Benefits of time dilation / length contraction pairing?

[JesseM]

While I applaud your dedication to rigour, I think you take it too far.

Just out of curiosity, I look http://www.fourmilab.ch/etexts/einstein/specrel/www/" and searched for the word "frame". It appears exactly once, in the introduction, in the phrase "frames of reference" in the context of describing the first postulate.

The entire paper is about what we now call frames, Einstein just doesn't use that term. When he introduces a "a system of co-ordinates in which the equations of Newtonian mechanics hold good" at the beginning of section 1, what do you think that is if not an inertial frame? And he talks about different systems of coordinates throughout the paper, sometimes just using the word "system" (it's clear he means coordinate system and not some other type of physical system from the context)--for example, part 3, where he actually derives the Lorentz transformation, is titled "Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former".

While trying not to go so far as committing an "appeal to authority", I do want to know why I am being held to such high standards of rigour (specifying that the Galilean boost addresses a single question about frames) when the genius who came up with Special Relativity in his own way didn't really mention frames at all?

Again, the whole paper is about frames. The precise word is irrelevant as long as the concept is understood; I'd be equally happy with saying the Galilean boost is about relating the coordinates of an event in one "system of coordinates" to the coordinates of the same event in another "system". Whatever wording you use, this is conceptually quite different from just telling you how the coordinate separation between two objects is changing in a single coordinate system.

I'm not saying you are wrong that the Galilean boost is about frames, and the Lorentz Transformation is about frames, I am just wondering if your demands are truly warranted.

Just the fact that I was confused for so long by the meaning of the equation x' = x - vt in post 295 shows that they are warranted; I'd rather not get into more lengthy discussions over such trivial stuff in the future. Even if you incorrectly described the equation as the Galilean boost, the problem could have been avoided if you had spelled out in words what each symbol meant physically; if you had said at the outset that x' was supposed to represent a separation in the same frame that x and t referred to, then I might have offered a quick correction about terminology but there wouldn't have been all the confusion about what you were trying to demonstrate with your equations. But the combination of not giving physical definitions of your symbols at the outset, using the term "Galilean boost", and writing your equation using exactly the same notation as is usually used for the Galilean boost naturally led me to draw the wrong conclusions about the physical meaning of the equation. Hopefully you agree that, spelled out in words, this:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

...is telling us something physically from this?

(distance in A's frame between B and object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

Would you be happy if there was an extra step added in which I address Galilean frames, say the equation is x' = x - vt and that there is also a kinematic equation x' = x - vt and while they talk about different things, the relationship x' = x - vt holds equally for whatever values of x and t you enter into it? (Since the conditions under which the equation holds are the same for x' = x - vt and for x' = x - vt).

As long as you define the physical meaning of whatever equations you use I'll be OK, although from a pedagogical point of view I don't really like the approach of using identical notation for two physically different equations. In any case, is it necessary to discuss Galilean relativity at all in your derivation? Isn't the kinematical equation the only one you actually make use of?

It may also help to point out that someone could easily have "discovered" the first equation before the discovery of the Galilei transformation, since the first equation doesn't involve multiple coordinate systems. Do you think it would be fair for this person to demand that the Galilei transformation be renamed after themselves, since they had already discovered the equation even though the physical meaning of what was being calculated was different and they hadn't even been thinking about the relation between multiple frames?

No, and my suggestion to rename equations was entirely facetious.

I understood it was meant to be facetious...but my point in the above comment was, if you agree this hypothetical pre-Galileo guy shouldn't get credit for the Galilei transformation despite writing down an equation like x' = x - vt, doesn't that mean you should also agree we shouldn't use the same terminology for his kinematical equation that we do for the spatial component of the Galilei transformation, even if they look the same symbolically?

I do wonder if you have the visual ability to see that what I am doing is not really invalid.

It has nothing to do with visual abilities, I get visually why it works out that the separation in A's frame between B and the object at position x is always going to be equal to the position coordinate assigned to that object in B's frame. The point is that the equations are telling you different things physically, and that since I naturally thought you were introducing x'=x-vt in post 295 to transform into B's frame, I was confused since under the Galilei transformation the light could not be moving at c in B's frame.

In short, are you happy with:

Introduce Galilean frames (hopefully already done by the education system)
Introduce a kinematic equation in the form x' = x - vt (partially done)

As I said I don't like using the same notation for two equations with different physical meanings, and I think from a pedagogical point of view it's more confusing than helpful.

Point out that both equations operate on the same conditions

By "operate on the same conditions", I take it you mean if we pick a given x,t in A's frame, we get the same value for the answer? That's fine as long as you point out the physical meaning of the "answer" is different.

PS Am I going to have to draw another diagram? I've already been thinking of the best way to try to show you that what I am doing is not as whacky as you seem to think it is. Part of the problem might be that I am an engineer, manipulating equations is partly what I do. As someone with more of a physics bent, you don't seem to like the actual use of equations (or what you might term "abuse of equations" )

Now that I understand the physical meaning of your symbols I don't object to "what you are doing" in the derivation so far, only to how you are explaining it. And I have no dislike of equations (a scurrilous charge for a student of physics, my good sir! ), I just need to be clear on the physical meaning of any variables/constants that appear in them.

 

[neopolitan]

I am happy to use the kinematic equation from the start. Historically, how long ago was that equation (or at least that relationship) first identified? (Part of my interest is to see how long ago we could have got to SR. If the kinematic equation is all that is required, rather than Galilean relativity, that might actually push the possible date back to before the 1100's given an early Islamic mathematician's work. "In dynamics and kinematics, Biruni was the first to realize that acceleration is connected with non-uniform motion, which is part of Newton's second law of motion." - http://en.wikipedia.org/wiki/Al-Biruni" Sadly, the coverage of this fellow's work is less visible to me than that for da Vinci, Galileo and Newton, but it seems to me that if Al-Biruni got so far as to consider non-uniform motion then uniform motion was probably understood. Al-Biruni's contribution to optics as claimed in the same article is interesting as well, apparently being among the first to consider the speed of light to be finite (but faster than sound) - it makes one wonder why someone like this has been pretty much invisible. There is what seems to be an inconsistency in that article, did people prior to Al-Biruni think that the speed of sound was infinite along with the speed of light? {Since Al-Biruni is credited with not only being among to consider the speed of light to be finite, but also the first to find that the speed of light is much faster than the speed of sound. I would have thought that infinitely fast is much faster than the speed of sound.} But this is merely an aside.)

Amusingly, I did try something like your:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

but with subscripts. I found it too unwieldy and distracting. So, I do see the value in it.

