RE: C. S. Morrison and Siegfried Bleher, RE: [Sadhu Sanga] Physics and qualia
Vasavada, Kashyap V
Dear Colin and Siegfried,
Siegfried: Thanks for exhibiting the actual equations, rather than stating in just words.
Colin : Sigfried’s calculation makes it clear that the smaller the period for acceleration and deceleration, the closer the final result will be to the SR formulas (2nd term in his equation). e.g
If accelerating and decelerating periods are say 0.05 instead of ¼ and constant velocity period is 0.9 (total time),
using Siegfried’s math I get for SR T= 1.67 T(0) and for mixed SR and accelerations T= 1.62 T(0). These are for v/c =0.8. For larger v/c time dilation will be much more. Also for extremely small periods of accelerations and decelerations, the answers will be essentially same as the SR results.
So the lesson is that the results of your imaginary experiment with A,B,C will depend on the actual data, accelerating and decelerating time for each , fraction of the total time for constant velocity, and the numerical values of velocities. Then we can put in numbers. If you double the speed of C so that he comes back in the same time to earth as B, C’s time dilation will be much more. If you give them the same velocities then taus will be different for the calculation.
Sigfried, good to hear from a fellow Maryland graduate. My guess is that you are much younger than me. I got my Ph.D in prehistoric time, 1964 in theoretical high energy physics.!! When did you graduate and what was your field of Ph.D. thesis?
In a previous e-mail, I asked you the same question about quantum mechanics as I asked Serge. Because of tons of e-mails, you may have missed it. If you cannot find it, I will send it again.
With Best Regards
Kashyap
From: online_sa...@googlegroups.com [mailto:online_sa...@googlegroups.com]
On Behalf Of Siegfried Bleher
Sent: Sunday, June 11, 2017 1:32 PM
To: Online_Sa...@googlegroups.com
Subject: RE: C. S. Morrison, RE: [Sadhu Sanga] Physics and qualia
Dear Colin and Kashyap,
I would add the following small modification to Kashyap’s answer regarding the twin paradox and the issue of acceleration. This is, in particular, in response to Colin’s intuition, given his example of the triplets,
that the relative velocity between earth and the rocket is the source of the main part of the difference in the twin’s ages, notwithstanding the influence of the rocket’s acceleration and deceleration. The way to account for the difference in age of each
member of the triplet is to integrate the proper time
for each of them, where
is the spacetime metric tensor. If we consider only the rocket (no large masses
around), then the metric tensor is the Minkowski metric
for flat spacetime. If we let
t be the time of the trip as measured by someone on earth, and
is the time as measured by the traveling twin (or triplet), then we have (from the expression for proper time above)
This includes acceleration and deceleration phases of the rocket, as well as any phase of constant velocity. To make things easy to calculate, suppose the rocket attains a fraction
of the speed of light
during a constant acceleration phase which, as observed from the rocket, occurs for the time
Assume the deceleration phase of the trip is also constant, and takes the same amount of time, so the total outward-bound time from rocket’s frame would be
where
is the time spent at constant velocity
Integrating the equation above gives the outward-bound time as
The total trip time is
(in the earth frame of reference). Smaller acceleration implies longer
(and shorter
), larger acceleration implies shorter
(and longer
). For example, suppose the rocket accelerates during ¼ of the outbound trip, travels at constant velocity equal to
for ½, and decelerates for the remaining ¼ of the outbound trip. Then
, and
So the total trip time from the earth’s perspective would take 2.826 times the time in rocket frame. My apologies for including the math explicitly, but I want
to point out there is time dilation during each phase of trip, whether the rocket is accelerating, decelerating or moving with constant velocity. There is, as Kashyap points out, acceleration to distinguish between the two frames of reference of earth and
rocket. But this would not account for the (presumably) longer time interval of the trip during which the rocket is traveling at constant velocity and accumulating the larger part of its time dilation. The thing that resolves the twin paradox in my mind is
how each measures distance. Distance as measured in the earth frame (i.e. the proper length) is measured between the earth and the distant destination, say planet B, when these are assumed stationary relative to each other and their locations are measured
simultaneously. On the other hand, the rocket is moving away from earth and toward planet B. So, whereas the two reference points are stationary in the earth frame, they are not stationary in the rocket frame, so in the rocket’s point of view the distance
to the destination is contracted (Lorentz contraction), and it takes him far less time to travel there than his twin on earth thinks. In fact, he (in the rocket) does think earth’s clocks are running slow compared with his, but the reason he thinks so is
that he has traveled much farther to earth’s perspective in between clock ticks on earth than he measures in his own frame. Now, if we want to include the influence of earth’s gravitational pull, or other large bodies between us and planet B, we would have
to integrate the proper time equation with an appropriate function for the metric tensor that accounts for the presence of these masses. The influence of other forces is also calculable, in principle, from Einstein’s equation, if we include these other forces
on the energy-momentum side of the equation. We would then have to derive the appropriate metric tensor from Einstein’s equation, and proceed to the proper time equation above.
