilean-case.
  All of the tangents to that circle are parallel.
  So, simultaneity in absolute in galilean-relativity.
  
  For Euclidean space, we don't interpret it as a" spacetime" because it doesn't have a physically-realizable causal structure,... but note that its tangents are also non-parallel, which is familiar from ordinary geometry.)

desmos.com/calculator/tqjkyhzn40

 

[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 suppose my thoughts on the relativity of simultaneity are that - I'd rather defver talking about it as long as possible, just because I know it confuses people.

My thoughts are just the opposite. I think it's the very first thing that ought to be introduced. Of course only qualitatively at first. It plants the seed of the notion that "time is relative" so that students know, at least vaguely, that there is something weird going on with time. Something new and different.

 

[PAllen]

My thoughts are just the opposite. I think it's the very first thing that ought to be introduced. Of course only qualitatively at first. It plants the seed of the notion that "time is relative" so that students know, at least vaguely, that there is something weird going on with time. Something new and different.

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.

 

[robphy]

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.

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, which is used to compute the spatial-velocity of the particle according to t^a). Decomposing a particle's 4-momentum to get its spatial part gets the "relativistic momentum". Decomposing the electromagnetic field tensor to get its spatial parts gets you the "electric and magnetic fields according to t^a".
So, banishing simultaneity suggests we should banish a bunch of other things as well.
But of course, "components of vectors and tensors" are tied to "measurements of spacetime-objects" by various observers.
So, that's why these measured quantities are "relative quantities".


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]).
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.

(*analogous unifying concept: no preferred direction in a 2D-vector space:
in 2D-space, the rotation matrix has no real eigenvectors,
in 1+1-spacetime, a Galilean boost has no timelike-eigenvectors, and a Lorentz boost has no timelike eigenvectors )


The "a-ha" moment for me in relativity
came when my professor operationally-defined how coordinates get assigned by an inertial-observer.
using radar-measurements (based on that observer's clock readings and the elapsed-time between light-signal emissions and receptions-of-their-echoes)...
... but he expressed everything in terms of the 4-velocities of the observer and the particle-being-measured
and the metric... in abstract-index notation.
So many things clicked together above.
No need to "find components" in some basis since the projection-operators encoded all of that.
No need to construct a "latticework of clocks and rulers" (as in Taylor and Wheeler).
No need for calculations with differentials and games people play with them.
It was all linear algebra encoding the geometry, neatly done.
And if you interpret the terms trigonometrically, one recognizes the hyperbolic-trig functions (usually called v=\tanh\theta \gamma=\cosh\theta and \gamma v =\sinh\theta and k=\exp\theta).

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".

(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.)