Do you think Relativity theory can explain all the paradoxes?

[Jonathan Scott]

However, as you said, the emitted proton is simultaneous with 2018 on the earth, but the 4LY long delay is a problem. So the proton won't record the information from 2018 is that what you are talking about?

All observers at a given location see the same images or signals arriving regardless of their velocity (although the velocity can of course have an effect on the received signal, such as Doppler shift or aberration).

Then we can use a wormhole between Earth and x so that light takes only a short distance and our proton will record the future of earth. Isn't that okay?

Even if such a wormhole were possible, the view through the wormhole would simply be as if Earth were that much closer (so it would reflect a more recent past), and would be a different view showing a different time from that seen without the wormhole.

 

[stevendaryl]

I'll give you another thought experiment, Instead of proton, use Jack and instead of Earth use Jill, both are 20 years old and are 4LY away. If Jack moves towards Jill (Please remove Acceleration), how old will be Jack when he reaches Jill?
Okay, just answer what will be their age when they both meet together?

The answer to the question "How old is Jill when Jack is 20?" is frame-dependent. If the answer is "20" in Jill's frame, then the answer will be "more than 20" in Jack's frame.

Let's look at it from both points of view:

Let $e_1$ = the event where Jack is 20.
Let $e_2$ = the event where Jill is 20.
Let $e_3$ = the event where Jack reaches Jill.

In Jill's frame, $e_1$ and $e_2$ are simultaneous, and Jack ages slower than Jill between the times of events $e_2$ and $e_3$. So Jill concludes that Jack is younger than she is when they get together.

In Jack's frame, $e_1$ takes place before $e_2$. So at the time of $e_2$, Jill is already older than Jack. Jill ages slower than Jack between the times of $e_2$ and $e_3$, but not enough slower to make up for Jill's "head start". So Jack concludes that he is younger than Jill when they get together.

To make it concrete, let's suppose Jack is traveling at 0.866 c. Then in Jill's frame:

1. When Jack is 20, Jill is 20.
2. It takes Jack 4.62 years to get to Jill.
3. Jack ages 2.31 years.
4. Jill ages 4.62 years.
5. Jack is 22.31 when he gets to Jill
6. Jill is 24.62 when Jack gets to her.

In Jack's frame,

1. When Jack is 20, Jill is 23.47
2. It takes Jack 2.31 years to get to Jill.
3. Jack ages 2.31 years.
4. Jill ages 1.15 years.
5. Jack is 22.31 when he gets to Jill
6. Jill is 24.62 when Jack gets to her.

 

[PeterDonis]

When the proton is launched in 2014 (now the frame of the proton is simultaneous with 2018 of Earth time)
Our imaginary proton has an inbuilt video recorder, so when the proton is just a few kilometres up in the sky of Planet x, they stops the proton with a device. So that when they check the video recording in it they can see the future of Earth in 2018! Right??

No. The proton's video recorder doesn't record events on Earth that are simultaneous with the current event on the proton; it records light coming to the proton from Earth, which contains time-delayed images of events on Earth. (This is another reason why "simultaneity" is not really a physically meaningful concept: information about distant events doesn't travel instantaneously.)

The light coming from Earth and reaching Planet X in 2014 (Planet X time) is from Earth 2010 (Earth time). The same light is hitting the proton just after it launches; the fact that the proton's speed has changed (so its "rest frame" has changed) doesn't change what light is reaching it from Earth. (It does change the frequency of the light: the Earth images will be highly blueshifted as seen by the proton's video recorder, so the recorder will have to be able to record very hard gamma rays instead of visible light.)

 

[Nugatory]

However, as you said, the emitted proton is simultaneous with 2018 on the earth, but the 4LY long delay is a problem. So the proton won't record the information from 2018 is that what you are talking about?
Then we can use a wormhole between Earth and x so that light takes only a short distance and our proton will record the future of earth. Isn't that okay?

Depends on what you mean by "okay"...

If wormholes exist (not at all likely, although as with unicorns and flying pigs we cannot prove that they don't exist) and if they behave the way you want them to (certainly possible, because we're making this up as we go) then yes, light from events on Earth in 2014 could reach planet X well before 2018, where the dates are all according to the synchronized clocks on Earth and planet X. That still doesn't tell you anything about anything that happens after 2014 until it's happened - we can't record light that hasn't been emitted.

