# Thread: Special relativity - Am I getting it right now?

1. ## Special relativity - Am I getting it right now?

After much help on the ATM forum I think I am seeing the basics fitting together. So I wondered if I might presume to show how it looks to me?

For me it all starts with what I agree is counter intuitive:

The Speed of Light

The speed of light. An innocuous phrase. Stating a seemingly obvious fact that light moves at a particular speed, in the same way that anything that moves has a speed.

Yet what does it mean?

The speed of light in a vacuum is constant; but relative to what?

The speed of light is constant and the same relative to any observer.

If we measure the speed of light between two points in space, without regard to whether those points are moving, then we are measuring how long it takes to travel a fixed distance without regard to any movement of the source nor of the recipient.

Take, for example the light from a star (that lies in the solar plane). As it orbits the sun the Earth is travelling at around 70,000 mph, sometimes towards that start and sometimes away from it. Yet, contrary to all common sense and expectations, the speed of the light from that star is always 'c', the speed of light.

How can that be?

Einstein gave us the answer when he said that time is not absolute. This new way of regarding time supported Einstein's Special Relativity, which showed that all could be resolved once one accepted that the measurement of time could depend on where it was measured from.

Einstein's Light Clock, which he defines as part of his 'thought experiments' serves to show how this works.

The light clock comprises no more than pulses of light sent to a distant mirror and thence reflected back to the light source. The mirror is a fixed distance so that the light returns in a set time, 1 second is commonly used.

Fig 1

But if the clock, in its Frame of Reference, is moving at velocity v, relative to an observer; that observer will see the clock travel the distance vt in the time t that the light takes to reach the mirror.

So, during the time, t, that the light travels to reach the mirror, the clock has travelled an additional distance,vt.

The Greek letter τ (tau) is commonly used to denote Proper Time, that is time measured by an observer on a standard clock that he is travelling with. A clock that he is stationary to.

But is the time τ the same as time, t, for the moving clock?

That is, is the time in the clock's Frame of Reference the same when measured by an observer, relative to whom, the clock is moving?

Fig 1 above, gives us the answer in no uncertain terms, for by the application of Pythagoras it tells us
τ² = (ct)² - (vt)²
τ = ct√ 1 - v²/ c² or
t = γτ

where c = 1
and γ = 1/√ 1 - v²/ c²

γ is also known as the Lorentz factor which is much used in Special (and indeed in General) Relativity

But that is just formulae, which are difficult to picture, so let us put some figures into the scene:

Fig 2.

Part 1 shows the traditional (Galileian) view of how the light pulse would appear for a clock moving at 0.6c.

In the second that the light would travel to the mirror, the clock will have moved 0.6 light seconds along the x axis, combining to create a diagonal path, that the Pythagoras theorem would give a length of: √1 + (0.6)² = √1.36 = 1.166 light seconds. Which is further than the light could travel in 1 second.

However, in Part 2 we can see the Relativistic view and how far the moving light will have travelled, along the diagonal path in that second; and we see it will have reached the point 0.8,0.6 in the coordinates, or Frame of Reference, of the observer for whom the clock is moving.

Yet an observer travelling with that clock would see the light reach the mirror, 1 light second away, in that time, so we know that in that second the light has reached the mirror.

This seeming paradox is what took the genius of Einstein to resolve.

Indeed we can see this in Part 2 of Fig 2. For the y axis which is measuring the distance travelled by light in light seconds can also represent the time in seconds of the stationary observer.

And we see that, while the light in the moving clock has travelled for 1 second, only only 0.8 seconds have passed for the stationary observer.

This means that, when the Moving clock has travelled for one second, it will have travelled 0.6 light seconds relative to the stationary observer; one light second along the diagonal path, in the moving frame of reference of the clock; yet only 0.8 light seconds in 0.8 seconds in the stationary observer's frame of reference.

And this leads us to the inevitable conclusion that the time it takes for the pulse of light to reach the mirror, is measured differently in the two frames of reference. An observer moving with the clock would measure 1 second, while an observer for whom the clock was moving would measure only 0.8 seconds.

So how well does this agree with the formula we deduced earlier?

