How Do Time Dilation and Space Expansion Affect Gravitational Redshift?

[yuiop]

OK, in the meantime I am just typeing up a post about what happens below the event horizon which might be of interest to you ;)

 

[yuiop]

... However, the question of validity appears more difficult, at least, to me. As a broad generalisation, I don’t like paradoxes. It tends to say to me that we don’t understand everything that’s going on. In part, this is why I like to look for physical interpretation that supports mathematical conjecture. However, as outlined in post #17 and the thread above, resolving the apparent disparity between what is perceived by the distant (D) and local (A) observers does appear to be problematic. I guess I can sum up my reservation by saying that I can accept the relative nature of time, but not a contradictory nature. Therefore, if (D) sees time stop in (A), I don't how you can simply run time on in (A) disregarding the implications in (D)

OK, now we get onto the more interesting stuff :)

One point of view, possibly the accepted point of view, is that time runs on the same deep down in the gravitational well as for an observer at infinity and it is just the delay in information getting to the distant observer that accounts for the apparent difference in clock rates. At the event horizon the delay becomes infinite and information about what is happening there, never gets to the distant observer.

Another point of view (mine) is that the black hole never completely forms into the classic point singularity of infinite density. You could call this version the assymptotic version. In this thread https://www.physicsforums.com/showthread.php?t=223730&page=2 in post #17 onwards the interior Schwarzschild solution was discussed and it was shown that once a mass collapses to r=Rs*9/8 that time starts to reverse at the centre of the mass. The implication is that some physical process (gravity or pressure) starts to re-distribute the matter and density distribution so that all the mass ends up as thin shell just outside the Schwarzschild radius. To outward appearances an assymptotic black hole is just like a normal black hole but it allows a small amount of highly red shifted radiation to escape which gives a simpler explanation for Hawking radiation. It also makes the information loss paradox a non-paradox which agrees with the conclusion reached by the scientists in the link you posted. http://www.physorg.com/news101560368.html

The interior solution applies for coordinates below the physical surface of the gravitational mass and takes the reduced closed mass below the surface into account. The exterior Schwarzschild solution only applies to regions outside the surface of physical mass. For a classical black hole with all the mass located at the cental singularity the exterior Schwarzschild solution is valid below the even horizon all the way down to close to the singularity. The only trouble is that the exterior solution appears to yield imaginary answers below the event horizon due to taking the roots of negative numbers. There is however another way to use the exterior solution that might circumvent this problem to a certain extent and allow us to peek below the horizon. The normal gravitational gamma factor is given as:

(Eq1) \frac{1}{\left(1-Rs/R\right) }

This can be re-expressed as

(Eq2) \frac{\sqrt{1-Rs/R_o} }{\sqrt{1-Rs/R}}

Where Rs is the Schwarzschild radius, R is the radius where the event being measured is located and Ro is the radius that the observer making the measurement is located at. It is easy to see that if the observer is located at infinity Eq2 reduces to the usual Eq1.

Now we know that normally an observer sees clocks below him running slower than his own clock and objects tend to fall from locations of low time dilation to locations of high time dilation. If the observer is below the event horizon (say at R = Rs*9/10) and he is observing a clock below him at R = Rs*1/2 then the time dilation factor would be:

\frac{\sqrt{1-10/9} }{\sqrt{1-2/1}} = \frac{\sqrt{-1/9} }{\sqrt{-1}} = \frac{ (1/3) i}{i } = 1/3

where i is the square root of (-1)

It can be seen in this example the clock below the observer is running faster than his own clock suggesting a gravitational potential that is in the opposite direction to the gravitational slope outside the event horizon. From this point of view is it possible that everything falls towards the event horizon whether below or above the event horizon? Of course this solution is based on the exterior solution that assumed a point singularity which we are saying never formed. The interior solution on the other hand suggests that as the black hole forms mass is redistributed before the event horizon forms so that all the mass is in a thin shell just outside the horizon and assymptotically and infinitely slowly falling towards the horizon. The advantage of the interior solution is that we can track the distribution of matter before the singularity forms. The assymptotic interpretation solves the problem of time stopping at the event horizon according to a distant observer while progressing normally according to a local observer because there is never any matter or observers at the event horizon. All the above is meant to be within the context of General Relativity and not meant to be a disproof of that theory. It can also be noted that einstein himself did not agree that General Relativity predicted black holes with singularities of infinite density at the centre.

 

[mysearch]

General Response to #32

Thanks for the outline of the possible internal workings of a black hole. It is something that I would like to pursue further once I have my understanding of the external workings on a firmer footing So many thanks for the links as I had not come across the details of the asymptotic proposal before.

However, I would say that I am somewhat sceptical, at this stage, that science has an adequate and coherent quantum model of matter to say what is really happening at/or below the event horizon. Main reply to #29 to follow.

 

[yuiop]

Thanks for the outline of the possible internal workings of a black hole. It is something that I would like to pursue further once I have my understanding of the external workings on a firmer footing So many thanks for the links as I had not come across the details of the asymptotic proposal before.

However, I would say that I am somewhat sceptical, at this stage, that science has an adequate and coherent quantum model of matter to say what is really happening at/or below the event horizon. Main reply to #29 to follow.

I guess we will also have to figure out the meaning and significance of tortoise coordinates and why they seem to to contradict the asymptotic model.

 

[mysearch]

Response to #29: Part 1 of 2

Kev, I have tried to respond to each section in #29 in-turn. However, apologises for the excessive length, i.e. parts 1 & 2, but I wanted to try to explain the outstanding issues based on my current understanding.

--------------------------------------------------------------
Section-1:
This section relates to bullets a, b, c as per post #14:
Yes, I agree. As per my post #17, the Schwarzschild metric leads to the equation:

\frac{dr}{dt} = \pm c/\gamma = \pm c \left(1-Rs/r\right)

\gamma = \frac {1}{\sqrt{\left(1-Rs/r\right)}}

This also confirms your table in #11. Of course, the contradiction cited in #17 still remains and questions both the meaning and validity of the equation, because it suggests that the coordinate speed of light [c] is zero at the event horizon.
--------------------------------------------------------------

Section-2:
Summarises kinetic & gravitational effects. Took me a while to orientate myself to your notation. However, if I have interpreted the notation correct, there are two that I would like to try and clarify:

Parallel kinetic length contraction: x' = x/y
Vertical gravitational length contraction: x' = x/g

The implication is that velocity and gravity both lead to length contraction, which is something I questioned. However, I wanted to be clear on the meaning of [x,x`]. The normal Lorentz transform as described in many reference, e.g. Wikipedia: http://en.wikipedia.org/wiki/Lorentz_transformation

x' = \frac{x-vt}{\sqrt{1-v^2/c^2}}

In the reduced form, where t=0, the `parallel kinetic length contraction` would appear to go to:

x' = \gamma(x) and not x' = x/y

My interpretation of this equation is as follows: [x`] is the length measure onboard the craft traveling at [v], e.g. 1 metre. While [x] is the equivalent length measured by the stationary observer. [x` > x] as the observed length is contracted due to velocity. So while we are saying the same thing in words, we appear to have different equations.

I suspect there is also a different interpretation associated with the `vertical gravitational length contraction`. It is my understanding, although this is not an assertion of fact, that the radial length expands under gravity due to the increased curvature of space as a larger gravitational mass [M] is approached. However, I would like to pursue the clarification of this point by stepping to the next section in part-2.

End of Part 1:

 

[mysearch]

Response to #29: Part 2 of 2

Section-3:
It is probably easier to replicate the 2 definitions you modified:

o As velocity [v] approaches the speed of light [c], time slows and length contracts in the direction of motion, at least, with respect to a `stationary` observer.

You change the words `time` and `length` to `clocks` and `rulers`, therefore I wanted to clarify whether there was an important physical implication in this change rather than semantics. In cosmological expansion, the overall volume of space expands, but atoms are unaffected by this expansion, i.e. atoms do not get bigger as cosmological space expands. However, I have always interpreted length/ruler contraction in a different way, i.e. conceptually the stationary observer would perceive everything in the moving frame of reference to contract in the direction of motion, even atoms. This is why [x`] is measured locally by a ruler as 1 metre, while the stationary observer perceives both the ruler and object being measured to both be contracted.

o On approaching a gravitational mass [M], time slows and length expands in the direction of gravitational pull, as a function of radius [r], at least, with respect to a `distant` observer.

Again, we have a similar change, but the term ` length expands` is changed to `rulers contract`. Based on similar assumption as outlined previously, I assumed that the local observer’s ruler would expand in-line with any expansion in space and as such, the local observer would not perceive any stretching of the ruler as its was orientated along the radial direction. However, a distant observer would perceive both the ruler and object being measured to both be expanded.