(Yes, with a bit more work you will be able to find all sorts of strange and wonderful paradoxes if you assume that wormholes exist and behave the way you want them to. However, these paradoxes aren't going to take you far in thread entitled "Do you think relativity theory explains all the paradoxes?". It takes only a modest amount of slicing with Occam's razor to see that problem isn't with relativity theory, it's with what you're assuming about the behavior of the things that you're calling "wormholes".)

 

[GrayGhost]

No. The proton's video recorder doesn't record events on Earth that are simultaneous with the current event on the proton; it records light coming to the proton from Earth, which contains time-delayed images of events on Earth. (This is another reason why "simultaneity" is not really a physically meaningful concept: information about distant events doesn't travel instantaneously.)

The light coming from Earth and reaching Planet X in 2014 (Planet X time) is from Earth 2010 (Earth time). The same light is hitting the proton just after it launches; the fact that the proton's speed has changed (so its "rest frame" has changed) doesn't change what light is reaching it from Earth. (It does change the frequency of the light: the Earth images will be highly blueshifted as seen by the proton's video recorder, so the recorder will have to be able to record very hard gamma rays instead of visible light.)

PeterDonis,

Your summaries are very good here. Trojan666ru need only realize the difference between "as the remotely located Jill presently exists in Jack's spacetime system" versus "as per EM images of Jill received by Jack at Jack's own location". I could explain this via Minkowski figures, although the dynamic acceleration makes it a tad bit tedious.

Regarding your highlighted statement above ... here are my comments:

1) coordinates are not aprior in nature.

2) During Jack's own proper acceleration ... Jill's digression "backwards in time per Jack" in Jack's spacetime system, and Jill's clock ticking faster than Jack's own clock per Jack (a deduction also requiring doppler shift considerations), are not the result of any energy expenditure on Jill's part, nor any energy being imparted unto Jill. It's all due only to the change in Jack's own sense-of-simultaneity (sense-of-NOW) as the result of his own frame transitioning during his own proper acceleration. IOW, it's all due to dynamic changes in Jack's own POV due to changes in his own state of motion.

Was there any other reason in your mind, for the highlighted statement above?

The reason I mention this, is because while all the above holds true ... it is also true that Jill indeed digresses backwards in Jack's time per Jack. This cannot be an illusionary effect of sort, assuming we accept that Einstein's 1-way light speed is in fact c (which seems to be unprovable). While coordinates are man made, their relations are governed by physics. I mean, to say simultaneity is not a meaningful physical concept, is to say that measurable relativistic effects are not meaningful physical concepts, yes? It seems to me that "relative angular orientation differentials between spacetime systems" in the continuum be physically meaningful, even though (1) coordinates are apriori in nature, and (2) dynamic re-shifting of remote observers are due strictly to the observer's own change in POV due to his own frame transitioning.

Thank You,
GrayGhost

 

[Dale]

it is also true that Jill indeed digresses backwards in Jack's time per Jack.

No, this is not true. There is no coordinate system where this is correct, regardless of acceleration.

 

[PAllen]

The reason I mention this, is because while all the above holds true ... it is also true that Jill indeed digresses backwards in Jack's time per Jack. This cannot be an illusionary effect of sort, assuming we accept that Einstein's 1-way light speed is in fact c (which seems to be unprovable). While coordinates are man made, their relations are governed by physics. I mean, to say simultaneity is not a meaningful physical concept, is to say that relativistic effects are not meaningful physical concepts, yes? It seems to me that "relative angular orientation differentials between spacetime systems" in the continuum be physically meaningful, even though (1) coordinates are apriori in nature, and (2) dynamic re-shifting of remote observers are due strictly to the observer's own change in POV due to his own frame transitioning.

Thank You,
GrayGhost

Adopting one way speed of light postulate in no way fixes these conclusions. To wit, I adopt it, I analyze SR and GR geometrically with a concept of complete 'existence' of 4-d manifold, yet I disagree that these assumptions force this and other conclusions. As Jack changes motion, Jack has many possible physically motivated notions of simultaneity. Just a few are:
1) Set up a family of observers which appear stationary with respect to me at all times. Then common clock readings for this family of observers defines simultaneity.
2) Use Einstein's simultaneity convention with assumption of light speed isotropy (even for accelerated motion).
3) Re-interpret the whole of existence based on what would be measured by in a physically realized inertial coordinate system that was always moving the way I am at this moment. Do so, whenever my motion changes. Insist that this is reality.