Anyway, since with the riders, you don't object with my derivation so far, perhaps I should try to sum up where we are right now, before we go further. If I do that, being as rigorous as I can, are you happy to work from that point onwards?

cheers,

neopolitan

 

[JesseM]

I am happy to use the kinematic equation from the start. Historically, how long ago was that equation (or at least that relationship) first identified?

Not sure, I guess as soon as people came up with the notion of measuring how distances between things change with time (which would require at least somewhat fine-grained clocks), along with the concept of speed as distance traveled/time elapsed, they could have realized that the distance between a thing moving at constant speed v and a stationary thing would be shrinking at v times the time elapsed. Maybe this would come up in seafaring or something, even if it wasn't written as an algebraic equation. On the other hand, to write an equation like x - vt for a distance between a moving object and a stationary one, you need some notion of assigning objects position coordinates on a coordinate grid (or at least a coordinate line), and of choosing your origin so the moving object starts at position x=0 at time t=0, don't know if people would have thought in those terms until Descartes invented Cartesian geometry (incidentally, Galileo was about thirty years older than Descartes so I'd guess he never actually wrote the 'Galilei transformation' in algebraic terms, even if it's implicit in his work that he was saying the laws of physics would be invariant under this transformation).

Amusingly, I did try something like your:

(position in B's frame of object at position x in A's frame at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

but with subscripts. I found it too unwieldy and distracting. So, I do see the value in it.

Yeah, it would be unwieldy to include the full description in the symbol itself, but it would be helpful if each time a new symbol is introduced, you could say something like "define x' as the position in B's frame of the event that had coordinates x,t in A's frame", something along those lines.

Anyway, since with the riders, you don't object with my derivation so far, perhaps I should try to sum up where we are right now, before we go further. If I do that, being as rigorous as I can, are you happy to work from that point onwards?

Sure, as long as being rigorous means defining the physical meaning of any new terms you introduce.

 

[neopolitan]

Can we use this notation?

x'_a = the separation (hence the x) between B and an Event (hence the prime) according to A (hence the _a)

This perhaps should be expanded a little to specify clearly that unprimed means "between A and an Event". And capitalisation of Event is used to clarify that I am referring to a specific event, not just any event.

So you have two frames (A -> Event and B -> Event), as many perspectives as you like (_a, _b, _c, _d - but we will only use two) and we will have two dimensions (x and t).

I might also need to clarify something that I have firmly in mind about my t values.

When we draw a spacetime diagram, we can draw an interval between, say, A and an Event in A's future. But in the real world (another engineering trait coming out perhaps), A will not know about that Event until enough time has elapsed for the Event to take place (in the A frame) and for a photon to travel from the Event to A.

For that reason, I see utility in moving the origin of the t_a axis to be simultaneous with the Event in the A frame. If I do that then, as a consequence, x_a = c.t_a.

Notionally, A and B are colocated at (0,0) - now this does not have to physically take place because we calculate intervals so any initial offset will cancel out.

Are you happy with this? Clarifying what I am asking:

1. Are you happy with the notation regime suggested?
2. Are you happy with the concept that A and B are notionally colocated at (0,0), but they don't actually have to be colocated at that time or place?

cheers,

neopolitan

 

[JesseM]

Can we use this notation?

If x'_a is meant to be a variable, as opposed to the separation between B and the Event at some specific time, can we write it as x'_a(t)?

I might also need to clarify something that I have firmly in mind about my t values.

When we draw a spacetime diagram, we can draw an interval between, say, A and an Event in A's future. But in the real world (another engineering trait coming out perhaps), A will not know about that Event until enough time has elapsed for the Event to take place (in the A frame) and for a photon to travel from the Event to A.

For that reason, I see utility in moving the origin of the t_a axis to be simultaneous with the Event in the A frame. If I do that then, as a consequence, x_a = c.t_a.

In other recent posts (such as 308) you used t_a to refer to a specific time interval (between A and B being colocated and the photon passing A) in A's frame, so it's potentially confusing to refer to "the t_a axis"--I assume you just mean the t-axis in A's frame? Whereas in the equation x_a = c.t_a, does t_a still refer to that time interval I mentioned? And when you talk about moving the origin to be "simultaneous with the Event in the A frame", do you just mean the origin has the same time coordinate as the Event (i.e. the t-coordinate of the Event in A's frame is 0), not that the Event is actually at the origin? So it's still true that the event has coordinates x=x_a, t=0 in A's frame, and it's still true that the photon reaches A at x=0, t=t_a? If so I don't really understand why you talk about "moving the origin", since this is exactly how things were before.

Notionally, A and B are colocated at (0,0) - now this does not have to physically take place because we calculate intervals so any initial offset will cancel out.

Do the origins of their coordinate systems still coincide at a time coordinate of 0 in both systems? If so, of course it is not necessary for observers at rest in these coordinate systems to be located at x=0 in each system, they can be at any position coordinate we like. But in this case I'd like a redefinition of t_a--does it refer to the time in A's frame that the photon passes x=0, or the time it passes A, or something else?

1. Are you happy with the notation regime suggested?
2. Are you happy with the concept that A and B are notionally colocated at (0,0), but they don't actually have to be colocated at that time or place?

See above, I'm not sure I understand what you're saying here.

 

[neopolitan]

I use t_a as a time interval in the A frame between an Event, and the reception of the photon coincident with that Event (or spawned by that Event).

I started writing the summary and lost it all. Very annoying.

What I mean about moving the origin of the axes is that the Event can take place whenever. But despite that, we shift the origin of the axes so that the Event is simultaneous in the A frame with (0,0).

In other words, we can work backwards. We get a photon today and discover from other reliable sources that the photon was released a distance of 10 light years away (in our frame), so we shift the origin of our t axis to back when the photon was released (in our frame) making today t = 10 years. Equally, we can be told that in three years from now, a photon will be released from the same location. We can shift the origin of our t axis forward 3 years from today, making today t = minus 3 years (knowing that the photon won't reach us until t = 10 years.

So I can shift the origin of the t-axis backwards or forwards as I like, which means I can consider any event, at any time.

If either of A and B were to not be located at the origin of their frame of reference, I would make it B.

I'm a bit perplexed by the idea that x'_a wouldn't be variable with different values of t_a.

To the same extent that x' in the equation x' = x - vt is variable with t, so to is x'_a in the equation x'_a = x_a - vt_a variable with t_a.

Also, I do believe that in the standard Lorentz Transformation x' is variable as you vary t (and indeed x).

I would agree that I would have to write x'_a(t_a) = x_a - vt_a, if I routinely saw the Lorentz Transformation written as:

x'(x,t)= \gamma.(x - vt)
t'(x,t)= \gamma.(t - vx/c^2)

But I don't.

I feel as if you are demanding more than is justified. I can do as requested if I must though.