As such, this still seems to be an open issue that I would like to clarify.

--------------------------------------------------------------

Section-4:

Jorrie and I were of the opinion that the radius calculated by a local observer by measuring circumference would agree with radius measured by the observer at infinity.

Yes, I agree. By definition, the calculated radius or coordinate-r corresponds to that measured in flat spacetime. Wasn’t too sure about the next bit, as I would have thought we could simply say that the circumference was measured without reference to any velocity. However, I agree that horizontal/tangential circumference would not be affected by gravity. At this point, I would like to introduce two definitions, although you will probably disagree with the second:

Coordinate-r = circumference/2\pi
Spatial-r = \gamma(coordinate-r)

The implication of the second definition suggests that the radial distance expands, although it is probably not measurable by the local observer for the following reasons starting with your words:

Yes, the local observer can never measure the change in the proper length of his ruler. I prefer to think that the observer at infinity sees the ruler as length contracted when orientated vertically.

While you begin with `Yes`, I don’t think you will agree with my statements at the end of section-3. I was assuming space expands in the radial direction due to the increasing curvature of space. However, I was also assuming that the local ruler would also be affected and therefore the increase in the spatial-radius is not measurable locally, only observed at a distance. So, to conclude, let me try to apply my assumptions to the 2 observers under consideration, i.e. the distant observer (x, t) and the shell observer (x`, t`) within the gravity well using my assumptions:

x` > x, i.e. x’ = \gamma(x)

t` < t, i.e. t&#039;= (t)/ \gamma

The velocity [v] is a function of distance over time:

v' = x'/t' = \gamma(x)/ (t)/ \gamma = \gamma^2 (x/t)

As such, [v’] is greater than [v] by \gamma^2 as required by the very first equation. Again, apologises for such a long posting, but as stated, I wanted to try and address all the open issue from my current understanding.

 

[yuiop]

--------------------------------------------------------------

Section-2:
Summarises kinetic & gravitational effects. Took me a while to orientate myself to your notation. However, if I have interpreted the notation correct, there are two that I would like to try and clarify:

Parallel kinetic length contraction: x' = x/y
Vertical gravitational length contraction: x' = x/g

The implication is that velocity and gravity both lead to length contraction, which is something I questioned. However, I wanted to be clear on the meaning of [x,x`]. The normal Lorentz transform as described in many reference, e.g. Wikipedia: http://en.wikipedia.org/wiki/Lorentz_transformation

x&#039; = \frac{x-vt}{\sqrt{1-v^2/c^2}}

In the reduced form, where t=0, the `parallel kinetic length contraction` would appear to go to:

x&#039; = \gamma(x) and not x' = x/y

My interpretation of this equation is as follows: [x`] is the length measure onboard the craft traveling at [v], e.g. 1 metre. While [x] is the equivalent length measured by the stationary observer. [x` > x] as the observed length is contracted due to velocity. So while we are saying the same thing in words, we appear to have different equations.

I think we are odds with the notation. It might be clearer to introduce the term "proper length" which has a clear meaning and re-write my statement:

Parallel kinetic length contraction: L' = Lo/y = (X2' -X1') = (Xo2-Xo1)/y
Vertical gravitational length contraction: L' = Lo/y (X2' -X1') = (Xo2-Xo1)/g

where Lo is the proper length measured by the local observer at rest with rod/ruler.

You seem to have x' as the proper length which might be where we differ.

Refering to the wikipedia reference http://en.wikipedia.org/wiki/Lorentz_transformation

they have t&#039; = \frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}

it should be clear that the primed time coordiante is not the proper time. For example at v =0.8c a proper time interval of 1 second in the rest frame of 1 second is 1.6666 seconds in the primed frame which is moving relative to the clock. Similarly the primed x' coordinate is not a proper distance coordinate.

We also know that at v=0.8c a measurement of 1 light second for the proper length Lo of a rod in the rest frame of the rod transforms to a measurement of 0.6 lightseconds in the primed frame. Time transforms to a larger value (dilation) while length transforms to a smaller value (contraction).

For time t&#039; = \frac{t-vx/c^2}{\sqrt{1-v^2/c^2}}

T&#039; = (t2&#039; -t1&#039;) = \frac{(t2-t1)-v(x2-x1)/c^2}{\sqrt{1-v^2/c^2}}

For x2=x1=0

T&#039; = \frac{(t2-t1)}{\sqrt{1-v^2/c^2}} = To*\gamma



For length x&#039; = \frac{x-vt}{\sqrt{1-v^2/c^2}}

L&#039; = (x2&#039; -x1&#039;) = \frac{(x2-x1)-v(t2-t1)/c^2}{\sqrt{1-v^2/c^2}}

Now to measure length in the primed frame we cannot simply say t2-t1=0 because the times t1 and t2 are simultaneous in the rest frame but not in the primed frame. So we have to use the reverse transformation: See post #4 of this thread --> https://www.physicsforums.com/showthread.php?t=235322 for the calculation.

The end result is:

L&#039; = (x2&#039; -x1&#039;) = (x2-x1)*\sqrt{1-v^2/c^2} = Lo/\gamma

I suspect there is also a different interpretation associated with the `vertical gravitational length contraction`. It is my understanding, although this is not an assertion of fact, that the radial length expands under gravity due to the increased curvature of space as a larger gravitational mass [M] is approached. However, I would like to pursue the clarification of this point by stepping to the next section in part-2.

End of Part 1:

Say an observer measures the radius of the orbit to be R by measuring the circumference with rulers or timing an orbital period. When the local observer tries to measure the radius directly along the radius using rulers that are length contracted according to the infinity observer, he will measure a greater radial distance than the distance R which both the local and infinity observer agree on. As you can see in this case length contracted rulers corresponds to expanded distance and this may be another source of confusion. It should be noted that the contraction of the rulers is a measurement according the infinity observer while the expansion of the radial distance is the direct measurement of the local observer.

This also confirms your table in #11. Of course, the contradiction cited in #17 still remains and questions both the meaning and validity of the equation, because it suggests that the coordinate speed of light [c] is zero at the event horizon.

In the asymptotic proposal this contradiction is resolved because actually reaching the event horizon is unatainable just as as reaching the speed of light is unatainable.

 

[yuiop]

Section-3:
It is probably easier to replicate the 2 definitions you modified:

You change the words `time` and `length` to `clocks` and `rulers`, therefore I wanted to clarify whether there was an important physical implication in this change rather than semantics. In cosmological expansion, the overall volume of space expands, but atoms are unaffected by this expansion, i.e. atoms do not get bigger as cosmological space expands. However, I have always interpreted length/ruler contraction in a different way, i.e. conceptually the stationary observer would perceive everything in the moving frame of reference to contract in the direction of motion, even atoms. This is why [x`] is measured locally by a ruler as 1 metre, while the stationary observer perceives both the ruler and object being measured to both be contracted.

I changed the word "length" to "rulers" to emphasize that I was talking about a physical object possibly of infinitesimal size, where as you seem to be using the word "length" to mean "distance". In the Bell's spaceship paradox the distance between the two accelerating rockets remains constant in the unaccelerated reference frame while it expands in the reference frame of one of the rockets. Rulers oboard one the rockets appear to contract in the unaccelerated frame while remaining constant in the the frame of one of the accelerating spaceships. So in the accelerated frame Length (rulers) contract and distance expands. The subtle difference between length and distance is the root of the that paradox that confuses many people. Physical objects length contract according to an observer with relative motion with respect to the object but the same is not true for distances.

Also, as I said before you are interpreting the primed x' to mean a local proper measurement of an object, when the primed variables usually refer to measurements made in a frame with motion relative to the rest frame of the object. However, I am not sure if there is any hard and fast rule or convention on this, but we should be consistent.

Again, we have a similar change, but the term ` length expands` is changed to `rulers contract`. Based on similar assumption as outlined previously, I assumed that the local observer’s ruler would expand in-line with any expansion in space and as such, the local observer would not perceive any stretching of the ruler as its was orientated along the radial direction. However, a distant observer would perceive both the ruler and object being measured to both be expanded.

As such, this still seems to be an open issue that I would like to clarify.

Same comments as above.

Section-4:

Yes, I agree. By definition, the calculated radius or coordinate-r corresponds to that measured in flat spacetime. Wasn’t too sure about the next bit, as I would have thought we could simply say that the circumference was measured without reference to any velocity. However, I agree that horizontal/tangential circumference would not be affected by gravity. At this point, I would like to introduce two definitions, although you will probably disagree with the second:

Coordinate-r = circumference/2\pi
Spatial-r = \gamma(coordinate-r)

I disagree with the second for a different reason than you probably imagine. Local rulers length contract according to Lo/gamma as measured by the observer at infinity. If by R_spatial you mean the distance measured from the centre of the body to the orbital radius under consideration then it is greater than R_coordinate but not by gamma. The rulers used to measure the radial distance will be length contracted to different degrees depending on how deep they are within the gravity well so the total distance would have to arrived at by integrating the total of all the individual infinitesimal rulers and the result will be much greater than simply R_coordinate*gamma. Close to an event horizon you would need an amount of local rulers than tends towards infinite to measure distances.

