# Gravitational Redshift

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

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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

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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.

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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.

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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.

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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).

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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?

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.

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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.

Gold Member
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.

Gold Member
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.

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?

Gold Member

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 travelling 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 travelling 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?

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 travelling 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 travelling 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.

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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)

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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 everthing 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.

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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 ;)

Gold Member

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.

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.

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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.

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 couldnt be simpler.

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 couldnt be simpler.

I am refering to the classic twins paradox where both twins measure each other's clocks to be ticking slower, but when they get back together they realise that only one has really been ticking (ageing) slower because one returns younger than the other. So basically they both determined each other's clocks to be ticking slower than their own but in fact one of them was not "really" ticking slower.

There are enough ongoing threads on the twin's paradox in this forum, that there is no need to discuss it in detail in this thread.

each sees the other do exactly what it would expect it to do. its only a paradox to those that dont understand how drastic an effect loss of simultaneity has. not sure why you think that means that one cant determine its rate.

by its 'real' rate you seem to be implying its absolute rate but then you say that such an absolute rate cant be introduced. if you dont mean the absolute rate then you must mean the relative rate and that is easily determined.

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If you want a link to the speed of light change due to acceleration or gravity I can give you one. Also, based on the varying speed of light and Euclidian geometry, I have posted a new relativistic gravity law. The speed of light within the gravitationl field quantifies the change in total mass. The ratio of the speed of light at the end location to the start location is equal to the total mass ratio between the start location and the end location, so that in fact the frequency doesn't change within the gravitational field.

granpa,

the increase in time caused by the twin paradox is quantified by allowing the speed of light to change in an accelerating field. It turns out that the spatial gradient of the speed of light equals the force per unit mass experienced by the accelerating twin divided by the speed of light.

Gold Member
General Response to #59, 50, 61, 62

OK, I understand the argument that every triplet is in its own frame of spacetime, but it would seem that you could, at least, rationalise these different perspectives by mapping all of them into a conceptual frame of reference.

In the triplet paradox there is no ambiguity about the start/stop points in spacetime and therefore it would seems to provide be a reasonable definition of a ‘conceptual’ frame in this case. The triplets A, B & C all wear wristwatches that allow them to measure their own local ‘proper time. As far as I can see, the watches of the 2 triplets in motion, e.g. B & C, always tick at the same rate, which is slower than A.

Footnote:
From #60: how fast a given clock is ticking in any given frame. it depends only on the relative velocity. its couldnt be simpler.
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The reason for extending the twin paradox was to compensate for velocity & acceleration as both B & C undergo identical journeys - only the direction differs. As such, I am assuming the equations of special relativity given identical answers.

While the example given in the Impure Twin Paradox thread#26 may mathematically resolve the ambiguity of time, i.e. time in B & C appearing to run slower from the other perspective, this description seems only to add to the confusion. Certainly, I cannot picture B and C both aging less/more than the other at the same time in any meaningful way.

Regarding the issue of space contraction. In order for the speed of light to be invariant, B & C perceive the distance covered to be less than A, although identical to each other. Of course, at the end of the journey, B & C must reconcile the distance with A. Again, this can be done by B & C recognising they were the ones moving at a higher relative velocity with respect to A and the conceptual frame of reference chosen.

While I am making no inference between the conceptual frame and any absolute frame, it would seem that the conceptual frame could be extended to any size, e.g. sun in the solar system or galactic centre in the Milky Way etc. So while the premise that all frames of reference are equally valid may be right, it seems that most paradoxes only arise when we swap between these frames. Therefore, would it not be sensible, from a teaching perspective, to first establish a more intuitive frame of reference and then subsequently discuss the relative ‘distortions’ that lead to perceptual paradoxes?

However, I would like to hear whether there are any major objections to this approach from people who clearly know more about this subject than me.

Gold Member
Quick Reponses to #63, #64

If you want a link to the speed of light change due to acceleration or gravity I can give you one

Yes, I would be interested. Forgive my ignorance, but where does this idea sit within the mainstream view of relativity?

mysearch

I have given a link in one or 2 previous posts on different threads, so you can just browse my last posts till you find it, because I think it is really against the rules for me to give links. As far as the speed of light change, I don't really know if it could be considered mainstream or not, but I think the answer may be not. It doesn't matter, because in the end it boils down to mathematics. It is always possible to view an accelerating observer from a non-accelerating reference frame. The accelerating observer will encounter photons or particles, and the relative velocity at the time of encounter determines what the accelerating twin experiences. It is possible to use a varying speed of light to satisfy this requirement.

As far as a varying speed of light in a gravitational field (your post #44) you must distinguish between the total mass and the frequency. The rest mass is multiplied by the Lorentz factor to give the total mass. The lorentz factor quantifies spatial frequency (ie wave crests per distance) not time frequency. To account for time frequency the Lorentz factor is multiplied by the speed of light. This is an important distinction when using a varying speed of light. In a gravitational field the spatial frequency changes, but this is compensated for by a change in the speed of light, so the time frequency does not change.