We said that: t = γτ
and when v = 0.6c, γ = √1 – (0.6)² = √1 – 0.36 = √ 0.64 = 0.8

Therefore: t = 0.8τ
τ = 1.25 t

or 1 second in the clock's Frame of Reference, measured from a moving (relative to the clock) observer, is the same duration as 0.8 seconds, measured by an observer in that Frame of reference.
Last edited by Grimble; 2012-Jun-13 at 08:15 AM. Reason: bigger picture images

2. Yes, if I have read it right, it all sounds good to me. I think you have it.

Take, for example the light from a star (that lies in the solar plane). As it orbits the sun the Earth is travelling at around 70,000 mph, sometimes towards that start and sometimes away from it. Yet, contrary to all common sense and expectations, the speed of the light from that star is always 'c', the speed of light.
This is the only part I would be careful with. Special Relativity involves only inertial frames of reference (moving along a straight line with constant speed), while a gravitational field is non-inertial, which is where General Relativity comes in, although the locally measured speed is always still c, regardless of whether one is moving toward or away from the source, yes. You done good.

EDIT - Just to make things clearer, the Galilean version would be the same as the Relativistic version as measured from the rest frame, the rest frame and moving frame each measuring things as you describe in the first image, but what one frame deduces the other frame measures, based upon a Galilean universe with no time dilation or length contraction, might be similar to what you describe in the second image, right. The difference between what the rest frame and moving frame measure is where SR comes in, by deducing what each must measure of the other's rulers and clocks in order for both inertial frames to measure c for light, in the manner in which you have done, since of course, rulers and clocks are the only tools available to directly measure this speed to begin with.
Last edited by grav; 2012-Jun-05 at 09:35 PM.

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Space and time are inseparable so the c in relativity can be thought of as a simple dimensional constant giving us the amount of time we can expect to find in a given amount of space. Any two events separated by 300,000 km are also separated by one second of time no matter what. We know the value of c to a high degree of accuracy and it is used with great success with such things as GPS and it serves as a universal constant that experimenters anywhere in the world can use to determine the length of a meter or the time of a second as they are defined by international convention. As Herman Bondi said in 1964, so far as he was concerned, the discovery of c was no more than the discovery of the length of the standard meter in standard seconds. Our units of space, time, and c are all mutually defined and observations confirm the correctness of their use in relativity.
Judging from your OP, I would say you get all this but you are asking why our understanding of c goes off the rails when we begin to think of c as both a constant and the speed of light. I say this is a good question. The uses of c as both a constant and a speed ends in paradox and c acts nothing like a speed. Our units of space, time, and c are all mutually defined so we can't even use them to measure the speed of light. Consider the absurdity of trying to measure the speed of light over the distance of a light year.

4. Thank you for an interesting reply.
It certainly raises issues to think about.
It makes me wonder whether measurement is not similar in many ways to relativity. In that it is all relative, when we measure we are comparing against a standard, against a scale, against another definition. Another scientific value that is defined in terms of yet more scientific quantities.
The speed of light then comes to be something more, for if time and space are indeed defined it terms of each other, then the relationship between them becomes a self contained definition where all that can change are the units used.

Or is this all just philosophising?
Last edited by Grimble; 2012-Jun-13 at 07:00 PM. Reason: spelling

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Originally Posted by Grimble
It makes me wonder whether measurement is not similar in many ways to relativity.
We measure a one second time delay between events for every 300,000 km of separation. That is the connection between measurement and relativity.
Originally Posted by Grimble
The speed of light then comes to be something more, for if time and space are indeed defined it terms of each other, then the relationship between them becomes a self contained definition where all that can change are the units used.
That is why all observers measure the value of c the same despite their velocities. Velocity can not change the value of c because c is a self contained constant and not a speed. Any observer who measures a distance simultaneously measures the amount time one can expect to find in that distance or vice versa. We can change the units we use but the ratio of space to time remains a constant c for all observers independent of all velocities. Different observers may measure different amounts of space or time between events relative to their point of view but they will always find the ratio of space to time to be c because c is a dimensional constant and not a speed. When Roemer measured the 'speed of light' in 1676, he was measuring the the varying distances between Earth and Jupiter and the amount of time delay that accompanies every measurement of distance. He was measuring the value of c as a dimensional constant but he was not measuring the speed of light. The speed of light is something other than c and possibly immeasurable.