The implication of the second definition suggests that the radial distance expands, although it is probably not measurable by the local observer for the following reasons starting with your words...
I was assuming space expands in the radial direction due to the increasing curvature of space. However, I was also assuming that the local ruler would also be affected and therefore the increase in the spatial-radius is not measurable locally, only observed at a distance.

Well, I agree local radial distance expands but I will add that that local observers can measure it in the sense that they measure a greater radial distance than the infinity observer when they use their local rulers. They can not however, measure the length contraction of their own local rulers or rods. I hope we can agree on the difference between length (rulers) and distance as I have defined it. In the gravitational case distance expands and length is constant according to local observers while distance is constant and length contracts according to a distant observer.

 

[mysearch]

Initial Response to #37 & #38

Kev: As always you have responded with some very convincing arguments.

The subtle difference between length and distance is the root of the paradox that confuses many people.

I believe this may also be the root of my confusions, so I want to take some time to reflect on the all the issues you have raised with respect to proper length and distance. I also need to think about the `Bell's spaceship paradox` that has really got me scratching my head.

I really appreciate the help you have extended and will respond on any open issues, if any, as soon as possible. Thanks.

 

[mysearch]

Response to #37: Lorentz Transforms

I thought it might be easier to try and close specific issues one by one rather massing all the issues together, especially as the Bell's spaceship paradox has me doubting everything at the moment.

As you may have realized, I have been trying to self-learning about relativity over the last couple of months, mainly from Internet sources; so have simply adopted the notation I commonly saw in use, e.g.
http://en.wikipedia.org/wiki/Lorentz_transformation
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/ltrans.html

Both these sources show the two main transforms as follows:

x&#039; = \frac{x-vt}{\sqrt{1-v^2/c^2}} = \gamma(x-vt)

t&#039; = \frac{t-vx/c^2}{\sqrt{1-v^2/c^2}} = \gamma(t-vx/c^2)

This suggested to me that [x’>x] with [x’] being a length measured by the moving observer and [x] the equivalent length measured by the stationary observer. As such, [x] is contracted with respect to [x’].

You seem to have x' as the proper length which might be where we differ.

Wasn’t aware that I was making any inference to proper length or proper time, simply trying to follow what I thought was accepted notation. However, by the same arguments [t’>t], but to be consistent, [t’] would also have to be a measure of time as perceived by the moving observer and [t] an equivalent measure with respect to the stationary observer. I interpreted this to mean that the tick of the clock in the moving frame would be perceived to be longer than in the stationary observer. If we use your example:

v=0.8c and \gamma=1.6666
t’=1.6666 and t=1
t’ > t

Therefore, I understood this to infer that a clock in the moving frame of reference runs slower than one in the stationary frame. I realize this interpretation possibly creates a distinction between the rate of time, i.e. the tick of the clock, and the duration or the proper time interval. However, it allows [x’, t’] to both be simply viewed as quantities associated with the moving frame, i.e. no inference to proper length or proper time. Therefore, it appears we were using different notation, which then led to different interpretations. Using your notation:

L’ = Lo/y so that Lo = yL’ it implies Lo > L’

This appears equivalent to the previous notation x&#039;=\gamma(x) and, if so, we appear to have general agreement on the Lorentz transform regarding length. However, at first glance, there appears to be a difficulty with the time transform:

T’ = To*y or To = T’/y which implies To < T’

This was why I initially raised the query against your form, but you subsequently highlighted the point that possibly clarifies the situation:

It should be clear that the primed time coordinate is not the proper time

As discussed above, this is probably true, but I never put this inference on [t’]. Therefore, I wanted to check whether my explanation of [t’] resolves the different perspectives of both styles of notation? If so, have we now converged to an agreement or am I still missing any of the points you were making?

 

[yuiop]

...

As you may have realized, I have been trying to self-learning about relativity over the last couple of months, mainly from Internet sources; so have simply adopted the notation I commonly saw in use, e.g.
http://en.wikipedia.org/wiki/Lorentz_transformation
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/ltrans.html

Both these sources show the two main transforms as follows:

x&#039; = \frac{x-vt}{\sqrt{1-v^2/c^2}} = \gamma(x-vt)

t&#039; = \frac{t-vx/c^2}{\sqrt{1-v^2/c^2}} = \gamma(t-vx/c^2)

This suggested to me that [x’>x] with [x’] being a length measured by the moving observer and [x] the equivalent length measured by the stationary observer. As such, [x] is contracted with respect to [x’].

Hi mysearch,
this is a self learning curve for me too and I find these discussions with you a welcome oportunity to clear up the concepts for myself too. Your comments make me realize I do not have some ideas as rigorously defined as I imagined I had, so bear with me if I adapt as I go along and sometimes appear to contradict things I said earlier ;) First of all the use of the word "length" in your comment "[x’] being a length measured by the moving observer and [x] the equivalent length measured by the stationary observer. As such, [x] is contracted with respect to [x’]" might be a source of confusion because x' is a coordinate and not a length. [x,t] or even [x',t] are the coordinates of a momentary event and that event does not have to belong to either the S or S' reference frame. It is difficult to define if an observer is at rest with or moving with respect to a single instantaeous event so it was perhaps misleading of me to claim that x or x' is proper measurement. Similarly, x can not contract with respect to x' because it a point coordinate with no width. We can only talk of length or length contraction with reference to a pair of coordinates such as (x2-x1) to define a spatial interval and we can only talk of time dilation with respect to a pair of time coordinates such as (t2-t1) to define a temporal interval. Only intervals can have a clearly defined rest frame. A single coordinate such as [x.t] is a point in space and time and can not length contract or time dilate. The quantities L and T that I used for length contraction and time dilation are intervals and a different animal from coordinates x and t.

...
I interpreted this to mean that the tick of the clock in the moving frame would be perceived to be longer than in the stationary observer.

Your sentence leaves a lot of room for misunderstanding. If you meant "the tick (interval?) of the (moving?) clock (as measured?) in the moving frame would be perceived (by the observer co-moving with the clock?) to be longer than (the interval measured?) in the (frame of the) stationary observer." then that is not correct. The interval measured by the stationary observer (that the clock is moving relative to) will be longer than the interval measured by the observer co-moving with the clock.

...
Therefore, I understood this to infer that a clock in the moving frame of reference runs slower than one in the stationary frame.

This is true, but it does not coincide with your above statement.

...
I realize this interpretation possibly creates a distinction between the rate of time, i.e. the tick of the clock, and the duration or the proper time interval.

It's an important distinction to make. If by "tick of the clock" you meant clock rate then you should have used slower or faster to compare measurements rather than longer or shorter.
The stationary observer considers the clocks moving relative to him as running slower than his own clocks. The interval measured between two events by his own clocks can be longer or shorter than the interval measured by an observer moving relative to him. For example say we have a rocket moving relative to our observer (Anne). A flash at the centre of the rocket is observed to take one mllisecond to reach both ends of the rocket by an observer (Bob) onboard the rocket. Anne will measure a much greater interval than one millisecond for the flash at the centre to catch up with the front of the rocket and a much shorter interval than one milisecond for the flash to go from the centre of the rocket to the rear. As you can see it is not simply a case of saying an interval in Bob's frame will be measured as longer in Anne's frame.

...
Therefore, it appears we were using different notation, which then led to different interpretations. Using your notation:

L’ = Lo/y so that Lo = yL’ it implies Lo > L’

...
T’ = To*y or To = T’/y which implies To < T’

The above two statements are true with the understanding that:

Lo is a measurement of the proper length between two spatially separated markers that are at rest with the observer making the measurement and L' is the measurement of the distance between the two markers by an observer moving relative to the markers by comparing the simultaneous spatial coordinates of the two markers.

To is the interval between two events at the same location by an observer at rest with the spatial location of the two events and T' is the interval between the same two events made by an observer moving relative to the two events. The observer moving relative to the two events will not consider the events to have occurred in the same place.

L’ = Lo/y and T’ = To*y are what is commonly meant by the expressions "length contraction" and "time dilation". Dilation is alternative word for expansion and indicates the inverse nature of length contraction and time dilation.

Are we any closer to consensus yet?

 

[dude222]

The gravitational redshift is entirely due to a change in the speed of light with locations in the gravitational field.