Done right, none of these lead to wrong predictions. However I, and many here, reject the idea that (3) represents 'reality', or is in any way preferable to the others.

 

[GrayGhost]

it is also true that Jill indeed digresses backwards in Jack's time per Jack.

No, this is not true. There is no coordinate system where this is correct, regardless of acceleration.

DaleSpam,

I should have said the above this way ...

The event marked by Jill's remote location "when Bob first initiates his proper acceleration", digresses backward in time per Jack's spacetime system during his rapid proper acceleration.

Indeed, Jill is always in the present per Jack, which is why Jill's clock ticks faster than Jack's (as it exists in Jack's system") during Jack's proper acceleration.

That said, consider my prior post with that verbal enhancement, which is what I actually had intended.

Adopting one way speed of light postulate in no way fixes these conclusions. To wit, I adopt it, I analyze SR and GR geometrically with a concept of complete 'existence' of 4-d manifold, yet I disagree that these assumptions force this and other conclusions. As Jack changes motion, Jack has many possible physically motivated notions of simultaneity. Just a few are:

1) Set up a family of observers which appear stationary with respect to me at all times. Then common clock readings for this family of observers defines simultaneity.

2) Use Einstein's simultaneity convention with assumption of light speed isotropy (even for accelerated motion).

3) Re-interpret the whole of existence based on what would be measured by in a physically realized inertial coordinate system that was always moving the way I am at this moment. Do so, whenever my motion changes. Insist that this is reality.

Done right, none of these lead to wrong predictions. However I, and many here, reject the idea that (3) represents 'reality', or is in any way preferable to the others.

PAllen,

Well, I do understand your point. However, option 3 would allow the non-inertial POV to agree with all inertial POVs on the readout of clocks at any event, because the non-inertial POV is actually defined by the assumption that he may drop into inertial motion at his present location thus becoming an inertial POV. Of course, if he does not drop into an inertial motion, light signals cannot cannot validate option #3 because POE events are dynamically changing in their location (per he) during his proper acceleration.

It seems to me that #1 or #2 could not agree with inertial POVs on clocks at events, even though such conventions will produce solns. I would think that the goal is to have all POVs (inertial and non-inertial) agree on the readout of clocks at any and all events, and not just agreement per those of a particular non-inertial frame. Are you saying that #1 and #2 produce solns compatible with always inertial POVs?

Thank You,
GrayGhost

 

[PAllen]

PAllen,

Well, I do understand your point. However, option 3 would allow the non-inertial POV to agree with all inertial POVs on the readout of clocks at any event, because the non-inertial POV is actually defined by the assumption that he may drop into inertial motion at his present location thus becoming an inertial POV. Of course, if he does not drop into an inertial motion, light signals cannot cannot validate option #3 because POE events are dynamically changing in their location (per he) during his proper acceleration.

Who cares whether an accelerating observer suddenly stopping accelerating agrees on the clock readings for all events in the universe with someone whose past is completely different. I'm not saying you can't do that, but I don't consider it plausible, desirable, or in any way implied by the postulates of relativity.

It seems to me that #1 or #2 could not agree with inertial POVs on clocks at events, even though such conventions will produce solns. I would think that the goal is to have all POVs (inertial and non-inertial) agree on the readout of clocks at any and all events, and not just agreement per those of a particular non-inertial frame. Are you saying that #1 and #2 produce solns compatible with always inertial POVs?

Methods #1 and #2 are far more standard than the #3 in both SR and GR. Method #1, with a few more technical assumptions, becomes Fermi-Normal coordinates which are typically considered the most physically meaningful definition of an accelerated frame in both SR and GR. Method #3 is never used in GR (and can't be, because there is no such thing as a global inertial frame), yet SR is supposed to be special case of GR.

Einstein never once in his life used method #3 but had no problem getting tricky predictions of SR correct.

 

[PeterDonis]

It's all due only to the change in Jack's own sense-of-simultaneity (sense-of-NOW) as the result of his own frame transitioning during his own proper acceleration.

