What is a metric for uniformly moving frame?

[Nugatory]

the observed state of the distant moon in this example will, and must be the same for both observers

But you have asked for something different: The observed state of the moon now, and thanks to the relativity of simultaneity, "now" for one observer is not the same time as "now" for another observer. They will of course agree about the angle that a particular light signal makes as it strikes the moon's surface and is reflected, but they will find that event happens at different times.

And of course this is after backing out the effects of light travel time. The observer ten light-years distant will correctly conclude that the light left the moon ten years before it reached his eyes, and the observer twenty light-years distant will correctly conclude that the light left the moon twenty years before it reached his eyes... but those will not in general be the same time in their respective frames if they are moving relative to one another.

 

[wil]

This will be exactly the same moment of time - before and after the transformation, and any transformation.

You can apply any convention of time measurement, for example:
t = 80000000 or t' = -6777777 - it is the same moment of time, but in two different time keeping methods.

Any convention of simultaneity can't give insight into future, nor to the past.

 

[Dale]

This will be exactly the same moment of time - before and after the transformation, and any transformation.

You can apply any convention of time measurement, for example:
t = 80000000 or t' = -6777777 - it is the same moment of time, but in two different time keeping methods.

You are missing the point of relativity of simultnaety. Not only are the numbers different for the time assigned to a given event, but also the set of events that occur simultaneously with the given event.

 

[Ken G]

Yes, or put differently, a key lesson of relativity is that it is not "the same moment" because there is no unambiguous global meaning of "the same moment" in the first place. That is the physical basis of the "spacetime" concept-- there is only "here and now", and they come together-- so there is no unambiguous meaning to "there and now."

 

[wil]

No. This is the change of a time coordinate only, means: the numbers which observer uses to label the events, nothing more.
The events, facts alone are always fixed in its original place.

This is probably an ideal analogy to the tensors in GR:
change of coordinates, changes nothing in a geometry, and in physics.

 

[Dale]

Yes, simultaneity is just a labeling and changes nothing in the physics.

The quantities you have been asking about depend on the labeling. That is what it means when I said that they depend on the reference frame.

 

[Ken G]

Right, we are all agreeing that the coordinate choices can't matter-- as long as we choose coordinates that locally correspond to things we can measure (so our coordinates mean something), we can extend them elsewhere any way we like, and equip that extension with instructions for how to convert to the measurements that someone else in that elsewhere place can do (so they still mean something). The concept of simultaneity is thus a purely coordinate construct, we only thought simultaneity meant something more than that when we didn't realize it was all just coordinates.

The problem is, it sounds like wil wants to say, unless I misread, that "things are really simultaneous, and we just can't tell because we have different coordinates". But actually we should say "whether or not we choose to regard events as simultaneous is an arbitrary function of our chosen coordinates, all that physically matters are the invariant causality relationships, so if event A cannot effect B and B cannot effect A, then there is some perfectly valid coordinate system that can regard them as simultaneous without any changes to physics as we know it."

Framed like this, it means that pre-relativity, we used to think simulataneity was a very narrow class of events because we used to think signals could travel at any speed, so either A could affect B or B could affect A for essentially any events A and B except that very narrow class of "truly simultaneous" events. But when signal speeds were found to be limited by c, it means the class of events that cannot have any causal connection between them got much wider, and all of them can equally be regarded as simultaneous, simultaneity is in effect everything outside the light cone.

 

[wil]

The quantities you have been asking about depend on the labeling. That is what it means when I said that they depend on the reference frame.

The problem I asked is really frame independent, like the lunar phases - exactly!

This is a comparison of the speed of local processes in the two particular systems,
this means we have two reference frames only, not more.

You can represent a time measured by my wrist watch using other time unit, but what that changes?

I measure, in some process, 1200000000 the cesium transitions, and thus this is a global fact - in any other frame the number of the transition (of my cesium) must be perfectly the same!

On the Earth lives now, say: 7 billions of humans;
the 'now' means an event defined wrt some objective fact, for example the precise position - coordinates of the solar system in the Galaxy.

The question is: how many humans lives on the Earth, in the same moment of time, wrt other reference frame?