 

[mysearch]

Response to #41:

Hi Kev,
Sorry, I haven’t had much time this weekend to really work on all the issues you raised in #37/#38, especially the Bell’s paradox. I have found a paper by 2 Japanese guys that looks very interesting but haven’t had time to read it properly, as yet. Like you, I find it very useful to have my current understanding challenged, because often I find that I have made initial assumptions that appear logical, but cannot always be defended. In this respect, I try to remember the quote:

For every problem, there exists a simple and elegant solution, which is absolutely wrong. J. Wagoner

However, I wanted to try to move along the discussion regarding the Lorentz transforms, if possible, because it seems to be an important foundation.

First of all the use of the word "length" in your comment "[x’] being a length measured by the moving observer and [x] the equivalent length measured by the stationary observer. As such, [x] is contracted with respect to [x’]" might be a source of confusion because x' is a coordinate and not a length.

I did mean length when talking about [x, x`], but you are right to highlight this issue, as it is not rigorous. I have simply assumed the following:

x = x1-x0 and x’ = x1’-x0’ such that:

x&#039;=x1&#039;-x0&#039;=\gamma((x1-x0)-v(t1-t0) ) = \gamma(x-vt)

In this respect, [x1, x0] are the coordinate offsets in a given frame of reference, while [x] is the measured separation or length between these coordinates. In this respect, your adoption of [L & Lo] is probably more correct, but I think we are trying to say the same thing.

Your sentence leaves a lot of room for misunderstanding. If you meant "the tick (interval?) of the (moving?) clock (as measured?) in the moving frame would be perceived (by the observer co-moving with the clock?) to be longer than (the interval measured?) in the (frame of the) stationary observer." then that is not correct.

Apologises for the confusion. I was simply trying to rationalise what might appear to be a contradiction between the rate of time and the duration. The rate that a clock runs in the stationary frame, e.g. [r], is faster than the moving frame, e.g. [r’], such that we might say [r > r’]. This is what I meant by the ticking of the clock. For example, in both frames, each observer perceives time passing at the rate of 1 second per second. However, if we assume \gamma=2, there will be 2 ticks of the clock in the stationary frame for 1 tick in the moving frame, hence r > r’, meaning that the stationary observer perceives the rate of time in the moving frame to be ½ second per second. However, the reverse of this logic means that 2 seconds [t’] in the moving frame is only measured as 1 second in the stationary frame [t], hence t’ > t. This was my interpretation of the orientation suggested by the Lorentz transform

t&#039;= \gamma(t-vx/c^2) = \gamma(t) when x=0

I hope this is enough to clarify what I was trying to say, as overall, I believe we are essentially saying the same thing regarding the Lorentz transforms, but have adopted different notation to express the concepts stemming from them, although you might still disagree. At this stage, I have deliberately avoided using proper distance and proper time, as I still need to respond to the many doubts you raised in my mind about such concepts, aka the Bell paradox, along with all the equivalent effects under gravity. Hopefully, I will have some more time tomorrow.

 

[mysearch]

Response to #42:

Sorry, I am not sure that I fully understood the implications of your statement. Within all local frames, the speed of light is [c]. As such, the gravitational redshift might be interpreted as either an energy loss to the gravitational field (Newtonian) or a frequency shift due to time dilation (relativity), which in-turn affects energy by virtue of E=hf. Is there another accepted explanation?

 

[yuiop]

Hi Kev,
Sorry, I haven’t had much time this weekend to really work on all the issues you raised in #37/#38, especially the Bell’s paradox. I have found a paper by 2 Japanese guys that looks very interesting but haven’t had time to read it properly, as yet.

I posted somthing about Bell's spaceship paradox here (post#121)https://www.physicsforums.com/showthread.php?t=210634&page=9 that I hope you will find useful.

I love your quote of J Wagoner "For every problem, there exists a simple and elegant solution, which is absolutely wrong." I got a good chuckle from that. Might steal it and use it as a sig ;)

However, I wanted to try to move along the discussion regarding the Lorentz transforms, if possible, because it seems to be an important foundation.

I did mean length when talking about [x, x`], but you are right to highlight this issue, as it is not rigorous. I have simply assumed the following:

x = x1-x0 and x’ = x1’-x0’ such that:

x&#039;=x1&#039;-x0&#039;=\gamma((x1-x0)-v(t1-t0) ) = \gamma(x-vt)

In this respect, [x1, x0] are the coordinate offsets in a given frame of reference, while [x] is the measured separation or length between these coordinates. In this respect, your adoption of [L & Lo] is probably more correct, but I think we are trying to say the same thing.

The moving observer using the equation x&#039;=x1&#039;-x0&#039;=\gamma((x1-x0)-v(t1-t0) ) = \gamma(x-vt) and using t1-t0=0 will arrive at the conclusion that objects with relative motion length expand. The problem with that conclusion is the assumption that when t1 and t0 are simultaneous in S that they are simultaneous in S' too which is not correct. To correct the equation so that the measurements of both ends of the moving rod are made simultaneously then t0+(x1-x0)*v/c^2 should be substituted for t1 to get

x&#039;=x1&#039;-x0&#039;=\gamma((x1-x0)-v(t0+(x1-x0)*v/c^2-t0) )

x&#039;=x1&#039;-x0&#039;=\gamma((x1-x0)-v((x1-x0)*v/c^2) )

x&#039;=x1&#039;-x0&#039;=((x1-x0)-v((x1-x0)*v/c^2) )/\sqrt{1-v^2/c^2}

x&#039;=x1&#039;-x0&#039;=((x1-x0)(1-v^2/c^2) )/\sqrt{1-v^2/c^2}

x&#039;=x1&#039;-x0&#039;=(x1-x0)\sqrt{1-v^2/c^2}

x&#039;=x1&#039;-x0&#039;=(x1-x0)/\gamma

which is a more meaningful useful equation because the measurement is made when t1'=t0'.

The corresponding time transofrmation of an interval is:

t&#039;=t1&#039;-t0&#039;=\gamma((t1-t0)-v(x1-x0)/c^2)

t&#039;=t1&#039;-t0&#039;=(t1-t0)*\gamma

when x1=x2

 

[yuiop]

Hi mysearch,

I have an alternative derivation of x &#039;=x1&#039;-x0&#039;=(x1-x0)/\gamma using the invariant interval that might help convince you.

Starting with

(1) t=(t&#039;+vx&#039;/c^2)\gamma

Substitute the Lorentz transformation for x' in terms of x in (1) [this will be helpful later].

(2) t=(t&#039;+v((x-vt)\gamma)/c^2)\gamma which simplifies to

(3) t= \gamma t&#039; + vx/c^2

Now using the invariant interval S^2 = \Delta x^2-c^2\Delta t^2 =\Delta x&#039;^2-c^2\Delta t&#039;^2

(4) (x1&#039;-x0&#039;)^2-(ct1&#039;-ct0&#039;)^2 = (x1-x0)^2-c^2(t1-t0)^2

Since we require (x1'-x0') when t1' = t0'

(5) (x1&#039;-x0&#039;)^2= (x1-x0)^2-c^2(t1-t0)^2

Substitute the expression for t in terms of t' obtained from eq (3)

(6) (x1&#039;-x0&#039;)^2= (x1-x0)^2-c^2(\gamma t1&#039; + vx1/c^2-(\gamma t0&#039; + vx0/c^2)^2)

(7) (x1&#039;-x0&#039;)^2= (x1-x0)^2-c^2(\gamma (t1&#039;-t0&#039;) + v/c^2(x1-x0))^2

Since it has been required that t1'=t0'

(8) (x1&#039;-x0&#039;)^2= (x1-x0)^2-c^2(v/c^2(x1-x0))^2

(9) (x1&#039;-x0&#039;)=(x1-x0)\sqrt{1-v^2/c^2}=(x1-x0)/\gamma

 

[mysearch]

Response to #45/46

Hi Kev,
Glad you liked the quote, its a good reminder and thanks for the input. However, I thought we might try to anchor our respective derivations in a practical example because I am still not sure whether we are interpreting the notation in different ways, but meaning the same thing? Let’s start with the reference you provided to the Lorentz Transformation, as I have been quoting these all along, and then construct an example based on the most basic postulate of special relativity, i.e. the invariance of the speed of light in all inertial frames, i.e.

c = x/t = x’/t’