Gold Member
Response to #57: Part 1 of 2

This first part essentially introduces an example and its purpose, while the second part tries to provide a physical interpretation of the relativistic effects due to gravity.

Quote from #57
If you want a "physical image" of what is happening then you have to look towards something like Lorentz Aether Theory which is the same mathematically as Special Relativity but puts everything in a different philosophical and more physical context.

It is true that I am looking for a physical interpretation of relativity, but I do not see why sticking with the standard model should preclude this goal. Therefore, I have deliberately set up the following example in order to highlight some physical interpretations and to see what objections are raised. The example is only considering the relative implications of gravity between 2 stationary observers, i.e. no relativistic velocity complications. The attached diagram (ex68.jpg) represents our 2 observers (A) & (B), where (A) is deep within the gravity well, i.e. $$\gamma=8$$, while (B) is effectively infinitely removed from the central mass [M].

Note: the values of $$[\gamma]$$ are chosen to be illustrative of the effects on spacetime rather than necessarily being realistic.

o In Figure-1a, we see the photon path BAB. The photon emitted at (B) is ‘red’ but then blueshifted on the path B-A, while on the return; the photon is emitted as blue but then redshifted on the path A-B. The implication being that the frequency shift is essentially symmetric on the paths B-A and A-B.

o In Figure-1b, we see the photon path ABA. The photon emitted at (A) is ‘blue’ but then redshifted on the path A-B, while on the return; the photon is emitted as red but then blueshifted on the path B-A. Again, the implication being that the frequency shift is symmetric on the path A-B and B-A.

The inference of $$[x_1..x_7]$$ is simply that multiple observers can exist between A-B corresponding to each value of $$\gamma$$. Each observer determines the value of [c=1] based on their own local measurements of distance and time [t], such that [c=s/t]. However, as reflected in Figure-2, the measurement of distance at each point needs to be physically interpreted, as does time, if the value of [c] is also to convey some physical meaning. This said, the local speed of light is always assumed to conform to:

 $$c = s_B/t_B = s_A/t_A = 1$$

However, it is clear that the value of [c] determined by each local observer will depend on the value of and [t] inserted. While this may seem obvious, what is not always obvious is what the observer is actually measuring. As a very general statement of relativistic effects, elapsed time can either be shorter or longer, as can distance. As such, there are 4 permutations and therefore possible answers to equation , although not all may be physically meaningful.

Figure-2 reflects standard theory by illustrating that [ds] will expand relative to [dr] due the geodesic curvature of space as [Rs] is approached. The diagram is reflective of the range of $$[\gamma=1..8]$$ used within the example. As such, [ds] is 8x greater than [dr] at (A), but whether the observer at (A) physically realises this aspect is tabled as an open question. Likewise, theory also predicts that time dilates as [Rs] is approach. As such, 1 unit of elapsed time in (B) would equate to 8 units in (A) given the relative values of $$[\gamma]$$.

While we are familiar with the relativistic curve ranging from unity to infinity as [Rs] is approached, it is useful to note that relative time and distance only change linearly with $$[\gamma]$$. So, with reference to figures 1a and 1b, each segment of the path AB or BA that corresponds to a unit change in $$[\gamma]$$ will also cause a proportional change in time or distance. However, it is again highlighted that this may not be apparent to the local observer. Therefore, it is suggested that any physical interpretation needs to consider 2 frames of reference:

o Local Observer
o Conceptual Observer

The conceptual observer has the advantage of perceiving both [dr, ds] and $$[dt, d\tau]$$. The physical interpretation is now taken up in part-2.

#### Attachments

Gold Member
Response to #57: Part 2 of 2

Part-1 outlines the example, including reference diagrams, and its purpose; therefore it is the logical starting point of this discussion. Part-2 now considers what, if any, physical interpretations can be drawn. As previously introduced, the local speed of light is always assumed to conform to:

 $$c = s_B/t_B = s_A/t_A = 1$$

If we consider equation  with respect to observer (B), who is conceptually at infinity, i.e. no spacetime curvature, where $$[\gamma=1]$$, we might consider [c] expressed in unit distant and time producing a unit value of [c]:

 $$c = s_B/t_B = 1/1 = 1$$

However, the suggestion is that if these unit measures were at [A] with $$[\gamma=8]$$, equation  would become:

 $$c = s_A/t_A = 8/8 = 1$$

However, the values inserted in  are only perceived from the conceptual frame and not by the local observer at (A). Locally, at (A), time $$t_A$$ is still considered as unit time, as the physical perception is that the rate of time is still 1 second per second. As such, the local values of equation  would be:

 $$c = s_A/t_A = 1/1 = 1$$

In sense, we are seeing the constancy of the speed of light in both the local frame and the conceptual frame, which might be generally written as either:

 $$c = dr/d\tau = 1$$ Local frame: min-r/min-t
 $$c = ds/dt = 1$$ Conceptual frame: max-r/max-t

What equation  implies is that if [c=1] in all local frames, then the local (proper) time is always dilated and the local perception of radial distance has also to be the coordinate-radius [dr] not [ds].