6. Originally Posted by Bob Angstrom
We measure a one second time delay between events for every 300,000 km of separation. That is the connection between measurement and relativity.
Not that the presence of a particle is necessary, but two events separated by 300,000 km will have one second time delay between them if a massless particle travels directly between them and coincides with both events when they occur. Otherwise, two events with 300,000 km of separation can have any amount of time delay between them, occurring simultaneously for instance.

7. Originally Posted by Bob Angstrom
The speed of light is something other than c and possibly immeasurable.
I don't quite follow this last statement, for as speed is the ratio of distance to time, a 'dimensional constant' as you referred to it, then is that not the definition, the speed of light being a constant and a ratio that is known to a great precision, then how can it be 'other than 'c', be immeasurable?

But my pondering was more about how time and space are defined? If they are only defined in terms of one another, if every sort of measurement is only defined in terms of other measurements, or in terms of physical bodies under set conditions, that are defines in terms of other measurements that include their speed relative to some other specified body or point, then how is anything defined except ratios?

As I said maybe this is really philosophy

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Originally Posted by grav
Not that the presence of a particle is necessary, but two events separated by 300,000 km will have one second time delay between them if a massless particle travels directly between them and coincides with both events when they occur. Otherwise, two events with 300,000 km of separation can have any amount of time delay between them, occurring simultaneously for instance.
I think if we were able to directly observe your “otherwise” condition it would be in violation of special relativity. Can you give an example?

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Originally Posted by Grimble
I don't quite follow this last statement, for as speed is the ratio of distance to time, a 'dimensional constant' as you referred to it, then is that not the definition, the speed of light being a constant and a ratio that is known to a great precision, then how can it be 'other than 'c', be immeasurable?
A ratio of distance to time is simply a ratio. It is not necessarily a speed no matter how precisely it is known and, if it is a dimensional constant rather than a speed, it makes no sense to use it as a speed. That is my answer to your questions in the OP.
A speed requires that there be a something with a vector that we can observe to be speeding.
Originally Posted by Grimble
But my pondering was more about how time and space are defined? If they are only defined in terms of one another, if every sort of measurement is only defined in terms of other measurements, or in terms of physical bodies under set conditions, that are defines in terms of other measurements that include their speed relative to some other specified body or point, then how is anything defined except ratios?
For lack of a better definition, I define space as Nature's way of keeping everything from happening in the same place and time as Nature's way of keeping everything from happening at once.
The constant c is a ratio that lets us use measurements of space and time interchangeably because we know the relationship between them just as we know the ratio between kilometers and miles. We can use a ratio to convert one measurement to another but we can't use a ratio as a speed if it is not a speed. For example, we cant go faster than I.6 kilometers per mile.

10. Originally Posted by Bob Angstrom
I think if we were able to directly observe your “otherwise” condition it would be in violation of special relativity. Can you give an example?
Sure. Like I said, two events occuring at the same time for instance, like the lightening strikes in the train analogy. Events separated by one light-second distance will be separated by exactly one second if a massless particle coincides with both events, because massless particles travel at a speed of exactly c. Or if causally connected somehow, the effect will occur one second or more after the cause, but not less, because the speed of information cannot be greater than c, since any particles that carry information between the events cannot travel faster than c.

11. Originally Posted by grav
Not that the presence of a particle is necessary, but two events separated by 300,000 km will have one second time delay between them if a massless particle travels directly between them and coincides with both events when they occur. Otherwise, two events with 300,000 km of separation can have any amount of time delay between them, occurring simultaneously for instance.
I think we need to clarify this, especially to tie this into Grimble's original thread. First off, with all the talk of 300,000 km, we should state that we're assuming a given Frame of Reference, since the distance between two events is FoR-dependent. With this FoR, then we don't actually need a (massless) particle to make the trip since the events will have coordinates in our FoR, and we can work out everything from those.