In the example, (A) will be the stationary frame and (B) the moving frame. The velocity of (B) with respect to (A) is 0.866c, giving a Lorentz factor \gamma=2. Let’s assume both frames are synchronised at the spacetime coordinates [x0, t0] and [x0’, t0’], but verify this assumption via the Lorentz transforms:

x0^{\prime} = \gamma (x0-vt0) = 2(0-(.866c)0) = 0

t0^{\prime} = \gamma ( t0-vx0/c^2) = 2(0-(.866c)*0/c^2) = 0

At this point of synchronisation, observers in (A) and (B) see a pulse of light fired at a target, which is stationary with respect to (A) and 1 lightsecond away. Now both must see the pulse of light moving away at [c] and we might readily calculate that (A) sees the pulse hit the target after 1 second, but what happens in (B)? Just for reference, light covers 2.99E8 metres in 1 second and 0.866c = 2.59E8m/s:

x1^{\prime} = \gamma (x1-vt1) = 2(2.99E8-(2.59E8)1) = 0.8E8 metres

t1^{\prime} = \gamma ( t1-vx1/c^2) = 2(1-(2.59E8)*2.99E8/c^2) = 0.267 seconds

At first, the Lorentz transforms appear to have given an anomalous answer as it suggests that the event that occurred after 1s in (A) occurs after only 0.267s in (B). However, I believe this corresponds to the `line of simultaneity` between the 2 frames, although this is more easily understood via the attached spacetime diagram. Even so, we might wish to confirm the velocity of light [c] as determined by the observer in (B) to check for a mistake, i.e. 0.8E8/0.267 = 2.99E8m/s. As such, we appear to have correlated the constancy of [c] via the Lorentz transforms, so how do we correlate the distance?

Of course, if (B) considers itself stationary, then the target must have a closing velocity of 0.866c with respect to (B). As such, the target starts from an offset comparative to [x1, x1’] and moves towards the light pulse in the (B) frame. So the total distance is the distance covered by the light pulse, i.e. [0.8E8m] plus the distance covered by the target traveling at 0.866c in the same time, i.e. 0.267s, which gives us a figure of 0.692E8m. Therefore, the comparative distance in (B) to the 1 lightsecond (2.99E8) covered in (A) is (0.8E8+0.69E8=1.49E8), which then corresponds to an expected length contraction proportional to \gamma=2.

x’ = (x)/\gamma = 2.99E8/2 = 1.49E8m such that x’ < x

The time calculated in (A) corresponded to the time for the light pulse to cover 1 lightsecond, (2.99E8m). As such, a comparative time [t’] in (B) would be that needed for light to cover 1.49E8m, i.e. 1.49E8/c=0.5s. As such, we now have a value of [t’] that proportional to \gamma=2, where t` < t.

In part, I believe the confusion over whether [x`<x] or [x’>x] depends on how you interpret the orientation, although I suspect theory dictates how this should be done. In the previous example, the length in (A) equalled 2.99E8m, i.e. 1 lightsecond, which at the speed of light is covered in 1 second. In the example, the Lorentz transforms calculate the length and time measured in (A) into the comparable length and time in (B), i.e. 1.49E8m and 0.5seconds based on \gamma=2. This appears to correspond to length contraction and time running slower in (B) compared to (A). However, what this also appears to imply is that a length in (B), e.g. 2.99E8m, is contracted to 1.48E8m to an observer in (A) because of (B) relative velocity with respect to (A).

This said, if we ignore the ambiguity of the wording in the last paragraph, are the figures in the example consistent with what we have both been trying to say?

 

[yuiop]

Hi mysearch,

Your calculations, diagram and line of simultaneity seem O.K.

...

x^{\prime} = (x)/\gamma = 2.99E8/2 = 1.49E8m such that x’ < x

This is Ok but your use x' to symbolise (x1'-x0') is confusing because x' is also used to sybolise the Lorentz transform x{\prime} = \gamma (x-vt1)

Perhaps it would be clearer to use something like \Delta x&#039; to symbolise x1'-x0' or just use x1'-x0' explicitly rather than just x' to avoid confusion with x{\prime} = \gamma (x-vt1)

It is also worth noting that \Delta x&#039; &lt; \Delta x is not always true unless \Delta t&#039; =0 In the case when t'=0 I have been using the notation L' as shorthand for \Delta x&#039; when \Delta t&#039; = 0 and it is always true that L' = Lo/y where it is understood Lo is the proper measurement along the x axis.

Hi Kev,
...
The time calculated in (A) corresponded to the time for the light pulse to cover 1 lightsecond, (2.99E8m). As such, a comparative time [t’] in (B) would be that needed for light to cover 1.49E8m, i.e. 1.49E8/c=0.5s. As such, we now have a value of [t’] that proportional to \gamma=2, where t` < t.

Your calculation that the interval t1'-t0' is less than t1-t0 for a light signal to travel from one location to another in frame A (or S) is OK but it is not a comparison of clock rates because the interval t1'-t0' is made using a single clock in frame B (or S') while the same interval is measured using two spatially separated clocks in frame A. The time interval t1'-t0' is not always less than t1-t0 as can be seen by doing the calculation for a light signal going in the opposite direction from x1 to x0. What we can always say is that if a time interval in frame A is measured in frame A by a single clock in A then B will measure that time interval to be greater. The notation I used to specify the measurement of a time interval in frame A as measured by a single clock in frame A is To and that time interval will always be measured as T' = yTo in frame B. T' and To is more about the relative rate clocks run at in different frames than a comparison of interval between events

To summerise:

x^{\prime} = \gamma (x-vt)
t^{\prime} = \gamma ( t-vx/c^2)

and

L' = Lo/y
T' = To*y

Where Lo and To are measurements of a ruler or clock by an observer at rest with the ruler or clock and L' and T' are measurements made by an observer moving relative to the ruler or clcok.

From the invariant interval

\Delta x&#039;^2 -c^2\Delta t&#039;^2 = \Delta x^2 - c^2\Delta t^2

it is fairly easy to see that \Delta x&#039; &lt; \Delta x is not always true and neither is\Delta t&#039; &lt; \Delta t

 

[mysearch]

Response to #48:

I think we are beginning to converge, although I accept that some of my notation is still probably incorrect and confusing, so will try to avoid this in the future, e.g.

This is Ok but your use x' to symbolise (x1'-x0') is confusing because x' is also used to symbolise the Lorentz transform x{\prime} = \gamma (x-vt1)

If we basically agree on the example calculation and the spacetime diagram, I feel most of the other problems are mainly to do with understanding the orientation of the specific example under discussion. Therefore, given the other valid questions raised in #37/38, I would now like to return to this aspect of the discussion, because I am still unsure about the issue of space contraction/expansion in a gravitational field. However, before addressing these issues I wanted to get a better handle on the issue of the Bell's spaceship paradox. In my defence, I will cite the following Chinese saying:

One who asks a question is a fool for five minutes; one who does not ask a question remains a fool forever.

Thanks

 

[mysearch]

Response to #37/#38

This posting is trying to get clarification or verification of a number of concepts raised throughout this thread, in particular postings #37/#38.

o SpaceTime Interval
o Proper Time
o Proper Length

The attachment st1.jpg is used as an initial reference to a Minkowski or spacetime diagram. Based on this geometry, the proper time is defined as:

[1] (Proper Time)^2 = (t^2 - x^2/c^2)

Within the causal past and future, [t] is normally greater than [x], while the interpretation of proper time is given below. Of course, in the case of (e3), [x] can be greater than [t], which would require the square root of a minus number. Therefore, in this case, it seems that the alternative definition of the spacetime interval is often preferred:

[2] (Space Interval)^2= (x^2-(ct)^2)

In essence, both these quantities seem to represent the spacetime separation in units of time or distance. The following definitions of `proper time [To]` and `proper length [Lo]` may not be rigorous, but hopefully anchors some meaning to a specific example. These definitions align to the statements made in #37/48, i.e. T = To*y and L = Lo/y, but it is believed that some caution needs to be exercised in how these statement correspond to time dilation and length contraction as implied by the Lorentz transform.

Definition of Proper Time [To]
Proper time [To] might initially be defined as the `wristwatch` time of an observer moving between 2 events with velocity [v]. This time would differ from the time [T] of a stationary person watching the observer move between the 2 events such that:

[3] T = To*\gamma

See attachment st2.jpg for the equivalent spacetime diagram. The diagrams alludes to 2 events (A) and (B). A stationary observer measures the time (5) and distance (3) taken by a moving observer to reach (B) from which the proper time corresponding to the wristwatch time of the moving observer is calculated to be 4. This suggests that time for the moving observer is running slower than the stationary observer. On the basis that the velocity [v] is invariant between both the stationary and moving observers, we might also determine the distance/length between (A) and (B) as determined by both observers, i.e.

Stationary Observer: x=vt=0.6*5=3
Moving Observer: x’=0.6*4=2.4.

The implication being that the length with respect to a moving observer is contracted. However, we will see how this statement might be misleading in some examples.