 

[Ken G]

The point is, those 1200000000 cycles are measured by you between two events, A and B, that are happening to you, and so your clock is present at A and B. Anyone else can agree that you did indeed measure 1200000000 cycles between those events, but the problem is, what do they measure as the number of cycles between those events? There is no unambiguous way to answer that question, because events A and B did not happen to them, they happened to you, and their clock was not present at both A and B. So anyone else that wants to say how many cycles they would reckon occurred between A and B has to first find events C and D that do happen to them, and they have to claim that C is simultaneous with A and D is simultaneous with B, and only then can they associate their cycles between C and D with your cycles between A and B. Since there is considerable freedom to match C and D against A and B, while still honoring all the causality connections that physics needs to be able to explain, there is this thing known as the "relativity of simultaneity."

There is no way around this-- there simply is not an unambiguous way to "correctly" match C and D to A and B, such that comparison of numbers of cycles means anything more than just an arbitrary coordinate system. This is also the reason that if one uses the Einstein simultaneity convention to match events C and D to A and B, then both people think the other one will measure fewer cycles between the events in question, as they will not be the same events. You will think the other person has chosen the wrong two events C and D to match up with your A and B-- you will think they should have used events C' and D', and so they should have gotten fewer cycles than 1200000000 between C' and D' when in fact they found more than 1200000000 between C and D.

 

[Dale]

This is a comparison of the speed of local processes in the two particular systems

The clocks were only co-located at one moment. For any other moment they are separated, not local, and therefore any comparison requires a choice of simultaneity convention. Therefore it is frame variant.

As I said several times above, the only frame invariant way to compare the clocks is for them to be co-located at two events. Otherwise they cannot be local for both.

 

[wil]

The problem is, it sounds like wil wants to say, unless I misread, that "things are really simultaneous, and we just can't tell because we have different coordinates". But actually we should say "whether or not we choose to regard events as simultaneous is an arbitrary function of our chosen coordinates, all that physically matters are the invariant causality relationships, so if event A cannot effect B and B cannot effect A, then there is some perfectly valid coordinate system that can regard them as simultaneous without any changes to physics as we know it."

We can use any convention of time measurement, but I'm afraid the convention will be always limited to some subset of reality.

The claim of type: 'if event A cannot effect B and B cannot effect A'
is highly dependent on our knowledge.

We can discovery in the future same C event, which joins indirectly and causally these events: A with B: A -> C -> B, thus the two events will be no longer arbitrarily ordered, means our assumed time convention can lost its applicability.

 

[Nugatory]

I measure, in some process, 1,200,000,000 the cesium transitions, and thus this is a global fact - in any other frame the number of the transition (of my cesium) must be perfectly the same!

You are correct that that fact is frame-independent.

You and your cesium clock are traveling through space-time on some timelike worldline. At some point on that worldline you zero your clock and start it counting. At some later point on that worldline the counter reads 1200000000, you correctly say that 130 milliseconds has passed. Everyone, regardless of frame and relative velocity, will agree that your clock read zero at the first event (you started it running) and read 1200000000 at the second event (you stopped it running). They will also agree that the interval along your worldline between these two events is 130 milliseconds (check my division, please), that's how much you aged between the two events, it's how much time you experienced between them. That's your proper time and it is also frame invariant.

Now consider me, equipped with a similar cesium clock but traveling along a different worldline than you, and moving relative to you. Which event on my worldline is "at the same time" as the event when you started your clock? Which event on my worldline is "at the same time" as the event when you stopped your clock because it had reached 1200000000 cycles? I'll say that two things happened at the same time if they have the same $t$ coordinate, and so will you - but where did these $t$ coordinates come from? They are frame-dependent, and as a result any method for associating points on my world line with yours is also frame-dependent.

You can see this effect even with special relativity in flat spacetime: It's the way that two observers in inertial motion relative to one another can both correctly find that the other one's clock is running slow. Using your coordinates, the endpoints of the measurement interval along your worldline are separated by 1200000000 cycles, but when we identify the two points on my worldline that have the same $t$ coordinates as your endpoints, the interval between them is less than 1200000000. And the exact same thing happens when we start with the endpoints of my 1200000000 cycle interval and find the corresponding points on your worldline using my $t$ coordinates - they correspond to an interval that is different from and shorter than your 1200000000-cycle interval.