Can you confirm that I absolutely must specify that x'_a varies as t_a varies?

cheers,

neopolitan

 

[neopolitan]

Summarising where I think we are at (including corrections in an attempt to be more rigorous).

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if a photon is released from a distance of x_a away from A and it takes a period of t_a to reach A, then:

x_a = ct_a

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'_a = x_a - vt_a

or

x'_a(t_a) = x_a - vt_a

which is, in words:

(separation between B and the location of the Event, in A's frame, when a photon from the Event reaches A, in A's frame) = (location of the Event, in A's frame) - (velocity of B in A's frame)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Then, we want to know how things look in B's rest frame.

x'_b = ct'_b and
x_b = x'_b + vt'_b {or x_b(t'_b)}

which are, in words:

(location of Event, in B's frame) = (speed of light)*(time interval between the Event and reception of photon from the Event, in B's frame) and

(separation between B and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between the Event and when a photon from the Event reaches B, in B's frame)

Since

t_a = time interval between the Event and when a photon from the Event reaches A, in A's frame

and

t'_b = time interval between the Event and when a photon from the Event reaches B, in B's frame

we have no expectation that t_a = t'_b

Happy with that?

cheers,

neopolitan

 

[JesseM]

I use t_a as a time interval in the A frame between an Event, and the reception of the photon coincident with that Event (or spawned by that Event).

Again, are you assuming A is at position x=0 in A's own rest frame? If not, when you say "reception of the photon" do you mean when the photon crosses A's worldline, or when the photon crosses the x=0 axis?

What I mean about moving the origin of the axes is that the Event can take place whenever. But despite that, we shift the origin of the axes so that the Event is simultaneous in the A frame with (0,0).

"Take place whenever" relative to what coordinate system? If you're using a coordinate system where it takes place at t=0, then it doesn't take place whenever, and if that's the starting point of your proof then everything else in the proof follows from that assumption and whatever conclusions you reach cannot simply be assumed to still hold if the Event is located somewhere else besides t=0 (if that's what you're getting at, I'm not sure). If it helps, suppose you end up proving that if in A's frame the spatial and temporal intervals between the Event at t=0 and some second event are x and t, then the spatial interval between these same pair of events in B's frame is gamma*(x - vt) and the temporal interval is gamma*(t - vx/c^2). In that case, even though your proof started from the assumption that the Event occurred at t=0 in A's frame, it would be easy to prove a lemma of the type I talked about back in post 249:

I suppose you could prove a lemma that shows that the distance and time intervals between a pair of events in a given coordinate system will be unchanged in a second coordinate system with the origin at a different location but which is at rest relative to the first (i.e. a simple coordinate transformation of the form x' = x + X₀ and t' = t + T₀ where X₀ and T₀ are constants).

So with this lemma added, you could then show that the relation between the intervals in A's frame and the intervals in B's frame will be the same even if you move the origins so that the Event is at some totally arbitrary set of coordinates. This lemma could be added to the very end of the proof. However, this will still not necessarily mean your proof is fully general; if in your proof you assume that the first Event and the second event (which together define the intervals you're dealing with in each frame) both lie along the path of a light ray, then you haven't proved that the same relation would hold for a pair of events with a timelike or spacelike separation.

In other words, we can work backwards. We get a photon today and discover from other reliable sources that the photon was released a distance of 10 light years away (in our frame), so we shift the origin of our t axis to back when the photon was released (in our frame) making today t = 10 years.

Why assume the origin was somewhere else to begin with? You don't even have to pick the placement of your axes until you've already received the photon, and at that point it's easy to position them so that the Event 10 light years away occurred at t=0, if that's all you're worried about. When dealing with SR problems you don't really have to concern yourself with these sorts of practical issues, just assume either an omniscient perspective on spacetime, or assume all coordinates are assigned indefinitely far into the future when all the events in the region of spacetime you're interested in are already known.

If either of A and B were to not be located at the origin of their frame of reference, I would make it B.

If B is not at the origin of its own frame, then does that mean B is not necessarily colocated with A at t=0 in A's frame? If it's not, then don't you have to modify the equation

I'm a bit perplexed by the idea that x'_a wouldn't be variable with different values of t_a.

t_a is a constant in any given physical scenario, is it not? It's the time coordinate of when the photon passes A, right? So if x'_a represents the distance between B and the position x_a as a function of time, this distance is varying with the time coordinate t in A's frame, not varying with t_a (unless you are using t_a to represent A's time variable as well as the specific time the photon passes A, something I requested you not do in my last post because it'd be confusing). On the other hand, if you just want to define x'_a as the distance between B and x_a at the specific time t_a when the photon passes A (or alternatively, at the specific time t'_a when the photon passes B), that's fine with me, in this case x'_a would be a constant rather than a variable. But you seemed to want x'_a to represent a distance that could vary with time rather than a distance at a specific time in post 306 when you said:

Note that in A's rest frame, the distance between the location of YDE and A does not change - therefore x_a does not change with time, but x'_a does (because x'_a is the distance between the location of YDE and B, according to A).

To the same extent that x' in the equation x' = x - vt is variable with t, so to is x'_a in the equation x'_a = x_a - vt_a variable with t_a.

Again, in any specific physical setup isn't t_a a constant? Of course you can vary the physical setup itself, but that's not what I meant when I said I thought you were making x'_a a variable--I thought that even given a particular setup (a particular choice of position x_a for the Event on the photon's worldline), x'_a represented the changing distance between B and x_a as a function of time in A's coordinate system, not the distance between B and x_a at some specific time like t_a (I based this on your comment from post 306 above).

Also, I do believe that in the standard Lorentz Transformation x' is variable as you vary t (and indeed x).

If you can vary x and t, sure, but if you pick some specific physical event then x', x, and t for that choice of event are all constants, echoing my comment about t_a and x_a being constants for a particular choice of physical setup in your scenario.

Can you confirm that I absolutely must specify that x'_a varies as t_a varies?

See above for a clarification of my meaning. If you want x'_a to be the distance between B and x_a at a specific time corresponding to some event in your setup like the photon passing A at t_a or the photon passing B at t'_a (and remember that you had actually defined x'_a in the latter way in our earlier discussions, not the former), then there's no need to call it a variable. But your comment in post 306 seemed to insist that x'_a is defined in such a way that it changes with time rather than being the distance between B and x_a at some specific time.

 

[neopolitan]

Again, are you assuming A is at position x=0 in A's own rest frame?

Yes


As for the rest, I'm a bit confused why something that seems so obvious to me is confusing for you.

I'll try again.

In A's rest frame, A is at rest.

In A's rest frame, B is not at rest.

In A's rest frame, the separation between where the Event takes, took or will take place and A is constant.