Definition of Proper Length [Lo]
Proper length (Lo) might initially be defined as the length of an object observed by a person at rest with respect to the object. This length would differ from the contracted length [L] observed by a person moving with respect to the object such that:

[4] L = Lo/\gamma

Based on the previous example in attachment st2.jpg, we saw the stationary observer measure the distance between events (A) and (B) to be 3. By the definition of proper length above, the length measured by the observer stationary to (A) and (B) was the proper length, i.e. Lo=3. As such, we could calculate the contracted length [L] from equation [4] as 2.4.

However, the proper length is said to be invariant on the basis that it always represents the maximum length seen by any observer. Possibly, the easiest way to visualise this is by assuming (A-B) to be the length of the spaceship in which the moving observer travels. Initially, before launch, both observers perceive the length to be 3 units. After take-off, (A-B) are now stationary with respect to the moving observer, not the stationary observer, as per our previous example. By definition, the proper length is invariant and a maximum, e.g. 3, and therefore the contracted length is that observed by the stationary observer, e.g. 2.4

Let’s take this line of logic one step further and assume that there are 2 identical spaceships. One used by the moving observer and one left behind with the stationary observer. Now, even though there is no ambiguity about which observer is actually moving relative to the another, the definition of proper length seems to imply that the moving observer perceives his spaceship to be 3 units in length, while perceiving the length of the spaceship on the ground to be only 2.4 units. While, the stationary observer on the ground perceives the lengths to be the other way round.

Proper Time Caveat?
If the moving observer’s was accelerating between events (A) and (B), would the proper time, i.e. wristwatch time, be shorter than that measured by a non-accelerated (inertial) wristwatch?

If yes, does the equivalence principle between acceleration and gravity, allow this additional time dilation to be interpreted in terms of an equivalent gravitational effect in conjunction with velocity?

Proper Length Caveat?
If yes, is there an equal implication of there being an extra effect on space curvature due to this acceleration as per gravity?

If curvature increases with gravity, does this mean that light has to travel a greater distance than implied by the coordinate-radius?

 

[yuiop]

Hi mysearch,
I have attached a slightly modified version of your first diagram to the aid the discussion. (hope you don't mind).

From the attached diagram:

e1 and e(0,0) are causally connected and said to be "time like separated". Ther is no reference frame in which they can be considered to be simultaneous. Same goes for events e(0,0) and e2 and e(0,0) and e3''.

e3 and e(0,0) are "space like separated" events as they can not be casually connected. e3 and e(0,0) are example of where delta(x) > delta(t)*c and in this case delta(t) =0. It is also true that delta(x)/c < delta(t) in this case. A reference frame in which two space like separated events are considered to be simultaneous can always be found. e4 and e(0,0) are also space like separated.


e(0,0) and e3' are "light like separated" and the two events can always be connected by a light signal according to any observer.


Ref: http://en.wikipedia.org/wiki/Spacetime

...
Let’s take this line of logic one step further and assume that there are 2 identical spaceships. One used by the moving observer and one left behind with the stationary observer. Now, even though there is no ambiguity about which observer is actually moving relative to the another, ...

There is always an ambiguity of which observer is moving relative to the other in realtivity :P

...
the definition of proper length seems to imply that the moving observer perceives his spaceship to be 3 units in length, while perceiving the length of the spaceship on the ground to be only 2.4 units. While, the stationary observer on the ground perceives the lengths to be the other way round.

Correct. Each observer will consider their own spaceship to be shorter than the other.

...
Proper Time Caveat?
If the moving observer’s was accelerating between events (A) and (B), would the proper time, i.e. wristwatch time, be shorter than that measured by a non-accelerated (inertial) wristwatch?

If yes, does the equivalence principle between acceleration and gravity, allow this additional time dilation to be interpreted in terms of an equivalent gravitational effect in conjunction with velocity?

The comparison is a little tricky because the accelerating observer has velocity as well as acceleration. In a comparison of the proper times of two observers leaving location A simultaneously and arriving at location B simultaneously, with one observer moving with constant velocity and the other accelerating continuously then the the accelerating observer experiences more time dilation and less proper time. The accelerating observer takes a longer curved path "through" spacetime. So yes, I would tend to agree with your conclusion.

...
Proper Length Caveat?
If yes, is there an equal implication of there being an extra effect on space curvature due to this acceleration as per gravity?

I would tend to say yes.

...
If curvature increases with gravity, does this mean that light has to travel a greater distance than implied by the coordinate-radius?

Depends who defines the distance the light has to travel. Local observers in the gravity well would measure the radial distance as greater than the coordinate-radius if they tried to measure it directly because they would effectively be using length contracted rulers. For the same reason an observer on the perimeter of a rotating disk would measure a greater radius than a non rotating observer.

It is also relevant to the Bell's spaceship paradox where the observers co-moving with the rockets measure the separation distance of the rockets to greater than the constant separation that the non accelerated observer measures. (The co-moving observers are using length contracted rulers).

 

[yuiop]

Proper Length Caveat?
If yes, is there an equal implication of there being an extra effect on space curvature due to this acceleration as per gravity?

If curvature increases with gravity, does this mean that light has to travel a greater distance than implied by the coordinate-radius?

A couple of thoughts to add to your deliberations.

Let A be an observer on the surface of a very dense gravitational body and B is an observer far out in space. They agree on their radiuses rA and rB as defined in by assuming R=circumference/2pi.

B sends a light signal down to A which is reflected from a mirror on the surface and returns to B. He notes that the round trip time for the light signal is greater than the 2(rB-rA)/c. Now B can draw two equally viable conclusions:

(1) The speed of light is constant everywhere and the "real" distance rB-rA is greater than that calculated by assuming the relationship R=Circumference/2pi.

(2) The speed of light slows down as it falls and takes longer to travel the fixed distance 2(rB-rA).

Now A sends a light signal up to B where it reflected by a mirror and returns down to A. He notes that the round trip time for the light signal is less than the 2(rB-rA)/c. Observer A also comes to two equally viable alternative conclusions:

(3) The speed of light is constant everywhere and the "real" distance rB-rA is less than that calculated by assuming the relationship R=Circumference/2pi.

(4) The speed of light speeds up as it climbs out of a gravity well and takes less time to travel the fixed distance 2(rB-rA).

Note that conclusions (1) and (3) contradict each other, while conclusions (3) and (4) are compatible.

Observers A and B conduct some more experiments measuring the redshift of signals and measuring the radial distance directly with rulers and pool all their information to reach the following conclusions:

(5) The coordinate speed of light progressively slows down by Gamma^2 as it falls and speeds up by Gamma^2 as it climbs.

(6) Clocks progressively slow down by a factor of Gamma, deeper in a gravitational well.

(7) Rulers progressively length contract more by Gamma the deeper you go in a gravitational well.

(8) The combined effect of conclusions (6) and (7) is that the local speed of light is always measured to be c.

(9) The Euclidean radius obtained from circumference/2pi is a true radius but the length contraction of rulers can make it appear that the radial distance does not satisfy the Euclidean relationship. The timed two way vertical speed of light is not consistent with the radial distance being greater than circumference/2pi.

====================================================================

A.T. posted this nice visulisation of the relationship of velocity, acceleration and curvature to length contraction here: http://www.adamtoons.de/physics/relativity.swf that might be worth having a look at.

 

[mysearch]

Response to #51

Kev,
Thank for both replies, both were very helpful and I will respond to both via separate posts starting with #51. I think we have broad agreement with the generalised spacetime diagram, i.e.

o e1 & e2 are both time-like and causally connected to e(0,0).
o e3 is space-like and not causally connected with e(0,0)
o e3, e3’ and e3’’ is a path moving through spacetime with respect to e(0,0)
o e3’ can become causally connected with an event (light beam) emitted from e(0,0) at some point in the future
o e3’’ can collide with e(0, t) at some time in the future.
o E4 is another space-like event, which if stationary with respect to e(0,0) will remain space-like and simply maintain the same position through time.

There is always an ambiguity of which observer is moving relative to the other in relativity

I know why you have highlighted this, but I would like to raise another example and ask your opinion. However, I will do this in a separate posting outlining a `triplet paradox`

Equally, as you have provided a good example of the caveats raised in your subdequent post ( #52), it makes sense to respond to all your comments in my response to #52. However, as you have highlighted, some of the caveats raised have relevance to the Bell paradox, which I am still trying to understand. In many ways, these questions are simply trying to confirm/reject some initial assumptions before addressing this paradox head on.

 

[mysearch]

Response to #52

Kev,
Thanks for the example; I find it really helpful to have something tangible to reference. All to often, standard ‘explanations` simply regurgitate pages of mathematical derivations and buzzwords, when a simply example would be far more helpful. The link to the simulation was also very useful, but I am not sure whether I am totally in synch with its results – see footnote as this is not the focus of my response.