 

[wil]

The clocks were only co-located at one moment. For any other moment they are separated, not local, and therefore any comparison requires a choice of simultaneity convention. Therefore it is frame variant.

As I said several times above, the only frame invariant way to compare the clocks is for them to be co-located at two events. Otherwise they cannot be local for both.

The setup is frame invariant:

O----O'
A-----------B <--- v C,

One observer with a clock is at a point O, which is fixed at r = r0 - in the metric.
He measures a distance AB = h, which is small h/r0 -> 0, this is to simplify the problem only, because now we can assume the v ~ const between A and B.

Alternatively the O can be in the middle position of O' - for a better approximation.

Second observer with a clock C falls freely in the gravity, and notices two events:
he is in position of the point A, and next at B - these both are frame independent,
that means these are the facts, like a bomb explosion, or a breaking of glass, due to direct contact with hammer in our hand.

Both clocks measure his own proper times, and we can compare these times;
the result is unambiguously and frame independent.

Try to compute this using GR, and the result will be easily verifiable in the described setup -
we have quite good clocks already, thus there is no problem.
This is an example of a real experiment to verify the GR theory.

 

[Dale]

O is not colocated with B and O' is not colocated with either A or B. Therefore you will have to use some frame dependent simultaneity convention to map the time of A and B to events on the world line of O or O'.

Are you familiar with spacetime diagrams?

 

[wil]

You can see this effect even with special relativity in flat spacetime: It's the way that two observers in inertial motion relative to one another can both correctly find that the other one's clock is running slow. Using your coordinates, the endpoints of the measurement interval along your worldline are separated by 1200000000 cycles, but when we identify the two points on my worldline that have the same $t$ coordinates as your endpoints, the interval between them is less than 1200000000. And the exact same thing happens when we start with the endpoints of my 1200000000 cycle interval and find the corresponding points on your worldline using my $t$ coordinates - they correspond to an interval that is different from and shorter than your 1200000000-cycle interval.

Do not assume the result in advance - just check it experimentally.
In addition: unverifiable theories are useless.

And the last sentence should be the motto of the PF.

Goodbye, for all the pupils.

 

[Dale]

Do not assume the result in advance - just check it experimentally.
In addition: unverifiable theories are useless..

i agree completely.

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

 

[stevendaryl]

i agree completely.

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

Well, there are two kinds of physics discussions: One is about what a theory predicts, and the second is about experimental results. Very often, when there is a more-or-less standard theory that is relevant to a problem, people say: "It's impossible for such-and-such to happen", when they really mean "According to quantum mechanics (or Special Relativity, or the laws of themodynamics, etc.), it's impossible for such-and-such to happen".

Physics newbies often don't realize when people are talking about what the theory predicts, and when they are talking about what experiments show. A basic rule of thumb is that if anyone claims that something is impossible, it's almost always from the standpoint of some standard theory.

 

[Dale]

The setup is frame invariant:

O----O'
A-----------B <--- v C

Here is a spacetime diagram to show why the measurement you are trying to make is not frame invariant and why it depends on the choice of simultaneity convention. As you specified there is a clock C (red worldline) which moves between event A and event B. Meanwhile, observer O (blue worldline) is colocated with C at event A, but because C is moving relative to O they are not colocated at B.

The proper time from A to B on C's worldline is frame invariant. However, B does not occur on O's worldline. To compare the time on C's clock at B to the time at O's clock at B we must somehow map B to some event on O's worldline (black lines), which is called a simultaneity convention.

If we use O's simultaneity convention then we will map B to B? and conclude that O is running faster than C. If we use C's simultaneity convention then we will map B to B?? and conclude that O is running slower than C. If we use some 3rd party's simultaneity convention then we may map B to B? and conclude that O is running at the same rate as C.