In A's rest frame, the separation between where the Event takes, took or will take place and B is not constant.

x_a is the separation between where the Event takes, took or will take place and A in A's rest frame.

x'_a is the separation between where the Event takes, took or will take place and B in A's rest frame.

Therefore, x_a is constant and x'_a is not constant.

It seems so simple to me, I can't quite grasp why it warrants such a long post about it.


For a specific time in A's frame, x'_a is defined, not varying, but not quite a constant either (because to me a constant is only a constant if you can vary something and once you pick a specific time, you don't have anything to vary in A's frame anymore, so long as you continue to talk about the same Event).

cheers,

neopolitan

 

[JesseM]

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if a photon is released from a distance of xa away from A and it takes a period of ta to reach A, then:

x_a = ct_a

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between the Event and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'_a = x_a - vt_a

You should also add the assumption that the Event occurs at time coordinate t=0 in A's frame, and that A is located at x=0, since otherwise this substitution wouldn't work.

(separation between B and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between the Event and when a photon from the Event reaches B, in B's frame)

I assume you meant to write separation between A and the location of the Event in B's frame, right? After all, the separation between B and any given location in B's frame will be constant. Assuming that's what you meant, then it seems to me the equation would only hold if we assume that A and B are colocated at the same time as the Event occurs in B's frame. But that's obviously problematic, because we already assumed the Event was simultaneous with A and B being colocated in A's frame, and as we know they can't both be true in relativity.

 

[JesseM]

I'll try again.

In A's rest frame, A is at rest.

In A's rest frame, B is not at rest.

In A's rest frame, the separation between where the Event takes, took or will take place and A is constant.

In A's rest frame, the separation between where the Event takes, took or will take place and B is not constant.

x_a is the separation between where the Event takes, took or will take place and A in A's rest frame.

x'_a is the separation between where the Event takes, took or will take place and B in A's rest frame.

Therefore, x_a is constant and x'_a is not constant.

Yes, that's exactly what I originally thought you meant, until you made the confusing comment that x'_a varies with t_a, rather than saying it varies with t, which is something I was asking about (I know you don't like my habit of responding to your posts line-by-line, but your habit of responding to my posts in a 'gestalt' manner often means you don't answer the specific questions I ask about, and instead just repeat things I already understand without answering the questions I specifically asked for the purposes of clarifying). Again, I thought t_a was a constant (given a particular physical setup) just like x_a--t_a represents the time the photon passes A, and for a particular physical setup there's only one unique time that this happens. After all, you wrote x_a = c*t_a--if you call x_a a constant, then based on this equation you must call t_a a constant too. It's possible you are using the symbol t_a to represent both the abstract time variable in A's frame and the specific time coordinate when the photon passes A, but I already speculated in two previous posts that you might be doing this and asked you to please not use that sort of ambiguous notation if that's what's going on.

Then you also made the point that you shouldn't have to spell out that x'_a varies with time when x' is not written as x'(x,t) in the Lorentz transformation. I pointed out that given a particular choice of physical event, x and t are not variables, analogous to how given a particular choice of physical setup (distance between the Event on the path of the light ray and A at time t=0 in A's frame), x_a and t_a are not variables in your equations. Do you understand this point, and if so do you drop this particular argument for why it's unreasonable for me to ask you to write x'_a(t) in your equations?

For a specific time in A's frame, x'_a is defined, not varying

Sure, for any variable with a function of t, if you pick a specific time t it doesn't vary at that time! This is totally trivial, but it hardly proves that a quantity that varies with t is not a "variable". The point is, in a given physics problem with a well-defined physical setup, I'd call a symbol a "variable" if its value can change depending on the value of some other symbol, and a "constant" its value depends only on the setup. Is it my request that you differentiate between the two in your notation really so onerous? Just to help add some context, one of the reasons I make this request is that I anticipate that later in the proof you're probably going to want to talk about the value of x'_a(t) at a different time other than t_a, perhaps at the time t'_a in A's frame when the photon passes A; at that point it really could become genuinely confusing if you use exactly the same symbol, a problem that will be avoided if you write x'_a(t_a) for the first and x'_a(t'_a) for the second. And if I'm wrong, and your proof will never make use of x'_a at any time other than t_a, in that case making a big deal out of the fact that x'_a is a variable seems pointless, it would be much simpler just to define x'_a as the distance between B and the position of the event at the specific time t_a when the photon passes A.

 

[neopolitan]

Fixing (corrections and clarifications are highlighted - I accept that wherever I need to make a clarification this is as bad as being wrong, I don't intend to defend being wrong - or being so unclear as to necessitate a clarification);

We start with the kinematic equation for an observer A observing B who has a velocity of v towards a location which has a separation of x from A:

x' = x - vt

or

x'(t) = x - vt

which is, in words:

(separation between B and position x, in A's frame, at time t in A's frame) = (position x in A's frame) - (velocity of B in A's frame)*(time t in A's frame)

We then introduce the concept that if at t=0 a photon is released from a distance of x_a away from A and it takes a period of t_a to reach A and A is located at x=0, then:

x_a = ct_a

which is, in words:

(location of the Event, in A's frame) = (speed of light)*(time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame)

Using these values in the kinematic equation we have:

x'_a = x_a - vt_a

or

x'_a(t_a) = x_a - vt_a

which is, in words:

(separation between B and the location of the Event, in A's frame, when a photon from the Event reaches A, in A's frame) = (location of the Event, in A's frame) - (velocity of B in A's frame)*(time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame)

Then, we want to know how things look in B's rest frame. If at t'=0 the photon released as described before is at a distance of x'_b away from B and it takes a period of t'_b to reach B and B is located at x'=0, then:

x'_b = ct'_b and
x_b = x'_b + vt'_b {or x_b(t'_b)}

which are, in words:

(location of Event, in B's frame) = (speed of light)*(time interval between colocation of A and B and reception of photon from the Event, in B's frame) and

(separation between A and the location of the Event, in B's frame, when a photon from the Event reaches B, in B's frame) = (location of the Event, in B's frame) + (velocity of A in B's frame)*(time interval between colocation of A and B and when a photon from the Event reaches B, in B's frame)

Since

t_a = time interval between colocation of A and B and when a photon from the Event reaches A, in A's frame

and

t'_b = time interval between colocation of A and B and when a photon from the Event reaches B, in B's frame

we have no expectation that t_a = t'_b

I'm aware that the event simultaneous with t=0 and the event simultaneous with t'=0 are not simultaneous with each other. However, each event used is colocated with the relevant observer (notionally, if A and B are timing events, the only events they can give accurate time values to are "colocation of self with other observer" and "colocation of self with photon").

The point I have to make clear again is that we only know what happened at an event once information (or photon from the event) reach us. Then we work backwards.