Best wishes,
Siegfried
PS Kashyap, my PhD in physics also comes from University of Maryland!
From:
online_sa...@googlegroups.com [mailto:online_sa...@googlegroups.com]
On Behalf Of Vasavada, Kashyap V
Sent: Saturday, June 10, 2017 11:32 PM
To: C. S. Morrison <cs...@hotmail.co.uk>
Cc: Online_Sa...@googlegroups.com
Subject: RE: C. S. Morrison, RE: [Sadhu Sanga] Physics and qualia
Hi Colin,
First a correction: I applied the GPS case (with correct numbers) to the astronaut case without thinking. After sending the e-mail, I realized that the orbits are completely different. GPS satellites are at approx. distance of 16500 mi from earth’s center and International space station is approximately 4200 mi from earth’s center. Both SR and GR time dilations depend sensitively on distances. So in the astronaut case the GR effect is smaller than SR effect. The net result is indeed that the space twin is about 0.01 sec younger than earth twin in 1 year. My apologies to the young astronauts! The documentary you saw was ok!
Now about other matters. Question of other forces is quite interesting , but frankly nobody knows. Forces other than gravity have different dependence on space-time than gravity according to the current models and for all we know they may not distort space time. Only with a grand unified theory we will be able to tell. String theory has some kind of unification built in, but it is extremely difficult to calculate.
At every instant the satellite is moving tangentially to the orbit. So it is ok to apply SR time dilation equation with tangential velocity. The GR equation already knows about centripetal acceleration by equivalence . So it has no problem in applying gravitational time dilation.
Long time back, I saw a calculation of twin paradox with accelerations. I remember that the conclusion was that you get the same answer as naïve SR time dilation. If I find that calculation, I will let you know.
Generally when they study decaying particles, the particles are not rotating in circular accelerators any more, but rather are flying towards the detector. The rotating particles like protons and electrons are stable. So only SR is enough.
For twins the main aging is taking place during travel. For this period the travelling (accelerating) twin is certainly not in an
Inertial frame. When he comes back and the two compare their clocks then only they are in the same approximately inertial frame.
For C to come back in the same time as B , its velocity will have to be doubled and consequently accelerations will also have to be doubled. Thus the answer and time dilations will depend on the details of B’s and C’s travel.
Best Regards.
Kashyap
From: C. S. Morrison [mailto:cs...@hotmail.co.uk]
Sent: Saturday, June 10, 2017 11:53 AM
To: Vasavada, Kashyap V <vasa...@iupui.edu>;
Online_Sa...@googlegroups.com
Subject: RE: C. S. Morrison, RE: [Sadhu Sanga] Physics and qualia
Hi Kashyap
Thanks for putting numbers on the astronaut case. Your conclusions, which are in line with what I thought, seem quite at odds with those of the documentary (if I am remembering it correctly) which I think said the astronauts age less than us!
Anyway, I totally agree that special relativistic time dilation will apply to satellite data and the decay rates of cosmic rays and other fast-moving particles. That is because when we observe such things we are looking at the moving clock. The twins paradox seems different because when they compare their ages they are in the same inertial frame. It would be like firing off a stream of radioactive particles with half-lives so long that relatively few decay in the process round the LHC for a while, then slowing them down and capturing them in some material and seeing what effect this journey had on their half-life.