 

[wil]

Here is a spacetime diagram to show why the measurement you are trying to make is not frame invariant and why it depends on the choice of simultaneity convention. As you specified there is a clock C (red worldline) which moves between event A and event B. Meanwhile, observer O (blue worldline) is colocated with C at event A, but because C is moving relative to O they are not colocated at B.

The proper time from A to B on C's worldline is frame invariant. However, B does not occur on O's worldline. To compare the time on C's clock at B to the time at O's clock at B we must somehow map B to some event on O's worldline (black lines), which is called a simultaneity convention.

If we use O's simultaneity convention then we will map B to B? and conclude that O is running faster than C. If we use C's simultaneity convention then we will map B to B?? and conclude that O is running slower than C. If we use some 3rd party's simultaneity convention then we may map B to B? and conclude that O is running at the same rate as C.

This is obvious, because this setup is frame invariant, thus it must be simultaneity convention invariant also.

Because this fact the measurement is perfect - the ideal to veryfy any theory.

In that setup we measure a local time duration between two real events, not between two abstract coordinates, which are not a real entities.

 

[Ken G]

That was the point of it-- since as far as we now know, all setups are "frame invariant," relativity asserts that you cannot have a set up that is not frame invariant. Also, all measurements are regarded as "ideal" for verifying theories, why would we intentionally build in flaws into our measurements if we want to test something? Finally, all meaningful time durations are a "local time duration between two real events", since physics is an empirical mode of inquiry so we no not regard abstract durations between unreal events as "measurements."

I think all that is happening here is that you are adopting a stance that one particular frame is the one in which "true simultaneity" is established, and you are basically saying "well, if you just do measurements between real events, you can't tell that this is actually true." It is certainly true that nothing in relativity tells us that you have to be wrong-- relativity does not say that there cannot be one frame where the simultaneity is the true one and all other frames really have time slowed down, indeed that was the original interpretation supplied by Lorentz and Poincare. Einstein's interpretation did not change any of the equations, but it is regarded as a more streamlined interpretation, and is preferred expressly because Einstein removed the extraneous elements and just pointed out that what is "actually true" is only what we can demonstrate to be actually true with real measurements between real events.

Thus, it seems to me that all you are saying is that you don't agree with an interpretation that eliminates "abstract" elements and sticks to what can actually be demonstrated as true. This is the nature of interpretations-- if you are using all the same equations, you are using the same theory, but you are supplying a different justification for the equations, a different language for talking about them. Interpretations are not right and wrong, but they can be the preferred mainstream language as in Einstein's approach, or they can be regarded as clunky, arbitrary, and possibly even missing the key lessons of the theory, as many would say your interpretation is doing. All the same, tomorrow new observations could be made that require correcting relativity in a way that rules out Einstein's interpretation but allows yours, so we always have to bear this in mind. Still, in the absence of any such observations, you really cannot argue that your interpretation is the correct one, for you have no evidence for that, but there is clear evidence that your interpretation involves an arbitrary and unnecessary choice to designate one frame as the one with "true simultaneity", and arbitrary and unnecessary elements are generally disfavored in a good interpretation.

An analogy would be claims that the Earth is at the center of the Big Bang expansion. Any observer is free to make that claim, since that is exactly how things appear when any observer looks outward at an expansion that obeys the Hubble law. But we recognize it is just an arbitrary and unnecessary interpretation, because the nature of the expansion is such that any observer anywhere could equally make that claim, so why would we favor our own point of view? Instead, we should take it as a lesson that any observer could make that claim, and put the claim in perspective as a result. But that doesn't mean the Earth really isn't the center of the expansion, and an observation (say, of primordial gravitational waves or something) could conceivably be done that demonstrates it really is. So even though it could be the only surviving interpretation of some new theory that replaces our current one, it is not a favored interpretation of the theory we have now.

 

[Dale]

This is obvious, because this setup is frame invariant, thus it must be simultaneity convention invariant also.

The mapping between B and events on O's worldline is neither frame invariant nor simultaneity convention invariant.

In that setup we measure a local time duration between two real events, not between two abstract coordinates, which are not a real entities.

I have already shown that this is not true. Event B is not local to observer O.