12. Originally Posted by RobA
I think we need to clarify this, especially to tie this into Grimble's original thread. First off, with all the talk of 300,000 km, we should state that we're assuming a given Frame of Reference, since the distance between two events is FoR-dependent. With this FoR, then we don't actually need a (massless) particle to make the trip since the events will have coordinates in our FoR, and we can work out everything from those.
Right. I am referring to Bob Angstrom's statement

Originally Posted by Bob Angstrom
Any two events separated by 300,000 km are also separated by one second of time no matter what.
and stated again in post #5, which would be required if and only if a massless particle travelled directly between the two events and coincided with each when they occur. Otherwise, any amount of time can pass between two events in a given FoR where the events are measured to be separated by 300,000 km.

13. (bold mine)
Originally Posted by grav
Otherwise, any amount of time can pass between two events in a given FoR where the events are measured to be separated by 300,000 km.
Sorry, I disagree (or maybe not understand you correctly). In a given FoR, then each event will have a time coordinate within that FoR - so the time difference is determined to be the difference between these coordinates. The actual presence or absence of a particle making the trip adds nothing to this determination.

14. Originally Posted by Bob Angstrom
A ratio of distance to time is simply a ratio. It is not necessarily a speed no matter how precisely it is known and, if it is a dimensional constant rather than a speed, it makes no sense to use it as a speed. That is my answer to your questions in the OP.
I agree. I was saying that the speed of light is merely a ratio, until one gives it units. I was not saying that a ratio was necessarily a speed.
A speed requires that there be a something with a vector that we can observe to be speeding.
"something with a vector that we can observe to be moving relative to something else.
For lack of a better definition, I define space as Nature's way of keeping everything from happening in the same place and time as Nature's way of keeping everything from happening at once.
Yes, one could say that, but I was meaning the way they are defined as quantities.
The constant c is a ratio that lets us use measurements of space and time interchangeably because we know the relationship between them just as we know the ratio between kilometers and miles. We can use a ratio to convert one measurement to another but we can't use a ratio as a speed if it is not a speed. For example, we cant go faster than I.6 kilometers per mile.
I was saying that speed is a ratio, not that a ratio was a speed.
So we are in agreement?

15. Originally Posted by RobA
I think we need to clarify this, especially to tie this into Grimble's original thread. First off, with all the talk of 300,000 km, we should state that we're assuming a given Frame of Reference, since the distance between two events is FoR-dependent. With this FoR, then we don't actually need a (massless) particle to make the trip since the events will have coordinates in our FoR, and we can work out everything from those.
Enlighten me Rob; when you say distance if FoR-dependent is that referring to where they are observed from? For won't distances measured by an observer within that same FoR, be subject to the first Postulate and be the same compared to any IFoR? It is only when measured by a moving observer (or by transforming measurements from the FoR (using the Lorentz equations for that relative velocity), that length contraction will affect that measurement?

16. There has been an awful lot of toing and froing on this thread about the validity and relevance and reliability of the 300,000 km of separation. And all for my benefit?

see below:
Originally Posted by Bob Angstrom
We measure a one second time delay between events for every 300,000 km of separation. That is the connection between measurement and relativity.
Originally Posted by grav
Not that the presence of a particle is necessary, but two events separated by 300,000 km will have one second time delay between them if a massless particle travels directly between them and coincides with both events when they occur. Otherwise, two events with 300,000 km of separation can have any amount of time delay between them, occurring simultaneously for instance.
Originally Posted by Bob Angstrom
I think if we were able to directly observe your “otherwise” condition it would be in violation of special relativity. Can you give an example?
Originally Posted by grav
Sure. Like I said, two events occuring at the same time for instance, like the lightening strikes in the train analogy. Events separated by one light-second distance will be separated by exactly one second if a massless particle coincides with both events, because massless particles travel at a speed of exactly c. Or if causally connected somehow, the effect will occur one second or more after the cause, but not less, because the speed of information cannot be greater than c, since any particles that carry information between the events cannot travel faster than c.
Originally Posted by RobA
I think we need to clarify this, especially to tie this into Grimble's original thread. First off, with all the talk of 300,000 km, we should state that we're assuming a given Frame of Reference, since the distance between two events is FoR-dependent. With this FoR, then we don't actually need a (massless) particle to make the trip since the events will have coordinates in our FoR, and we can work out everything from those.
Originally Posted by grav
and stated again in post #5, which would be required if and only if a massless particle travelled directly between the two events and coincided with each when they occur. Otherwise, any amount of time can pass between two events in a given FoR where the events are measured to be separated by 300,000 km.
Originally Posted by RobA
Sorry, I disagree (or maybe not understand you correctly). In a given FoR, then each event will have a time coordinate within that FoR - so the time difference is determined to be the difference between these coordinates. The actual presence or absence of a particle making the trip adds nothing to this determination.
And yet you are each making the same points, though not seeing that clearly in each other's posts(?)