If A receives a photon at t_a from an event at t=0, then when did that same photon pass B? What is t'_b in terms of t_a and x_a?

What is x'_b in terms of t_a and x_a?

Can we work it out?

I think we can.

 

[neopolitan]

Yes, that's exactly what I originally thought you meant, until you made the confusing comment that x'_a varies with t_a, rather than saying it varies with t, which is something I was asking about (I know you don't like my habit of responding to your posts line-by-line, but your habit of responding to my posts in a 'gestalt' manner often means you don't answer the specific questions I ask about, and instead just repeat things I already understand without answering the questions I specifically asked for the purposes of clarifying). Again, I thought t_a was a constant (given a particular physical setup) just like x_a--t_a represents the time the photon passes A, and for a particular physical setup there's only one unique time that this happens. After all, you wrote x_a = c*t_a--if you call x_a a constant, then based on this equation you must call t_a a constant too. It's possible you are using the symbol t_a to represent both the abstract time variable in A's frame and the specific time coordinate when the photon passes A, but I already speculated in two previous posts that you might be doing this and asked you to please not use that sort of ambiguous notation if that's what's going on.

Then you also made the point that you shouldn't have to spell out that x'_a varies with time when x' is not written as x'(x,t) in the Lorentz transformation. I pointed out that given a particular choice of physical event, x and t are not variables, analogous to how given a particular choice of physical setup (distance between the Event on the path of the light ray and A at time t=0 in A's frame), x_a and t_a are not variables in your equations. Do you understand this point, and if so do you drop this particular argument for why it's unreasonable for me to ask you to write x'_a(t) in your equations?

Sure, for any variable with a function of t, if you pick a specific time t it doesn't vary at that time! This is totally trivial, but it hardly proves that a quantity that varies with t is not a "variable". The point is, in a given physics problem with a well-defined physical setup, I'd call a symbol a "variable" if its value can change depending on the value of some other symbol, and a "constant" its value depends only on the setup. Is it my request that you differentiate between the two in your notation really so onerous? Just to help add some context, one of the reasons I make this request is that I anticipate that later in the proof you're probably going to want to talk about the value of x'_a(t) at a different time other than t_a, perhaps at the time t'_a in A's frame when the photon passes A; at that point it really could become genuinely confusing if you use exactly the same symbol, a problem that will be avoided if you write x'_a(t_a) for the first and x'_a(t'_a) for the second. And if I'm wrong, and your proof will never make use of x'_a at any time other than t_a, in that case making a big deal out of the fact that x'_a is a variable seems pointless, it would be much simpler just to define x'_a as the distance between B and the position of the event at the specific time t_a when the photon passes A.

I don't intend to use a value of x'_a that is different from its value at t_a.

I don't intend to use a value of x_b that is different from its value at t'_b.

I don't intend use a value of x_a other than such that x_a = c.t_a

I don't intend use a value of x'_b other than such that x'_b = c.t'_b.

Does that make things easier?

If I do find myself using anything other than these, I will try to mark them accordingly (but I really don't think that I will).

cheers,

neopolitan

 

[sylas]

I don't intend to use a value of x'_a that is different from its value at t_a.

What does this even mean?

Is x'_a a constant, or a variable? If you mean a constant, why the heck are you speaking of its value at certain time? If you mean a variable, why the heck would you only use a single value?

Isn't (x'_a, t'_a) just the co-ordinates of the event "photon passes A" in the frame of reference of B?

 

[neopolitan]

Isn't (x'_a, t'_a) just the co-ordinates of the event "photon passes A" in the frame of reference of B?

The post you quoted was in response to a specific concern from JesseM that I would go changing the meaning of x'_a. I don't intend to.

He got to that because I noted at one point that (in general) while in the A frame x_a does not change, x'_a does. But my (specific scenario driven) derivation centres around events which lock in values of x_a, t_a and hence x'_a. So in the scenario I describe, x_a, t_a and x'_a all have one value. So fixed are they that in earlier posts we assigned them numbers.

In answer to the question you posed here, no.

x'_a is the separation between B and where the photon was at t = 0 in the A frame (x = separation, ' = between B and the Event, _a = in the A frame or according to A).

cheers,

neopolitan

It may be worth mentioning that I am keeping the prime from the kinematic equation x' = x - vt

While I understand that this may cause concern because while I focus on observer A, and that that would make a few people consider that primes refer to B's rest frame, those people might be happier to know that the equations I end up with are:

x'_b = gamma.(x_a - vt_a)
t'_b = gamma.(t_a - vx_a/c²)

which is a confluence of the primed is the B frame, unprimed is the A frame and _b is the B frame, _a is the A frame.

 

[sylas]

The post you quoted was in response to a specific concern from JesseM that I would go changing the meaning of x'_a. I don't intend to.

He got to that because I noted at one point that (in general) while in the A frame x_a does not change, x'_a does. But my (specific scenario driven) derivation centres around events which lock in values of x_a, t_a and hence x'_a. So in the scenario I describe, x_a, t_a and x'_a all have one value. So fixed are they that in earlier posts we assigned them numbers.

In answer to the question you posed here, no.

x'_a is the separation between B and where the photon was at t = 0 in the A frame (x = separation, ' = between B and the Event, _a = in the A frame or according to A).

cheers,

neopolitan

Thanks... but I am still finding this incredibly hard to follow.

You've said "x'_a is the separation between B and where the photon was at t = 0 in the A frame". Distance WHEN? At t=0 also? t according to whom? You say it has one value. But then you've also said that x'_a can "change" in the A frame? How can that possibly be?

I have trouble following along when you speak of a "location". What is a fixed location in one frame is not a fixed location in another. I think it would be clearer if you stick to "events", so that you can sensibly speak of one event in several different frames.

You scenario is this, isn't it? It involves three particles: A, B and photon. A and B are moving at constant velocity v relative to each other. The events of interest occur in this order.

• A passes by B (co-located).
• Photon passes by B.
• Photon passes by A.

Is that right? You've also added another event of photon being "emitted".

The distance between events A and B (photon passing by A and photon passing by B) as observed by the particles A and B are related by the Doppler shift factor, are they not? Multiply, or divide the distance by
\sqrt{\frac{c-v}{c+v}}
to get the distance for the other observer. The distance is greater for the particle that the photon passes by first.

Cheers -- sylas

 

[neopolitan]

sylas,

I do appreciate your interest, but you might notice that a lot has come before this. If I reply to you, Jesse will reply to my replies to you (it's happened before) and we will end up going over old ground again which is something I am trying to avoid.

In general, x'_a(t) is variable with t. Specifically, x'_a(t) is fixed with a fixed value of t=t_a.