As far as I can see the change in their half-life compared to that of an unmoved sample of those particles must be due to the accelerations they underwent rather than the periods of constant motion, no matter how short and sharp those accelerations were. In the LHC they would be getting centripetally accelerated all the time, so how can special relativity apply at all in such cases?
Here's what I am wondering. Instead of twins, let us consider identical triplets A, B and C. A stays at home, B travels to the nearest star and back, and C travels to a star twice as far away and also comes back undergoing exactly the same accelerations as B. I understand that B and C will both be younger than A. But will C be younger than B or will they be the same age?
Thanks,
Colin
Send from Huawei Y360
On 10 Jun 2017 14:57, "Vasavada, Kashyap V" <vasa...@iupui.edu> wrote:
Dear Colin.
Dear Colin,
First a non-controversial astronaut in an orbit case: When orbiting, there are both special relativistic and general relativistic effects. In fact this is very important for GPS! Since orbiting clock is moving with respect to the stationary clock on earth, special relativity says that the orbiting clock will slow down by 7 microseconds per day. But the orbiting clock is in a weaker gravitational field with respect to earth’s clock. So it will be faster by 45 microseconds/day. So the net effect is that the orbital clock will go faster than earth’s clock by 38 microseconds/day. This correction has to be made by engineers working on GPS, otherwise the GPS would be off by some 10 Km/day! There is a story that engineers while building GPS were reluctant to put in this relativistic factors in their calculations and they put them at the last moment!! Normally people do not think much for relativistic corrections because speeds with which we are concerned are much smaller than speed of light. Also earth’s gravity is quite weak. But this is a striking example of importance of relativity. As for our astronauts , they will unfortunately come back little older than us. They appear to be in bad shape when they come back any way!!!
Now the slightly controversial case of twins (only interpretation, not the actual effect). The actual effects are verified every day in high energy physics labs and cosmic rays studies in connection with decay of particles. For twins, people have done calculations both ways and get the same answer as the straight forward naïve special relativistic effect! This is no big surprise since after all special and general relativity have to be consistent. Personally I prefer the following argument. Calculations done in inertial, non-accelerating frames, are correct. So the twin in accelerating rocket cannot conclude that his stay at home brother is younger. You can also argue that accelerations can be done for as short time and distance as you like depending on your rocket power. So that should not have a major effect. On the other hand since we believe in GR, it should and does give the same result by taking actual accelerations into account.
Best regards.
kashyap
From: C. S. Morrison [mailto:cs...@hotmail.co.uk]
Sent: Saturday, June 10, 2017 7:32 AM
To: Vasavada, Kashyap V <vasa...@iupui.edu>;
Online_Sa...@googlegroups.com
Subject: Re: C. S. Morrison, RE: [Sadhu Sanga] Physics and qualia
Thanks Kashyap,
That is what I had been thinking myself. I was beginning to doubt this view as a result of certain posts on this list.
It seems therefore that any form of acceleration or deceleration must cause time dilation (to the same extent as the effect of gravity in GR). Am I right? After all, the time dilation due to the periods of constant relative velocity of the two twins is purely relative. Each twin would observe that the other's clocks were going slower. Hence the only actual slowing of the second twin's clock relative to the first twin's clock must take place during the periods of acceleration and deceleration that the second twin undergoes - no matter what force these are caused by.
That seems to me to imply that other forces do in a sense affect the curvature of spacetime just as gravity does. Is that not why Einstein was so keen to try (unsuccessfully as it turned out) to model the effects of other forces in terms of spacetime curvature?
Here's a related query. I heard on one of Jim Al Khalili's TV documentaries that time goes slightly slower for the astronauts on the space station. I would have thought it went slightly quicker due to the fact that they were several hundred miles further from the centre of earth's gravitational field. However, the program said no because the astronaut's far greater velocity relative to us makes it go slower. My question is: Shouldn't these astronauts observing us through their telescopes receding at tremendous velocity calculate that our time is going slower relative to them? And as we are deeper in the gravitational field, we should also be aging slightly less than them. As far as I can see, if they do genuinely age slightly slower than us, that extension of youth could only be due to the tremendous accelerations and decelerations required to get them there and back. Is that true?