Interpretation?
The local observer doesn’t directly perceive the effects of relativity and therefore always measures the minimum value of space [dr] and time $$d\tau$$

In contrast, the conceptual frame corresponding to equation  is referencing the maximum values, [ds] and [dt]. However, both appear to support the constant speed of light [c=1] based on equations  and . At this point, we might try to imply some physical interpretation to the frequency of the photon emitted and absorbed at any point. The following interpretation is forwarded on the assumption that modern relativity replaces the description of Newtonian gravity as a force with the concept of spacetime geometry. As such, gravitational redshift has to be physically interpreted in terms of time or space changes.

Interpretation?
When emitted from (A), the photon was said to be blue, i.e. high frequency, but this was only with respect to the local dilated time at (A). As the photon moves toward (B), the relative rate of time ticks faster, redshifting the frequency of the photon. In contrast, a photon emitted at (B) was said to be red with respect to its local time, so as the photon moves toward (A) the frequency is blueshifted.

While I understand the maths that have led to the conclusion outlined in the following quote, it is suggested this conclusion can be also be interpreted from the conceptual frame to be physically more meaningful.

From #57: 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.

With reference to the figures in the attachment in part-1, it is difficult to conceive that ABA > BAB, as to the photon(s), it represents the same physical path. Physically , the only thing that changes is the relative rate of time perceived by the 2 observers in (A) and (B) who may be determining the distance based on their local assumption that [c=1] and that the roundtrip elapsed time is either $$[t_A or t_B]$$. It is clear that if [c=1] and ABA equals BAB, then the local observers would have to agree on the roundtrip time, which they do not given the relative time dilation. However, it will be suggested that we can use the conceptual frame to get a more meaningful resolution.

If we start with observer (B) and figure-1a, with each segment corresponding to $$[\gamma=1..8]$$, we may also considered each segment to correspond to a unit distance traversed in unit time, such that:

 $$c = s_{BAB}/t_{BAB} = 16/16 = 1$$

However, we know that the observer at (A) must measure the roundtrip time against its local proper time, which the conceptual observer can see is related by:

 $$t_A = t_B*\gamma_A$$

However, locally, observer (A) must still determine [c=1] therefore:

 $$c_A = S_{ABA}/ t_B*\gamma_A = S_{BAB}/t_B = c_B = 1$$

If so, this suggests:

 $$S_{ABA} = S_{BAB}*\gamma_A$$

 $$S_{ABA} > S_{BAB}$$

This seems to confirm the conclusion reached in #57. However, how else might we physical interpret equation ?

Interpretation?
(A) and (B) disagree on the length ABA and BAB, not because the photon actually traversed 2 paths of different length, but rather the BAB has been calculated in terms of [dr], while ABA has been calculated in terms of [ds]. Therefore, from the conceptual frame, the suggestion is that ABA = BAB.

Finally, to be clear on one point, the conceptual frame is not inferring that any absolute frame exists; it is simply intended as a possible learning aid. While mathematically theorist may reject this approach, the level of debate over the meaning of relativity, in this forum alone, suggests that mathematical derivation alone is still subject to many different physical interpretations. As always, would appreciate any constructive feedback.

Hi mysearch,

..your conceptual reference frame seems reasonable. It seems to be the same as what I have been calling the coordinate measurements of the observer at infinity (observer D) Where I have been stating things like D says the speed of light at A is c/64 it is obvious that D can not make direct measurements of observations at A but by performing logical deductions of all the measurements made at A,B,C and D and builds up a conceptual image that seems to fit all the facts, that he then plots on a coordinate chart. References to the coordinate distance, time and speed of light are references to the conceptual reference frame you describe.

P.S
By the way I have you seen the thread I started on Bell's paradox here : https://www.physicsforums.com/showthread.php?t=236681 . Does it help?

P.P.S
Are you ready to go exploring event horizons yet?

Gold Member
Quick Response to #70

Guess the purpose of #68/#69 was to see if we could converge to some physical interpretation of the implication of relativity. Wasn’t sure whether you would agree with the ABA = BAB conclusion and that ABA > BAB was really an apple and oranges comparison.

Yes I had seen the Bell Paradox thread and was glad that you had raised it as a separate issue, because previously it seemed to be buried in other non-obvious threads. Unfortunately, I haven’t had too much time over the weekend to get into the details, but will add to the thread as soon as possible.

However, I did post a question in your Impure Twin Paradox thread. Hopefully, while you take a look at this, it will give me some time to get my ‘event horizon-hiking boots` on. Although I am not sure, as yet, whether I have yet got all the right equipment. 