Let me say what I think you (that is a general 'you') are saying:

A massless particle travelling at 'c', (possibly a photon?) will travel 300,000 km in a second.

If it is present at two events 300,000 km apart then, having taken 1 second to travel between them (at 'c'), we know that they happened, 1 second apart.
Otherwise - (i.e. if we take away the restriction that such a particle was present at both events) - there may be any delay or none between them.

And that a 'particle' real or imagined is not necessary but is only a guide to understanding it.

17. Originally Posted by Grimble
Enlighten me Rob; when you say distance if FoR-dependent is that referring to where they are observed from? For won't distances measured by an observer within that same FoR, be subject to the first Postulate and be the same compared to any IFoR? It is only when measured by a moving observer (or by transforming measurements from the FoR (using the Lorentz equations for that relative velocity), that length contraction will affect that measurement?
Hi Grimble - and first off, well done on getting to grips with all this

Now, remember that in SR, the only Frames of Reference are Inertial Frames of Reference. Yes, all observers within any FoR (ie. by definition, who are stationary relative to each-other) will agree on all measurements of distance and time. For the example we're working with, that's the bunch of people who measure the distance between the events as being 300,000km. Everyone else (ie. those in other Frames of Reference - so by definition, those moving relative to this first bunch) will measure the distance between the events as something other than 300,000km.
.

18. Originally Posted by Grimble
And all for my benefit?
Yep Though we also just like chatting!

Originally Posted by Grimble
And yet you are each making the same points, though not seeing that clearly in each other's posts(?)
A massless particle travelling at 'c', (possibly a photon?) will travel 300,000 km in a second.
True, although there aren't many massless particles other than the photon - really only the gluon (although some claim that there's still a case for some, or even all, neutrinos; personally I don't buy that).

Originally Posted by Grimble
If it is present at two events 300,000 km apart then, having taken 1 second to travel between them (at 'c'), we know that they happened, 1 second apart.
Yes (ish)! It's more accurate to say that if a Frame of Reference measures two events as 300,000km apart, and a massless particle was present at both, then in that FoR the events happened 1 second apart. That doesn't mean that all FoRs will agree that the events happened 1 second apart, though. For example, another FoR might measure the same events as being just 150,000km apart - so that FoR would measure them happening just 1/2 second apart.

Originally Posted by Grimble
Otherwise - (i.e. if we take away the restriction that such a particle was present at both events) ...
Yes, it's the "Otherwise" point I'm not sure about.

19. Originally Posted by RobA
(bold mine)

Sorry, I disagree (or maybe not understand you correctly). In a given FoR, then each event will have a time coordinate within that FoR - so the time difference is determined to be the difference between these coordinates. The actual presence or absence of a particle making the trip adds nothing to this determination.
Right, as you said, the time difference between the two events is determined to be the difference between their coordinates, but it is not "one second no matter what", which is all I am saying.

20. Originally Posted by Grimble
Let me say what I think you (that is a general 'you') are saying:

A massless particle travelling at 'c', (possibly a photon?) will travel 300,000 km in a second.

If it is present at two events 300,000 km apart then, having taken 1 second to travel between them (at 'c'), we know that they happened, 1 second apart.
Otherwise - (i.e. if we take away the restriction that such a particle was present at both events) - there may be any delay or none between them.

And that a 'particle' real or imagined is not necessary but is only a guide to understanding it
Exactly. You have the time dilation down, which is 1/3 of the workings of SR. You can determine the length contraction and the simultaneity difference between inertial frames in much the same way as you did for time dilation.

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Originally Posted by grav
Right, as you said, the time difference between the two events is determined to be the difference between their coordinates, but it is not "one second no matter what", which is all I am saying.
An exception to "one second no matter what" would be contrary to SR. So when do we see an exception?