But the implicit "sense-of-simultaneity" you are using here is only one of many possible choices--and as DaleSpam pointed out, this choice doesn't even work as a coordinate chart (because it ends up assigning multiple 4-tuples of coordinate values to the same spacetime event--a coordinate chart must be a one-to-one assignment of 4-tuples to events).

to say simultaneity is not a meaningful physical concept, is to say that measurable relativistic effects are not meaningful physical concepts, yes?

No. If you think the answer is yes, then please point me to the measurable relativistic effects that directly correspond to "simultaneity". For example: what direct observable that Jack can measure corresponds to his "sense of simultaneity" changing with regard to Jill (i.e., to the large change in which event on Jill's worldline is simultaneous with the present event on Jack's worldline)?

 

[Dale]

The event marked by Jill's remote location "when Bob first initiates his proper acceleration", digresses backward in time per Jack's spacetime system during his rapid proper acceleration.

But this is exactly what cannot happen in any coordinate system. A coordinate chart is mathematically required, by definition, to be one-to-one. This means that a coordinate chart can only map one event in spacetime to any point in the coordinate space, and can only map one point in the coordinate space to any event in spacetime.

The event marked by X cannot digress backward in time because that would map multiple points in the coordinate space to a single event. This would violate the one-to-one requirement of coordinate charts, so a mapping that does that cannot be considered to be a coordinate chart.

 

[GrayGhost]

But this is exactly what cannot happen in any coordinate system. A coordinate chart is mathematically required, by definition, to be one-to-one. This means that a coordinate chart can only map one event in spacetime to any point in the coordinate space, and can only map one point in the coordinate space to any event in spacetime.

The event marked by X cannot digress backward in time because that would map multiple points in the coordinate space to a single event. This would violate the one-to-one requirement of coordinate charts, so a mapping that does that cannot be considered to be a coordinate chart.

Dalespam,

But multiple points in 1 spacetime system do not map to any single event, wrt option 3 ...

The convention of option 3 must allow for events to relocate per the non-inertial POV (including backward in time), which is something that can never happen per always-inertial POVs. So, that's something that must be allowed to exist, if option 3 is considered "as possible". Per option 3 and in any instant of time considered, there is always only a 1:1 coordinate relation between both spacetime systems for any single event, even though the values of that event's coordinates dynamically change in (only) the non-inertial POV. Yes?

My position, is that a coordinate chart need not require an event to be fixed to a single numeric spacetime coordinate per the non-inertial POV. I agree there must be a 1:1 coordinate relation between both spacetime systems for any single event in any instant of time. So, option 3 requires that events move in the non-inertial POV, no matter what the current defintion of coordinate charts requires.

Thank You,
GrayGhost

 

[WannabeNewton]

My position, is that a coordinate chart need not require an event to be fixed to a single numeric spacetime coordinate per the non-inertial POV.

This is complete nonsense. Events don't move.

 

[PAllen]

My position, is that a coordinate chart need not require an event to be fixed to a single numeric spacetime coordinate per the non-inertial POV. I agree there must be a 1:1 coordinate relation between both spacetime systems for any single event in any instant of time. So, option 3 requires that events move in the non-inertial POV, no matter what the current defintion of coordinate charts requires.

Thank You,
GrayGhost

Since I threw out "option 3" as my best understanding of your (and some others) approach, I wouldn't try to redefine coordinate charts. I would say using option 3 means you change coordinate charts every time your motion changes. At any change of motion, you translate your prior chart to a new chart. Personally, I don't see any value to this approach, nor do I think it provides any insight at all about "reality", but (so far) I don't see any reason that it is not a logically consistent approach that can be use to make correct predictions.

 

[Dale]

The convention of option 3 must allow for events to relocate per the non-inertial POV (including backward in time), which is something that can never happen per always-inertial POVs. So, that's something that must be allowed to exist, if option 3 is considered "as possible".

Which is why option 3 is not possible as a valid non-inertial chart.

My position, is that a coordinate chart need not require an event to be fixed to a single numeric spacetime coordinate per the non-inertial POV.

Your position is not in keeping with mainstream scientific practice. See for example chapter 2 here:
http://arxiv.org/abs/gr-qc/9712019

If you take your approach then you lose much of the basic mathematical structure that you need. You can no longer do coordinate transformations, find neighborhoods of coordinates, differentiate wrt the coordinates, etc. That is why it is a bad idea and is expressly contrary to the mainstream approach. You gain no benefit from it, and the price is way too steep, particularly since almost all physics requires differentiation.