Your basic example:

Let A be an observer on the surface of a very dense gravitational body and B is an observer far out in space. They agree on their radiuses rA and rB as defined in by assuming R=circumference/2pi.B sends a light signal down to A which is reflected from a mirror on the surface and returns to B. He notes that the round trip time for the light signal is greater than the 2(rB-rA)/c. Now B can draw two equally viable conclusions:

Now the first conclusion/assumption seems reasonable enough starting point, but I would consider the issues a little further.

(2) The speed of light slows down as it falls and takes longer to travel the fixed distance 2(rB-rA).

When we discussed `gravitational redshift`, we are actually referring to the effects on a photon’s frequency as it moves out of the gravity well. If the photon is falling into a gravity well, the photon is said to undergo blueshift. This suggests that the effects on the photon are asymmetric in terms of its direction in the gravitational field. In terms of the Newtonian view, the photon would lose energy to the gravity field when moving away from mass [M], but gain energy when falling towards it. On this basis, would you not expect the effects of the velocity of light to be asymmetric?

Now normally, the assumption is that the velocity has to remain constant, because the media defines the velocity of a wave and energy defines the frequency (E=hf). However, if the gravitational field affects the permittivity and permeability of vacuum, then it might be possible for the velocity of light to change. However, if this were the case, would this not have knock-on implications on the frequency [f] as [c=f\lambda]?

Now A sends a light signal up to B where it reflected by a mirror and returns down to A. He notes that the round trip time for the light signal is less than the 2(rB-rA)/c. Observer A also comes to two equally viable alternative conclusions:

Again, there is the asymmetric/symmetric issue. It is my understanding that gravity causes a curvature of space. This curvature means that the geodesic path is ever-greater than coordinate-radius defined by r=c/2\pi. Based on simple geometry, one might assume that A-B = B-A, which implies the round trip distance is the same. However, there is the complication of the direction of gravity that transposes to acceleration towards and away from the mass [M]. Therefore, I am not sure whether A-B = B-A?

(4) The speed of light speeds up as it climbs out of a gravity well and takes less time to travel the fixed distance 2(rB-rA).

At one level this makes sense, as the curvature flattens when moving away from [M], therefore [c] appears to cover more coordinate-r in unit time. However, I not quite sure how to reconcile this view with the energy associated with the photon, i.e. photons lose (kinetic) energy to the gravitational (potential) field?

(5) The coordinate speed of light progressively slows down by Gamma^2 as it falls and speeds up by Gamma^2 as it climbs.

Certainly, the solution of the Schwarzschild metric for the distant observer supports this statement, although it is predicated on the assumption that local time for the photon has stopped and distance has contracted to zero. The premise of this standard assumption might be considered comparable to some of the physical ambiguities said to occur at the event horizon. However, statement (6) is the logical extrapolation of the previous assumption and apparently supported by experiments related to time dilation. I raise these points not to be argumentative, but simply to highlight where the `limits of inference`, separating fact from hypothesis, may exist.

(7) Rulers progressively length contract more by Gamma the deeper you go in a gravitational well.

Again, I am raising points for clarification, rather than forwarding my own speculative hypothesis. When length contraction is discussed in the context of velocity, the observer perceives all objects on board a moving spaceship to contract in the direction of motion, including local rulers. In a sense, the motion distorts the perception of everything including atoms. As such, the moving observer perceives no change and the speed of light remains invariant, i.e. c= x/t = x’/t’.

The implication of (7) is that ruler onboard a spaceship moving into a stronger gravity well also contracts, which then maintains the constancy of [c] by the same logic as above. However, the path that the spaceship is moving along is expanding in the radial direction in comparison to the coordinate-radius. We seem to have 2 different mechanisms, the first affects physical objects within the moving space, while the second only expands the perception of space, but not the objects within it?

It is probably best fore me to timeout at this point.

Footnote
For example, I set the initial velocity to 0, gravity=1(?) and it gave proper time = 0.79s after 1 sec and length contraction = 0.71 for a free-falling object? If free-falling from infinity, relativistic effects of gravity and velocity would be synchronised. However, I haven’t really checked the figures because I was too sure of the gravity units?

 

[mysearch]

Triplet paradox?

This `paradox` is raised in respond to a comment raised in #51.

There is always an ambiguity of which observer is moving relative to the other in relativity

The standard twin paradox appears to be based on the assumption that everything in an inertial frame is relative. As such, two twins moving relative to each other could argue that the other is aging less. It is my understanding that this paradox is normally resolved by showing that only one twin undergoes acceleration and therefore has a high relative velocity, at least, with reference to his other twin. However, there still appears to be the ambiguity that in some other wider frame of reference, the traveling twin could have actually been decelerating to a lower speed.

The triplet variant is just an extension of the twin paradox. One triplet stays on Earth, while the other 2 take identical journeys at the same relative speed as each other with respect to the stay-at-home triplet, but always in the opposite direction, i.e.

Triplet-1: A
Triplet-2: A-B-A-C-A
Triplet-3: A-C-A-B-A

So the 2 traveling triplets move out at relativistic velocity, turn around and return past (A) without stopping. At (A), the triplets all pass each other moving at different relativistic velocities, which implies that time dilation should be affecting the physical age of each triplet at different rates. So the question is:

What is the relative age of each triplet at the end of the journey?

 

[yuiop]

Response to triplet paradox.

This `paradox` is raised in respond to a comment raised in #51.



The standard twin paradox appears to be based on the assumption that everything in an inertial frame is relative. As such, two twins moving relative to each other could argue that the other is aging less. It is my understanding that this paradox is normally resolved by showing that only one twin undergoes acceleration and therefore has a high relative velocity, at least, with reference to his other twin. However, there still appears to be the ambiguity that in some other wider frame of reference, the traveling twin could have actually been decelerating to a lower speed.

The triplet variant is just an extension of the twin paradox. One triplet stays on Earth, while the other 2 take identical journeys at the same relative speed as each other with respect to the stay-at-home triplet, but always in the opposite direction, i.e.

Triplet-1: A
Triplet-2: A-B-A-C-A
Triplet-3: A-C-A-B-A

So the 2 traveling triplets move out at relativistic velocity, turn around and return past (A) without stopping. At (A), the triplets all pass each other moving at different relativistic velocities, which implies that time dilation should be affecting the physical age of each triplet at different rates. So the question is:

What is the relative age of each triplet at the end of the journey?

Assuming vB=vC=0.8c and distance dB=dC and proper times denoted as tA, tB and tC for A,B and C respectively then tB=0.6 tA and tC=0.6 tA.

If distances dB and dC and velocities vB and vC are not the same then the proper time experienced by B will be tB=tA*sqrt(1-(vB)^2) and the proper time experienced by C will be tC=tA*sqrt(1-(vC)^2). tC can also be expressed as tC=tB*sqrt(tA^2-4dC^2)/sqrt(tA^2-4dB^2).

There are no ambiguities for the relative proper times experienced when all observers start at the same point and return to the same point. The same is not true for spatially separated observers with relative motion.

 

[yuiop]

Now normally, the assumption is that the velocity has to remain constant, because the media defines the velocity of a wave and energy defines the frequency (E=hf). However, if the gravitational field affects the permittivity and permeability of vacuum, then it might be possible for the velocity of light to change. However, if this were the case, would this not have knock-on implications on the frequency [f] as [c=f\lambda]?

If you look at this table I posted way back in post #11

Consider the following example. There are 3 observers (A,B,C) at various heights in a gravity well. According to an observer at infinity the gravitational gamma factor g= 1/ \sqrt{1-Rs/R} is 8, 4 and 2 for A,B and C respectively with A being the deepest in the well. Say a photon is emitted with a frequency of 1 and wavelength of 1 at location A as measured by A.

The measurements of (frequency, wavelength, speed of light) made by the observer (D) at infinity at locations A, B, C, and D would be:

A' = (1/8, 1/8, 1/64)
B' = (1/8, 1/2, 1/32)
C' = (1/8, 2, 1/4)
D' = (1/8, 8, 1)

and the local measurements would be:

A = (1, 1, 1)
B = (1/2, 2, 1)
C = (1/4, 4, 1)
D = (1/8, 8, 1)

...

then you can see that the relationship f\lambda = c is maintained for all measurements A,B,C,D and A',B',C',D'. The local observers measure the frequency as reducing as the photon climbs upward while the the distant observer says that is an artifact of there gravitationally time dilated clocks. For example observer D can ask observer C to flash a timing signal for every second that passes on C's clock. D will see C's timing signals as arriving at a rate of 2 per second and concludes that C's clock is running faster than his own clock. Knowing that he could conclude that C sees the photon frequency as reuced because C is using a fast clock. Conversely for a falling photon C could say that the D sees the frequency as increased because D is using a slow clock.