Best wishes,
Colin
Send from Huawei Y360
On 10 Jun 2017 09:46, "Vasavada, Kashyap V" <vasa...@iupui.edu> wrote:
Dear Collin,
I read the following question addressed to Siegfried several days back. I have not seen Siegfried’s answer to that. It is possible he did not get to answering it or that he answered it and I missed it! In any case, since you also mentioned my name, let me try!
The twin paradox you are talking about arises in special relativity (constant speed). The acceleration directly does not come into the picture except that to return he has to reduce his speed to zero (by deceleration, negative acceleration) and then accelerate in the same direction he came from. You can imagine that he is far away from any star so that direct influence of gravity is zero. ( But In general relativity acceleration is equal to gravity anyway. So that either way you will get the same result.) The negative and positive accelerations are applied only to show that the twin staying home and the one travelling are not equivalent. The twin paradox is that if the rocket is moving with speed v with respect to earth, the earth is receding with speed –v with respect to rocket. You can describe physics in any frame. So someone may argue that the stay home twin should remain younger. But the accelerations makes them inequivalent and removes the so called paradox. I trust this will answer your query.
Best Regards.
Kashyap
From:
online_sa...@googlegroups.com [mailto:online_sa...@googlegroups.com]
On Behalf Of C. S. Morrison
Sent: Saturday, May 27, 2017 11:07 AM
To: Siegfried Bleher <sbl...@msn.com>;
Online_Sa...@googlegroups.com
Subject: RE: [Sadhu Sanga] Physics and qualia
Dear Siegfried
I would be really grateful if you or Kashyap could elucidate a related issue that I have with relativity theory. It concerns the famous twins paradox where one twin racing away from the earth close to light speed will perceive the other twins 'clocks' going slower. And the twin on earth looking through a powerful telescope will perceive the space twin's clock's also going slower by the same amount due to special relativity. I have always understood that when the space twin swings round a nearby star and returns to earth he will arrive back somewhat younger than his sibling due essentially to the effects of general relativity caused by the accelerations he undergoes.
My question is: What if he turns round by some means that does not involve gravity? What if his rocket still has enough fuel to slow him down and blast him back in the opposite direction? Will he still arrive back younger than his sibling? And if so does that not mean there is an element of spacetime curvature associated with any acceleration (even those experienced by particles in the centrifuge you have been discussing)?
Best wishes,
Colin
C. S. Morrison - Author of THE BLIND MINDMAKER: Explaining Consciousness without Magic or Misrepresentation.
https://www.amazon.com/Blind-Mindmaker-Explaining-Consciousness-Misrepresentation/dp/1541283953
https://www.amazon.co.uk/Blind-Mindmaker-Explaining-Consciousness-Misrepresentation/dp/1541283953
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On 27 May 2017 02:18, Siegfried Bleher <sbl...@msn.com> wrote:
Dear Vasavada (and Norm),
Nice response regarding special and general relativity and centrifuge. Reiterating that the high ‘g-forces’ inside the centrifuge are provided by the walls of the centrifuge, not by curved spacetime, I would add that, although the spacetime is not curved as it would be by a large mass yielding similar forces, there is still some minimal curvature due to the net mass-energy present in the spinning centrifuge.
Best wishes,
Siegfried
Sent from my Windows 10 phone
From: Vasavada, Kashyap V
Sent: Friday, May 26, 2017 4:42 PM
To: Online_Sa...@googlegroups.com
Subject: RE: [Sadhu Sanga] Physics and qualia
Hi Norm,
Some of the questions you asked John, Bruno and Chris, are straight physics questions, having nothing to do with consciousness etc. So I am hoping all of you would not mind if I jump in the conversation as a physicist.
The special and general theory of relativity of Einstein have been known for some 100 years and have passed critical experimental tests hundreds of times. Do you know your GPS depends on these calculations for accurate predictions of your location? So most physicists would agree that by and large they are correct. That does not of course mean that there are no difficult issues remaining. In particular merging them with quantum theory raises thorny issues. Also, in the case of origin of universe and in dealing with black holes, some outstanding problems remain.