 

[ghwellsjr]

From this long debate what i understood is only one thing.
The proton hits the Earth AFTER THE DEATH OF ROBERT from Earth's rest frame. That is sure & the proton ages only a few days
But when you are considering the rest frame of proton, you are varying time of the Earth inorder to solve the paradox.

How is that possible?
If the proton is at rest in it's frame and rest of the universe is moving, then how come the time on Earth suddenly jumps into 2018 from 2014?
I need to know what kind of math/relativistic principle is used here?

As has been pointed out, there is no standard way to define how the proton can be "at rest in its frame" but my favorite way is to use the radar method which works identically to the standard Inertial Reference Frames (IRFs) and yet also works seamlessly with non inertial reference frames of the type you presented.

You mentioned in your OP that:

Both the planets are synchronised in time by sending light pulses so that they can keep 2013 on both planets.

It's important that both planets send signals in order for the proton to determine the correct synchronization. So let's see how this works. Let's assume that Planet X and the proton at some point in time don't know the time on earth. They send a message at the speed of light to the Earth requesting them to immediately send a message back with the current time on the earth. The proton (and Planet X) start their stopwatch when the message was sent to Earth and stop their stopwatch when they receive the response from earth. Then they take half of the measured time and assume that Earth sent the response at that time and they go back and re-assign that time to their own time at the half-way point. They also assume that Earth was that same number of light-years away at the time it sent the response. Here is a spacetime diagram to show how they do this using the same scenario (0.97c and 4 ly) as I did before:

Note that the proton sent its request when its stopwatch read 0 and it receives the response from Earth when its stopwatch read 8 years so it assumes that Earth received its message and sent its response when its stopwatch was at 4 years and since the message from Earth said they were at the year 2008, the proton assigns 2008 to the point where its stopwatch was at 4 years. The proton also assigns all the other years before and after accordingly.

In the meantime, the proton has sent another message one year after the first one and similarly confirms that Earth's year 2009 matches its year 2009 (since it's halfway between 2005 and 2013). Likewise, the proton concludes that the Earth remains 4 light-years away:

The proton continues to send a new request to the Earth every year and continues to keep track of sent/received times so that it can construct a non IRF diagram:

Here is a complete list of the information the proton has gathered:

Message   Response   Calculated   Calculated    Earth's
Sent      Received   Time         Distance      Time
 2004     2012         2008          4           2008
 2005     2013         2009          4           2009
 2006     2014         2010          4           2010
 2007     2014.123     2010.56       3.56        2011
 2008     2014.247     2011.12       3.12        2012
 2009     2014.370     2011.68       2.68        2013
 2010     2014.493     2012.25       2.25        2014
 2011     2014.616     2012.81       1.81        2015
 2012     2014.740     2013.37       1.37        2016
 2013     2014.863     2013.93       0.93        2017
 2014     2014.986     2014.49       0.49        2018

Now it's a simple matter for the proton to make a spacetime diagram for its non-inertial reference frame. Note that I'm drawing the position of the Earth to the left of the proton just to maintain similarity to previous diagrams:

And just as a sanity check, we can show how all the signals that the proton sent and the responses that the Earth sent are just as valid in this non-inertial frame as they are in the inertial frame:

As you can see, there's no jumping of any times (clocks) and no times (clocks) going backwards. The math is simple, averaging two measurements and dividing the difference by two. The relativistic principle is that the proton assumes that the signals that it sends to the Earth take the same amount of time as the responses the Earth sends back. Very simple, isn't it?

 

[stevendaryl]

Since I threw out "option 3" as my best understanding of your (and some others) approach, I wouldn't try to redefine coordinate charts. I would say using option 3 means you change coordinate charts every time your motion changes. At any change of motion, you translate your prior chart to a new chart. Personally, I don't see any value to this approach, nor do I think it provides any insight at all about "reality", but (so far) I don't see any reason that it is not a logically consistent approach that can be use to make correct predictions.

I think that students are misled in introductions to Special Relativity with all the talk about observers. It's not really correct that things (like simultaneity, velocity, length, etc.) are relative to an observer; they are relative to a coordinate system. The connection with observers is just that if the observer is inertial, then there is a natural, most convenient, coordinate system for that observer. If the observer is non-inertial, then the notion of things being "relative to the observer" has almost no value at all.