Basically, in relativity there is generally no one single explanation for a given set of events and usually depends on the observers point of view which relativity considers to be equally valid even if they are seemingly contradictory and hence the title of this post.

At the risk of causing more confusion I will introduce another example:P

Imagine in a set of convenient units that Earth and Mars are separated by a distance of 4 lightyears as measured in the Earth-Mars rest frame. Jane travels from Earth to Mars at 0.8c and the journey takes her 3 years to get there as measured by her own clock. Jane concludes the Earth-Mars distance has length contracted to 2.4 lightyears in agreement with her journey time and velocity. Bob who remains on Earth says the Earth-Mars distance never contracted just because Jane decided to take a trip there, that the journey actually took Jane 5 years which he can prove with clocks on Earth and Mars that are synchronised with each other and that Jane has reached a false conclusion because she timed the journey with a slow clock. Jane counters that Bob's conclusion that the Earth-Mars distance did not not length contract is an artifact of the fact that the clocks on Earth and Mars are not synchronised (from her point of view) and that the clocks on Earth and Mars are running slow relative to her clocks.

Did the Earth-Mars distance really length contract? Who's clocks are really running slower. The truth is no one really knows and neither position can be proved. Relativity offers no single explanation for a set of events as seen by two different observers with realtive motion. Basically Relativity is saying "Here is set of equations that predicts what a given observer will measure and that is all that matters scientifically. Explanations are in the realm of philosophy." Different observers will make different measurements of a set of events and will have a different explanation for the set of events and all points of view are equally valid. No one is wrong or right, they just have different points of view. We discussed this in an earlier example where two spaceships of equal length when at rest with respect to each other will each consider the other ship to shorter than their own ship when they have relative motion. Each is absolutely correct to say their own ship is longer than the other ship in Special Relativity, even though these views seem contradictory.

If you want a "physical image" of what is happening then you have to look towards something like Lorentz Aether Theory which which is the same mathematically as Special Relativity but puts everything in a different philosophical and more physical context.

Again, there is the asymmetric/symmetric issue. It is my understanding that gravity causes a curvature of space. This curvature means that the geodesic path is ever-greater than coordinate-radius defined by r=c/2\pi. Based on simple geometry, one might assume that A-B = B-A, which implies the round trip distance is the same. However, there is the complication of the direction of gravity that transposes to acceleration towards and away from the mass [M]. Therefore, I am not sure whether A-B = B-A?

Coordinate geometry using r=c/2\pi says A-B=B-A.
Local measurement using rulers says A-B=B-A but the distance is greater than r=c/2\pi
Measurements sending light signals say the distance ABA is greater than BAB where A is higher if the speed of light is assumed to be constant.
Measurements sending light signals say the distance ABA=BAB if the speed of light varies according to c'=c/Gamma^2 and that by making this assumption then ABA=BAB=2(A-B)=-2(B-A)=(A-B)-(B-A).

So only the conclusion reached by assuming the speed of light is constant everywhere in a gravitational field gives an asymmetric result that ABA>BAB.

...
The implication of (7) is that ruler onboard a spaceship moving into a stronger gravity well also contracts, which then maintains the constancy of [c] by the same logic as above. However, the path that the spaceship is moving along is expanding in the radial direction in comparison to the coordinate-radius. We seem to have 2 different mechanisms, the first affects physical objects within the moving space, while the second only expands the perception of space, but not the objects within it?...

Its the "relativity of explanations" again ;)

 

[mysearch]

Triplet paradox: Response to #56

Hi Kev,
Started to have a look at your thread entitled `Impure twin’s paradox`. There appears to be a very knowledgeable debate going on about various aspects of this paradox. While I liked the quote in #10:

The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts.--Bertrand Russell

….unfortunately, idiots also have doubts and the wise can be full of themselves. So the majority are still left with the problem of deciding whom to believe. Post #20 raised a valid point about checking the conclusions of relativity with calculation:

Since it is entirely possible that no-one will be bothered to actually go through the calculations themselves, here is my working.

Unfortunately, the case for calculation was subsequently weaken in #27:

I made errors in post #20 in this thread (and some others, but #20 was probably worst). I hereby retract that post (and would delete it, but it is too late).

This is not intended as a snide remark, I am simply highlighting the fact that while calculations can appear logical they are not always right. However, I was particular interested in the link given in #26.

http://www.sysmatrix.net/~kavs/kjs/addend4.html

While I have not yet checked this example, although I intend to, it would appear that this practical explanation comes to the key conclusion that 2 observer in relative motion can perceive the clocks in another frame to be running slower, while at the end of the journey, only 1 is actually older:

By either reckoning, Terra's Earthbound clock aged the greater, and by the same amount... so there is no disparity. And more importantly, it corroborates the core concept of Relativity, that any observer can rightly claim his vantage to be stock still: it's the other guy who's moving.

Please correct me if my interpretation is wrong, as it seems to be the central issue of confusion for most people trying to understand relativity. Anyway, back to the triplet variant of this paradox. Based on your figures and equations, it appears that you are saying that the 2 ‘travelling` triplets, i.e. B & C, will be the same age at the end of the journey, if the distances, speeds and accelerations are all equal, albeit opposite in directions. I am also assuming that you agree that triplets B & C will be younger than the stay-at-home triplet (A).

Now before seeing the reference in Impure twin’s paradox #26 I would have concluded that B & C must maintain the same relative age for the entire journey, not just at the end, as the relativistic parameters are always identical, except for the direction. As such, I find it very difficult to envisage any situation where they could past each other and perceive time onboard the other spaceship running slower. In fact, I still do, but I guess I need to work through the example in #26 before clinging onto my intuitive assumptions.

Didn’t have much time today, so will try to respond to #57 tomorrow.

 

[yuiop]

While I have not yet checked this example, although I intend to, it would appear that this practical explanation comes to the key conclusion that 2 observer in relative motion can perceive the clocks in another frame to be running slower, while at the end of the journey, only 1 is actually older:

This illustrates that what is perceived about a clock moving relative to you does not necessarily reflect the reality. Each perceives the other's clcok to running slower but when they come together they find that is not true for both of them. In fact you can not say anything with certainty about a distant moving clock moving relative to you until that clock is at rest with you.

Please correct me if my interpretation is wrong, as it seems to be the central issue of confusion for most people trying to understand relativity. Anyway, back to the triplet variant of this paradox. Based on your figures and equations, it appears that you are saying that the 2 ‘travelling` triplets, i.e. B & C, will be the same age at the end of the journey, if the distances, speeds and accelerations are all equal, albeit opposite in directions. I am also assuming that you agree that triplets B & C will be younger than the stay-at-home triplet (A).

Now before seeing the reference in Impure twin’s paradox #26 I would have concluded that B & C must maintain the same relative age for the entire journey, not just at the end, as the relativistic parameters are always identical, except for the direction. As such, I find it very difficult to envisage any situation where they could past each other and perceive time onboard the other spaceship running slower. In fact, I still do, but I guess I need to work through the example in #26 before clinging onto my intuitive assumptions.

Didn’t have much time today, so will try to respond to #57 tomorrow.

If when B and C return to Earth they do not stop but continue with constant velocity they will notice that the elapsed times of their clocks are the same but they will also say the instantaneous rate of each other's clocks is slower than their own. I think we have discussed before the subtle difference between elapsed time and instantaneous clock rate.

Also both will agree that that the elapsed time on their clocks is less than the elapsed time on the clock of the observer that stayed on Earth. That is what I meant when I said in post#56 "Assuming vB=vC=0.8c and distance dB=dC and proper times denoted as tA, tB and tC for A,B and C respectively then tB=0.6 tA and tC=0.6 tA.". Perhaps I should have made it clear that tA is the elapsed time on A's clock and the elapsed times on B and C's clock will each be 60% of A's elapsed time.

Yet another example: A remains stationary while B accelerates away, slows down, speeds up again etc. There is absolutely nothing certain you can say about the real elapsed times on clocks A and B relative to each other until they come to rest with respect to each other. In other words you can not say with certainty who is really ageing faster when two observers have relative motion until they come to rest with respect to each other. Trying to do so is trying to introduce a notion of absolute time that does not work in relativity.

 

[granpa]

In other words you can not say with certainty who is really ageing faster when two observers have relative motion until they come to rest with respect to each other. Trying to do so is trying to introduce a notion of absolute time that does not work in relativity.

not sure what you are trying to say. what do you mean 'really' aging? it is a trivial matter to determine how fast a given clock is ticking in any given frame . it depends only on the relative velocity. its couldn't be simpler.