The scenario is framed such that x_a(0) = c.t_a, in other words a time interval of t_a after t=0 (in the A frame), a photon passes A since it was initially the right distance away to cover that distance in that time.

At the time at which the photon passes A, B has traveled a distance towards where the event took place and in A's frame that is:

x'_a(t_a) = x_a - v.t_a

(the separation between B and where the event took place at the time at which the photon from the event passes A, in the A frame) = (the separation between A and where the event took place) - (the distance that B has moved towards where the event took place in the time it took for a photon to travel from the event to A)

Note, I am not specifically writing this to explain to you, I am writing it in a format that JesseM has said is necessary for it to be explained.

Now, I suspect that the method for explaining to JesseM just possibly won't be as suitable for explaining to you. If that is the case, can I suggest that you go back into the earlier posts, find an explanation which seems to suit you more, and I can try to address your questions in a separate thread?

cheers,

neopolitan

 

[JesseM]

I don't intend to use a value of x'_a that is different from its value at t_a.

I don't intend to use a value of x_b that is different from its value at t'_b.

I don't intend use a value of x_a other than such that x_a = c.t_a

I don't intend use a value of x'_b other than such that x'_b = c.t'_b.

Does that make things easier?

In that case, why introduce the complication of saying x'_a is a variable but x_a is a constant? You could easily make x'_a a constant just by specifically defining it as the separation at time t_a when the photon passes A, not the separation at an arbitrary time t in A's frame.

Just as a reminder though, back when we were going through the numerical example where you put numbers to these values, you did define the symbol x'_a in terms of t'_a, the time the photon passed B (both x'_a and t'_a had the value 5).

 

[sylas]

Now, I suspect that the method for explaining to JesseM just possibly won't be as suitable for explaining to you. If that is the case, can I suggest that you go back into the earlier posts, find an explanation which seems to suit you more, and I can try to address your questions in a separate thread?

S'okay. I am perfectly comfortable with relativity and don't need it explained to me. I can see I am not helping here, and withdraw. Sorry for the distraction!

My main aim was to suggest, gently, that you are not doing a very good job of giving clear and unambiguous definitions of what you mean. I'm glad you didn't take offense at that; I wanted to say it without coming across as being too negative. But I still find it really hard to follow what you mean with notation or use of language, and I don't think this is just me, or because it is non-standard. The problem is that it is almost always ambiguous. You evidently have a clear idea what you mean. I don't.

It should be possible to express whatever it is you mean with less words and repetition. All you need is to avoid any potential ambiguity for what notation refers to; and then pretty much any of the regulars who have struggled to follow these threads will get it, IMO. It's not a problem of finding the "right" explanation for different people.

When explanations refer to the distance to a "location", rather than an event, there's a potential ambiguity as to what the location means in different frames and times. A "location" without an associated particle usually means a fixed distance co-ordinate, or worldline with zero velocity; but that depends on the observer and I often don't know what observer is intended. Referring to a specific event, however, is nearly always crystal clear.

The main answer here is that observers A and B measure different distances from the event "photon passes B" to the event "photon passes A". If the photon passes by B and then A, and if v is their relative velocity (+ve for moving apart), then the distance between these two events d_A for observer A is related to distance d_B for observer B by

d_A = d_B \sqrt{\frac{c-v}{c+v}}​

You can show this with the Lorentz transformations; and there may be other ways to get the right answer.

 

[JesseM]

Fixing (corrections and clarifications are highlighted - I accept that wherever I need to make a clarification this is as bad as being wrong, I don't intend to defend being wrong - or being so unclear as to necessitate a clarification);

OK, the clarified version looks clear to me.

I'm aware that the event simultaneous with t=0 and the event simultaneous with t'=0 are not simultaneous with each other. However, each event used is colocated with the relevant observer

Each event you use to define the time intervals, yes. The "Event(s)" on the photon's worldline used to define x_a and x'_b aren't colocated with the observers, of course.

If A receives a photon at t_a from an event at t=0, then when did that same photon pass B?

You mean, what time in A's frame did the photon pass B? This is a time you haven't defined a symbol for yet, although in the earlier discussion you defined this as t'_a.

What is t'_b in terms of t_a and x_a?

Note that t'_b and t_a/x_a don't refer to intervals between the same pair of events, so if you show a certain relation between these values it won't necessarily prove anything about the intervals in different frames between a single pair of events as in the the Lorentz transformation. Also note that in the Lorentz transformation equation it's not only assumed you're talking about intervals between a single pair of events, but it's also assumed that when calculating the intervals you're being consistent about the order in which you're taking the events. For example, x_a and t_a could both be understood as space and time intervals between the same pair of events (photon passing A) and (Event on photon's worldline that's simultaneous with A&B being colocated) even if you didn't choose to define them in terms of this pair, so if t_a was to be defined as (time coordinate of photon passing A) - (time coordinate of Event on photon's worldline that's simultaneous with A&B being colocated) in order to make it a positive number, then that means in order to be consistent we would have to define x_a as (position coordinate of photon passing A) - (position coordinate of Event on photon's worldline that's simultaneous with A&B being colocated), so if you assume this Event has a positive position position coordinate that would make the interval x_a negative according to the above definition. I think that in your notation you are just defining x_a as the absolute value of the distance between A and the Event, so it would be positive rather than negative; in this case the physical meaning of the equation you derive relating these quantities will be quite different from the physical meaning of the Lorentz transformation relating intervals between a single pair of events calculated using a consistent order for the events.

Just to check where you're going with this, do you intend to derive an equation that has the same physical meaning as the Lorentz transformation, or do you just intend to derive an equation which looks superficially similar but whose physical meaning is different?

 

[neopolitan]

OK, the clarified version looks clear to me.

Then shortly I will move onto the next stage. I will try to hold back (which means: write a reply, leave it for a while, then check it for typos then post)

I've started replying to other parts of your post but run out of time. Will try to address them later.

cheers,

neopolitan

 

[JesseM]

Then shortly I will move onto the next stage. I will try to hold back (which means: write a reply, leave it for a while, then check it for typos then post)

I've started replying to other parts of your post but run out of time. Will try to address them later.

OK, but in order to avoid getting into an extended discussion of the steps of your proof only to find that the ending conclusion isn't actually equivalent to the Lorentz transformation, can you state in advance the final equation(s) you intend to derive, including the physical definitions of any symbols appearing in the final equation(s) in terms of the setup you've outlined (space and time intervals involving a photon traveling towards A and B which passes them at particular points along with some Event or Events on its worldline, presumably)?

 

[neopolitan]

OK, but in order to avoid getting into an extended discussion of the steps of your proof only to find that the ending conclusion isn't actually equivalent to the Lorentz transformation, can you state in advance the final equation(s) you intend to derive, including the physical definitions of any symbols appearing in the final equation(s) in terms of the setup you've outlined (space and time intervals involving a photon traveling towards A and B which passes them at particular points along with some Event or Events on its worldline, presumably)?