22. Originally Posted by Bob Angstrom
An exception to "one second no matter what" would be contrary to SR. So when do we see an exception?
Two bombs that are 300,000 km apart explode at the same time, with zero time delay between them. Or bomb A might explode 3 seconds after bomb B, or bomb B might explode 1/10 of a second after bomb A, etc.

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Originally Posted by grav
Two bombs that are 300,000 km apart explode at the same time, with zero time delay between them.
But every observer, no matter where they are or what their direction or velocities may be, will see a time delay of one second for every 300,000 km of separation between themselves and each of the bombs as measured from their individual reference frames. This is why I say there is a one second delay for every 300,000 km of separation matter what. “At the same time” is meaningless when differences in distance are involved and “zero time delay” is only observed with zero distance.

24. Originally Posted by Bob Angstrom
But every observer, no matter where they are or what their direction or velocities may be, will see a time delay of one second for every 300,000 km of separation between themselves and each of the bombs as measured from their individual reference frames. This is why I say there is a one second delay for every 300,000 km of separation matter what. “At the same time” is meaningless when differences in distance are involved and “zero time delay” is only observed with zero distance.
Okay well, that's fine then, although you said between events, but didn't specify the act of observing as one of the events itself. In any case, though, that is true, sight requires light and photons travel at a speed of c, producing a time delay between the actual occurance of an event and its observation of one second over a distance travelled by the photons from an event to an observer of 300,000 km.

25. OK, but if the 300,00 km is the distance light will travel in 1 second, for any observer, in any FoR (that is their own or, time dilated, in another), then how big is a km? How big is a second?

Is there a 'standard' km, a 'standard' second? (Have I missed something?)

I have heard it said that "all seconds are the same" but if so then how could a time dilated second equal one that is not time dilated, how could one time dilated second equal one dilated by a different factor?

My logic shouts out to me that either all seconds are the same and dilated they have different quantities or the seconds have different sizes depending on the dilation?

I know one cannot say it is a Proper km as that depends on circumstance rather than size (or are all Proper times/distances the same?)

Is the non dilated second measured by an observer, within his own FoR, a 'standard', the same size for every FoR (after all, all conditions would then be the same, the same physical laws including the FoR being at rest)?

26. Originally Posted by Grimble
OK, but if the 300,00 km is the distance light will travel in 1 second, for any observer, in any FoR (that is their own or, time dilated, in another), then how big is a km? How big is a second?

Is there a 'standard' km, a 'standard' second? (Have I missed something?)

I have heard it said that "all seconds are the same" but if so then how could a time dilated second equal one that is not time dilated, how could one time dilated second equal one dilated by a different factor?

My logic shouts out to me that either all seconds are the same and dilated they have different quantities or the seconds have different sizes depending on the dilation?

I know one cannot say it is a Proper km as that depends on circumstance rather than size (or are all Proper times/distances the same?)

Is the non dilated second measured by an observer, within his own FoR, a 'standard', the same size for every FoR (after all, all conditions would then be the same, the same physical laws including the FoR being at rest)?
One second passes in the same way and a meter is the same for observers in any frame of reference, the proper times/distances that is, but each frame measures different times and distances between events. You can start to see this even without getting into length contraction just yet, with what you already have in your first diagram. The second part of the diagram shows the mirrors as they are moving. From the perspective of an observer travelling with the mirrors, though, the light has simply travelled directly from one mirror to the other, say 1 light-second distance between the mirrors, the proper distance, which would look the same as in the first part of the diagram. According to the stationary frame watching the mirrors travel away, however, observed the same as in the second part of the diagram, the light has travelled a greater distance than 1 light-second from one mirror to the other.

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Originally Posted by grav
Okay well, that's fine then, although you said between events, but didn't specify the act of observing as one of the events itself. In any case, though, that is true, sight requires light and photons travel at a speed of c, producing a time delay between the actual occurance of an event and its observation of one second over a distance travelled by the photons from an event to an observer of 300,000 km.
When I asked, “When do we “see” an exception.” I was looking for an observation. We know from SR that we observe a one second time delay for every 300,000 km of separation no matter what. And if we adhere to what is observed, photons are not a part of the observations. With EM related events, we see electrons gaining and losing energy. We do not see anything moving with a speed between source and sink. By, "No matter what." I mean c may be simply a dimensional constant and having nothing to do with the speed of light or photons.