With Galilean relativity, it makes sense to ask: "How old is Jill when Jack is 20?" Students learn that simultaneity is relative, so the question doesn't have a unique answer. They know that the answer can be different depending on whose point of view you're considering. But if an observer is noninertial, then the question "How old is Jill when Jack is 20, according to that observer?" still doesn't have a unique answer.

You can stipulate that "Jill's age, relative to Jack" means "Jill's age, relative to an inertial coordinate system in which Jack is momentarily at rest". This notion of relative age is highly artificial and has really weird properties. If Jill is far away from Jack, then Jill can age rapidly in a short amount of time, or can even "youthen", depending on Jack's acceleration.

 

[ghwellsjr]

I'll give you another thought experiment, Instead of proton, use Jack and instead of Earth use Jill, both are 20 years old and are 4LY away. If Jack moves towards Jill (Please remove Acceleration), how old will be Jack when he reaches Jill?
Okay, just answer what will be their age when they both meet together?

You already answered your own question in your first post:

When it is 2014 on planet x (on Earth too) they eject the proton towards the earth.
Just imagine that this proton has an 'inbuilt clock' in it (just imagine).
As soon as the proton leaves the accelerator, a stopwatch which which is attached to it activates.
Scientists on Earth expect the proton reaching the Earth on 2018 + some months.
Unfortunately a scientist (Robert) on Earth dies in 2017 and proton reaches the Earth on 2018.
The proton will have only aged an year or so.

So let's substitute Jack for the proton and put Jill on the earth:

When it is 2014 on planet x (and on Earth where Jill is at the age of 20) they eject Jack (who is also 20) towards the earth.
Just imagine that Jack has a clock with him (just imagine).
As soon as Jack leaves the accelerator, a stopwatch which is attached to him activates.
Scientists on Earth expect Jack reaching the Earth on 2018 + some months when Jill will be 24 + some months.
Unfortunately a scientist (Robert) on Earth dies in 2017 and Jack reaches the Earth on 2018.
Jack will have only aged a year or so, so he will be 21.

I'm curious: why did you ask a question that you already provided all the information needed to answer it yourself?

 

[GrayGhost]

Which is why option 3 is not possible as a valid non-inertial chart.

Your position is not in keeping with mainstream scientific practice. See for example chapter 2 here:
http://arxiv.org/abs/gr-qc/9712019

If you take your approach then you lose much of the basic mathematical structure that you need. You can no longer do coordinate transformations, find neighborhoods of coordinates, differentiate wrt the coordinates, etc. That is why it is a bad idea and is expressly contrary to the mainstream approach. You gain no benefit from it, and the price is way too steep, particularly since almost all physics requires differentiation.

Firstly, happy Turkey Day to all.

On another matter, let's say one uses radar time to determine the velocity of the target, using τ1 = ½( τ0+ τ2). Would not the non-inertial POV (of relative relativistic rate) often record superluminal motions?

Thank You,
GrayGhost

 

[WannabeNewton]

On another matter, let's say one uses radar time to determine the velocity of the target, using τ1 = ½( τ0+ τ2). Would not the non-inertial POV (of relative relativistic rate) often record superluminal motions?

It is possible for the coordinate speed of light to deviate from $c$ in a non-inertial frame and it is possible for the coordinate speed of time-like particles to reach $c$ but these quantities do not correspond to physical observables! The speed of light as measured at a given event using the rulers and clock of a local Lorentz frame comoving with the non-inertial frame at said event will always be $c$ and that of a time-like particle will always be less than $c$.

Let an observer $O$ have 4-velocity $\xi^{\mu}$ at an event $p$ in space-time. A time-like particle with 4-velocity $\eta^{\mu}$ passes by at $p$ and $O$ makes a measurement of it's speed. Well speed is change in spatial distance (according to $O$) divided by change in time (according to $O$) so consider a future event $q$ infinitesimally close to $p$ that is on the worldline of this time-like particle. How do we measure the spatial distance between these two events? Well we project the vector pointing from $p$ to $q$ onto the local simultaneity slice of $O$ at $p$ and take its length. Similarly, to find the time between $p$ and $q$ we project this vector onto the time-axis of $O$ and take its length. The vector pointing from $p$ to $q$ is just $\eta^{\mu}$.