Alright, I'll put some thought into that as well.

cheers,

neopolitan

 

[Rasalhague]

I haven't caught up with all the most recent posts yet, but I hope no one minds if I butt in with some general observations.

I just googled the expression "moving clocks run slow" (3190 hits), then tried "moving clocks run fast" (128 hits), several of the latter apparently referring to unorthodox theories and rejections of relativity, although not all of them. Presumably authors who present the slogan "moving clocks run slow" as a verbal equivalent to the time dilation equation are taking the word dilation to mean a bigger number when it refers to time, but a smaller number when it refers to space! On the other hand, Taylor and Wheeler in Spacetime Physics explicitly define dilation as a bigger number, and I think this is the more standard idea.

So if some (most?) of the authors who use the expression "moving clocks run slow" are, in fact, referring to the temporal analogue of length contraction, it seems strange that they avoid the obvious term "time contraction" and its formula. (Unfortunately Google results for "time contraction" are mixed up with pages about timing the contractions of labour, so I can't make a fair comparison.) The only book I've yet found to mention "time contraction" is What Does a Martian Look Like by Jack Cohen and Ian Stewart, about the possible forms that extraterrestrial life might take, rather than physics as such.

This preference for the expression "moving clocks run slow" is presumably as much a matter of convention as the preference for the expression "time dilation" and the pairing of the time dilation formula with that of length contraction. Its popularily suggests that it may be no less natural a way of conceptualising the relationship. All of this strongly inclines me to agree with Neopolitan's comments in the very first post of this thread: that the pairing of time dilation with length contraction is a source of needless confusion. Thanks to Jesse's diagram and explanation of some reasons why contraction is a more natural way to view what happens to the length of a physical object when viewed as moving as opposed to standing still, it seems to me that it would indeed be more logical to pair time contraction and length contraction (i.e. use the same terminology for time as we use for space), for the sake of comparing like with like, and of avoiding the false impression of asymmetry which the traditional pairing creates.

That said, there are genuine asymmetries between time and space, or between the ways we relate to them, which might lead us to view length contraction as more natural while either transformation (dilation or contraction) seems equally natural for time. Here is a list I came up with. It could be that some of the items are essentially the same as others, stated in different words. The fourth point about determinism is based on what Jesse said in an earlier post.

1. Persistence. Clocks and rulers are both objects with sharply defined spatial limits; they both persist in time. To make a thought experiment more symmetrical, we could imagine everlasting clocks, each static at the origin of their respective rest frames, and infinite rulers (each existing for one moment only, as defined in its own rest frame). While this may be convenient for the thought experiment, its unphysicality could point to a real difference in how we relate to time and space.

2. Degrees of freedom. Objects are free to move in either direction along a line through space, but their motion in time is unidirectional.

3. Speed. Speed is defined as length divided by time, regardless of which component we're calculating a change in.

4. Determinism. Physical laws predict events in limited space over unlimited time (with certain limits on accuracy); they have less power to predict events in limited time over unlimited space. We're used to ideas like “what goes up must come down”, but it's harder to imagine a universe where what goes up here would be a reliable guide to what goes up simultaneously somewhere else.

 

[Rasalhague]

Some thoughts on terminology and notation.

In my notes, I've taken to using the term input frame (or source frame) for the frame for which we know the coordinate values, and output frame (or target frame) for the frame for which we want to calculate the coordinate values. I've been using the terms left frame and right frame, respectively, for the frame moving left (i.e. in the negative x direction) past the other, and the frame moving right (i.e. in the positive x direction) past the other. The initials L and R stand for left and right. For example, "Clock L is at rest in frame L, the left frame, so frame L is clock L's rest frame. Clock L is moving in frame R, so frame R is a moving frame for clock L."

The terms input and output depend on the question asked, and may change their referents (change the frames they refer to) if a different question is asked. The terms left and right depend on the definition of the frames, and keep the same referents so long as the same frames are being used. The terms rest and moving are defined relative to a particular object; they may change their referents if a different object is discussed. Contraction and dilation questions can be asked of both time and space with any of these terms.

Some authors use primed and unprimed for what I'm calling input and output, but others use primed and unprimed for left and right. Others again use primed to refer to whichever frame is defined as moving in a particular example. These definitions don't necessarily coincide!

Some authors do as Wolfram Alpha does and use a subscript zero for the coordinates of the input frame if your question involves time dilation or length contraction, and for the output frame if your question involves time contraction or length dilation. Such coordinates change which frames they're referring to whenever you go from asking a dilation to a contraction question of the same coordinate, or from asking a contraction to a dilation question of the same coordinate. Wolfram Alpha calls dilated time "moving time", and contracted length "moving length".

http://www34.wolframalpha.com/input/?i=time+dilation
http://www34.wolframalpha.com/input/?i=length+contraction

To me this feels like an arbitrary switch in terminology purely for the sake of maintaining this artificial, traditional association of dilation exclusively with time, and contraction exclusively with length.

 

[JesseM]

I just googled the expression "moving clocks run slow" (3190 hits), then tried "moving clocks run fast" (128 hits), several of the latter apparently referring to unorthodox theories and rejections of relativity, although not all of them. Presumably authors who present the slogan "moving clocks run slow" as a verbal equivalent to the time dilation equation are taking the word dilation to mean a bigger number when it refers to time, but a smaller number when it refers to space! On the other hand, Taylor and Wheeler in Spacetime Physics explicitly define dilation as a bigger number, and I think this is the more standard idea.

There is no contradiction between the phrase "time dilation" and the phrase "moving clocks run slow". The "dilation" in question is not of the clock's rate of ticking, but of the period between a given pair of readings. For example, if a clock ticks forward by 10 seconds between two events on its worldline, but the time interval between these two events is 30 seconds in my frame, then that period of 10 seconds between the events as measured by the clock itself has been "dilated" by a factor of 3 from my perspective. But at the same time, if it takes 30 seconds of my time for the clock to tick forward by 10 seconds, obviously I can also say this clock is "running slow" from my perspective.