28. Originally Posted by grav
Right, as you said, the time difference between the two events is determined to be the difference between their coordinates, but it is not "one second no matter what", which is all I am saying.
Ah, OK - looking back, it's all a matter of what the word "Otherwise" is covering. Sorry - I read that as "if a particle makes the trip … Otherwise", and reading into that that we may be leading into a discussion of Space-like separations. However, I take it you were simply meaning another (pair of) events and/or another FoR. Yes, certainly, two events that have no causal relationship may be measured by a FoR as being 300,000km apart (if spacetime distance allows that amount!), and can be measured as happening at any time difference - maybe one at 3pm and the other at 4pm, or indeed both at 5pm.

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Originally Posted by Grimble
I have heard it said that "all seconds are the same" but if so then how could a time dilated second equal one that is not time dilated, how could one time dilated second equal one dilated by a different factor?
There are universally established conventions for determining standards of length and time but, since space and time can vary from one reference frame to another, a second in one reference frame may vary relative to a second in another reference frame. A fellow named Jay Cole explained how units of length and time are determined and answered questions similar to yours in another forum that is no longer on line. I liked his answers so I saved them because he explained some difficult issues better than I could. His explanations are slightly out of date from current standard methods but the principles remain the same. Note Minkowski's definition of c as the speed of the metric. The quote below is from Jay Cole.

“Suppose we adopt as our unit of time the period of a particular cesium transition and arbitrarily assign the value of the speed of light to be 30 cm/nsec. By placing a mirror at a distance from a light source and adjusting its position until the time it takes for a pulse of light to be reflected back is 2 nsec, we can then specify its distance as 30 cm: thereby defining the value of the meter.
But any other lab that adopts the same assignment for the speed of light, will arrive at the same value for the meter, since the speed of light is a universal constant.
What this shows is that the speed of light c and the units of time and distance are not independent: A choice of any two of the values, already determines the third. One consequence is that c cannot be determined by experiment - inasmuch as a measurement of c can only return the value already agreed upon. To get around confusions that might arise, the value of c has been set (CODATA 1983), once and for all, at 299,792,458 m/sec – exactly, so that data already on the books may be retained.
In order that this result hold for any kind of experiment, the "proper times” as measured by any (accurate) clock must pass at the same intrinsic rate, whatever its state of motion - a conclusion that was pointed out by Einstein's teacher, Minkowski. That is, c is not properly the speed of light but "the speed of the metric" - or rather, the speed of time. All the theoretical results of relativity then follow.
This clears up several misunderstandings that early presentations of relativity used to make. That is, a moving body does not become more massive, or acquire added momentum, or change size, when it passes by a "stationary" observer: It just "passes its time more slowly", making it appear "sluggish" and scrambling up the data we measure along its direction of motion.
But of course, if c is the speed of time, then time is not merely something we can read from a clock, but a physical flow of some kind. The Romans apparently had the right idea, when they proclaimed, Tempus Fugit.”- Jay Cole

30. Originally Posted by Grimble
OK, but if the 300,00 km is the distance light will travel in 1 second, for any observer, in any FoR (that is their own or, time dilated, in another), then how big is a km? How big is a second?
Is there a 'standard' km, a 'standard' second? (Have I missed something?)
Sorry, this has got a bit confusing, hasn't it ! No, you haven't missed anything, and all units like km and seconds keep their normal everyday values and meanings. I agree, I think we've all just got a bad case of crossed wires, caused ultimately by English being such a lousy language to talk techie in. That's why I was being a bit pedantic about terms in the other thread.

Originally Posted by Grimble
I have heard it said that "all seconds are the same" but if so then how could a time dilated second equal one that is not time dilated, how could one time dilated second equal one dilated by a different factor?
All seconds are the same - for observers in the same FoR. That, of course, also means that all km are the same for those observers (since it's simply how far light travels in that second), and also all laws of physics operate perfectly consistently. The fact that other observers in other FoRs measure their seconds differently is irrelevant to our definitions

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