Mathematically, projecting onto the local simultaneity slice of $O$ at $p$ just means projecting orthogonally relative to $\xi^{\mu}$ at $p$ which (from elementary linear algebra) is $\eta^{\mu} + \xi^{\mu}\xi_{\nu}\eta^{\nu}$. Similarly, projecting onto the time-axis of $O$ just means projecting onto $\xi^{\mu}$ i.e. $-(\xi_{\nu}\eta^{\nu})\xi^{\mu}$. The respective lengths are then $[ (\xi_{\mu}\eta^{\mu})^2 -1]^{1/2}$ and $\xi_{\mu}\eta^{\mu}$ so the speed of the time-like particle relative to $O$ is $v = \frac{[(\xi_{\mu}\eta^{\mu})^2 -1]^{1/2}}{\xi_{\mu}\eta^{\mu}} < 1$ in units where $c = 1$. It can similarly be shown that for light, the change in spatial distance divided by change in time (all relative to $O$ as usual) is always $c = 1$.

 

[Nugatory]

folks argued that SR was invalid and not mainstream because the euclidean metric did not relate 2 frames of relative motion, a new (Minkowski) metric did.

Which folks, and when?

 

[Dale]

let's say one uses radar time to determine the velocity of the target, using τ1 = ½( τ0+ τ2). Would not the non-inertial POV (of relative relativistic rate) often record superluminal motions?

No, I don't think that it ever would. But as WBN pointed out the v<c limitation applies only to inertial coordinates anyway.

 

[ghwellsjr]

On another matter, let's say one uses radar time to determine the velocity of the target, using τ1 = ½( τ0+τ2). Would not the non-inertial POV (of relative relativistic rate) often record superluminal motions?

First off, I don't know how you can determine the velocity of a remote target by using just τ1 = ½(τ0+ τ2). You also need to use x1 = ½(τ2-τ0), using compatible units such as years and light-years. And of course, you need to make several measurements and repeat the calculations to track the position of the remote target as a function of the local time and from that, the velocity in the non-inertial frame can be calculated using the radar method.

For example, in post #51, I went into all the details about how the proton can construct its non-inertial rest frame using the radar method and it determines that the Earth approaches it at a speed of 0.8c even though in the Earth's rest frame, the proton is traveling at 0.97c.

You will also note that light continues to travel at c in the non-inertial rest frame just like in the IRF's (because we define it to do that). In fact, You can take the information contained in the proton's non-IRF and re-construct the Earth's IRF.

So as far as I can tell, there are never any superluminal motions of massive objects and light always propagates at c when using the radar method because we are still using Einstein's second postulate.

Does this make sense to you?

 

[ghwellsjr]

I need to know what kind of math/relativistic principle is used here?

Did you understand my answer in post #51?

Very simple, isn't it?

Did it answer your question or does it not seem simple to you?

I hate to see questions go unanswered and unless you provide feedback, we'll never know how effective the answers were.

 

[GrayGhost]

All,

It is not my practice to not respond to the posts of others. I have been told by mentor that my interpretation of option 3, which envisions events shifting in (space)time, is not mainstream relativity, and should not be discussed in this forum. The reason, is because only 1 coordinate chart may be used per POV, and events cannot relocate (even if by POV rotation alone, during frame transitioning). As such, I respect that, and will make no further responses on this matter in this thread or in this forum. Also just to be clear, I have never suggested that events could move in the all-inertial scenario, and I am not adept in GR admittedly. Thanx for your time and thoughts on the matter.

Thank You,
GrayGhost

 

[AndrzejK]

This is what we gets from a protons reference frame, both predictions are according to relativity and both are real. But it's a paradox.
In the first event from Earth's rest frame the scientist Robert dies before seeing the proton and in the protons rest frame of reference The scientist is alive!
So how do relativity solve this paradox?
How can MS Diagram can solve this paradox?
I agree they can represent it on a diagram but it doesn't solve the paradox.
If it solves how can i visualize it, i mean i need a picture with definite result, a result that is agreeable for all observers

This is the same as twins paradox. You probably don't see the difference between proton's frame (non-inertial) and planets' frames (inertial).
The best (imo) explanation is here http://www.if.ufrgs.br/oei/santiago/fis02012/FirstCourseGR.pdf on the page 25.