So if some (most?) of the authors who use the expression "moving clocks run slow" are, in fact, referring to the temporal analogue of length contraction

No, I'm sure that not a single one of them is referring to this, "the temporal analogue of length contraction" is a fairly arcane idea I brought up for the sake of my discussion with neopolitan that would probably never be used in practice. The idea (illustrated in the diagram I drew that neopolitan posted in post #5) is that if you have two events on a clock's worldline that are separated by a time t (say 10 seconds again) according to the clock's own readings, and then you draw surfaces of simultaneity (surfaces of constant t) in the clock's own rest frame that pass through these two events, and then consider how those surfaces would look in the frame of an observer who sees the clock in motion (where the surfaces will be 'slanted'), and work out the time between these surfaces along the vertical time axis of this observer's frame, it will be less than the time of 10 seconds, even though the time in this frame between those two events on the first clock's worldline (which is what the regular time dilation equation gives you) is greater than 10 seconds. This is analogous to length contraction where you look at two lines of constant x in an object's rest frame that represent the worldlines of the object's endpoints, then switch to a different frame where the object is moving so these same lines are slanted, and consider the distance between these slanted lines along the horizontal space axis of this frame, which is the "length" of the object in this frame.

If this is hard to follow even after looking at the diagram, it's not really worth worrying about, since like I said the "temporal analogue of length contraction" is just an artificial concept I came up with for the purposes of showing that you could imagine something analogous to length contraction, it's defined in such a weird way that it's not a concept that anyone would actually be likely to find useful for any other purpose besides illustrating that such an analogous notion is possible.

 

[Rasalhague]

Some notes I made to get my head around the symmetry which the traditional pairing of time dilation and length contraction disguises. All criticism welcome!

Assume a (-1, 1)-dimensional Minkowski spacetime described by two reference frames moving relative to one another with speed u.

Time. Let clocks be synchronised at the intersection of the origins of the two frames so that the coordinates of this coincidence are t_{L} = t_{R} = 0 and x_{L} = x_{R} = 0. The clocks last for all time. Clock L is confined in space to the location (line of collocality/syntopy) x_{L} = 0, clock R to x_{R} = 0.

Length. Let two rulers have their zero ends lined up at the intersection of the origins of the two frames so that the coordinates of this coincidence are t_{L} = t_{R} = 0 and x_{L} = x_{R} = 0. The rulers extend through all space. Ruler L exists only at the instant (line of contemporality/synchrony) t_{L} = 0, ruler R at t_{R} = 0.

We can ask contraction questions of time or of length. We can ask dilation questions of time or of length. The contraction questions we ask of time are formally the same as those we ask of length (except for the difference in coordinate). The dilation questions we ask of time are formally the same as those we ask of length (except for the difference in coordinate).


1. CONTRACTION

\frac{1}{\gamma} = \sqrt[]{1-\left(\frac{u}{c}\right)^{2}} = \frac{1}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}

1.1. Time contraction.

1.1.1. At a moment defined in frame R, the frame where clock L is moving, clock L shows this fraction of the time shown by clock R.

1.1.2. At a moment defined in frame L, the frame where clock L is still, clock R shows this fraction of the time shown by clock L.

1.2. Length contraction.

1.2.1. At a location defined in frame R, the frame where ruler L is moving, ruler L shows this fraction of the length shown by ruler R.

1.2.2. At a location defined in frame L, the frame where ruler L is still, ruler R shows this fraction of the length shown by ruler L.


2. DILATION

\gamma = \frac{1}{\sqrt[]{1-\left(\frac{u}{c}\right)^{2}}} = cosh\left(artanh\left(\frac{u}{c} \right) \right)

2.1. Time dilation.

2.1.1. At a moment defined in frame R, the frame where clock L is moving, clock R shows this multiple of the time shown by clock L.

2.1.2. At a moment defined in frame L, the frame where clock L is still, clock L shows this multiple of the time shown by clock R.

2.2. Length dilation.

2.2.1. At a location defined in frame R, the frame where ruler L is moving, ruler R shows this multiple of the length shown ruler L.

2.2.2. At a location defined in frame L, the frame where ruler L is still, ruler L shows this multiple of the length shown by ruler R.

 

[JesseM]

1. CONTRACTION

\frac{1}{\gamma} = \sqrt[]{1-\left(\frac{u}{c}\right)^{2}} = \frac{1}{cosh\left(artanh\left(\frac{u}{c} \right) \right)}

1.1. Time contraction.

1.1.1. At a moment defined in frame R, the frame where clock L is moving, clock L shows this fraction of the time shown by clock R.

But that's just what is usually called "time dilation"--if some quantity is greater in the frame of the observer who sees the instrument moving than it is when measured by the instrument itself, that's caused dilation; if some quantity is smaller in the observer's frame, that's called contraction. In this case, the quantity is the time between two events on the clock's worldline; the time between these events will be greater in the observer's frame than as measured by the clock itself, so this is time dilation. It seems unnecessarily confusing to change this convention and to say that it's "contraction" if the quantity measured by the instrument is smaller than the corresponding quantity measured in the observer's frame.

 

[Rasalhague]

But that's just what is usually called "time dilation"--if some quantity is greater in the frame of the observer who sees the instrument moving than it is when measured by the instrument itself, that's caused [called] dilation;

Yes, it's called dilation, unless we're talking about length, in which case the same phenomenon is called contraction! Wouldn't it be less confusing to call the same thing contraction for both dimensions?

if some quantity is smaller in the observer's frame, that's called contraction. In this case, the quantity is the time between two events on the clock's worldline; the time between these events will be greater in the observer's frame than as measured by the clock itself, so this is time dilation. It seems unnecessarily confusing to change this convention and to say that it's "contraction" if the quantity measured by the instrument is smaller than the corresponding quantity measured in the observer's frame.

But when we talk about rulers, the terminology is traditionally reversed. There too the distance between the corresponding pair of events is greater in the observer's frame, so why do we not call that dilation?

 

[JesseM]

Yes, it's called dilation, unless we're talking about length, in which case the same phenomenon is called contraction!

How do you figure it's the "same phenomenon?" In the case of clocks, the time measured in our frame between two events on the clock's worldline is greater than the time measured by the clock itself between these two events, so we call it "dilation". In the case of rulers, the distance measured in our frame between the ends of the ruler is smaller than the distance measured by the ruler itself, so we call it "contraction". Seems like consistent terminology to me.

But when we talk about rulers, the terminology is traditionally reversed. There too the distance between the corresponding pair of events is greater in the observer's frame, so why do we not call that dilation?

In the case of length you aren't measuring distance between a single pair of events, you're measuring the distance between the endpoints of the ruler at a single moment in time of whatever frame you're using. The distance between the endpoints of the ruler at a single moment of time in the observer's frame is smaller than the distance between the endpoints of the ruler at a single moment of time in the ruler's own rest frame.

If you want to talk about the distance between a single pair of events in two frames, you're right that the distance is larger in the observer's frame where they're non-simultaneous than it is in the frame where the events were simultaneous (this is what I called the 'spatial analogue for time dilation'). But this is not the same thing as measuring the length of an object in two different frames, since "length" always means the distance between the endpoints at a single moment in time.