Now about your question on centrifuge: First a little refresher on freshman physics. Whether in centrifuge or when you are on a rotating merry-go-round or on a car rounding a curve, no real force is pulling you out. When your seat is rotating, you are required to have a force towards the center called centripetal force. Otherwise Newton’s law would require you to go straight tangentially! Because of lack of sufficient inward force, you have a feeling that you are being pushed out. You interpret that as a (fictitious, not real) centrifugal force! In the case of centrifuges the bottom or walls of tubes exert an inward force on the liquid which results in several ‘g’s depending on the speed. In fact Einstein concluded from this example and the other example of an elevator accelerating up and your feeling lot heavier that acceleration of reference frames and gravity must be equivalent. This is called the principal of equivalence. It was actually starting point of his general theory of relativity which became theory of gravity.
When you try to express a highly mathematical theory like general theory of relativity in terms of ordinary non mathematical human languages some issues surely come up. But most physicists accept the language that in presence of a mass, surrounding space, which was flat before, becomes curved. The space has many properties as if you are on a surface of a sphere and not on a flat plane.
Any one is welcome to have his/her own opinions. There should not be any censorship. But if you say theory of relativity is totally wrong, you will have hard time convincing overwhelming number of physicists!
From:
online_sa...@googlegroups.com [mailto:online_sa...@googlegroups.com]
On Behalf Of Norm Silliman
Sent: Thursday, May 25, 2017 9:34 PM
To: Online_Sa...@googlegroups.com
Subject: Re: [Sadhu Sanga] Physics and qualia
To John, Bruno and Chris,
John said;
1. First, we are writing electronic posts off the cuff on perhaps the most fundamental subject in science. There is little technical detail shared that could change minds. Mostly we end up sharing personal opinions.
How about an observation from a heretic.
Science / Physics studies have powered the past two industrial revolutions. Micheal Faraday gave us
his books that described how magnetism causes electricity [and versa versa] about 1840. This triggered
the telegraph revolution in America.
Nikola Tesla gave us alternating- current (which helped to light up the world) between 1890 and 1910.
Faraday's work gave the world about 300 patents: Tesla's work work gave the world about
500 patents.
Albert Einstein gave us the Theory of relativity. He worked on his theory for about 40 years.
Einstein's work gave the world about 1 discovery in the field of physics.
The work of Faraday and Tesla was during the time of the prevailing scientific understanding was
light and forces were transmitted with the help of the 'all prevailing aether', and little or no mathematical
proofs of theories.
The work of Einstein was during the time of the prevailing scientific understanding was
light and forces were transmitted by photons, and mathematical proofs required of all theories.
My pet peeve is Einstein's gravity:
Einstein's general theory of relativity explains
gravity as a
distortion of space (or more precisely, spacetime) caused by the presence of matter
or energy. A
massive object generates a gravitational field by warping the geometry of the surrounding spacetime.
Google's quick look
What distorts space around or inside a centrifuge that allows a centrifuge to generate a force of
several hundred 'gravity's'? [to make it worse, this force field if pointed out!]
Just asking...
Norm
On 5/24/2017 9:47 PM, john.kineman wrote:
Hello Bruno and Chris,
I've been looking for a good entry point into this discussion of physics, qualia, and mechanism. I made some foundational arguments in earlier quotes which I think are not generally accepted, or at least not evidently so in the responses and ongoing defense of mechanism as a potentially holistic paradigm. I want to continue to argue why I am convinced it is not a viable candidate, and I can see that Bruno has the opposite view. So the question is, what kind of evidence would be convincing either way? Otherwise, we are just spending time writing words that make our worlds comfortable.
If we can agree to address the question in a critical fashion, in which we look for the key deciding factor, this forum could make some significant progress. So, I will make some points here about what that criterion should be, and why I think it argues for a realist view of qualia and a qualified acceptance of mechanism. I'll number so the points can be easily referenced.
1. First, we are writing electronic posts off the cuff on perhaps the most fundamental subject in science. There is little technical detail shared that could change minds. Mostly we end up sharing personal opinions. This should be resisted as much as possible, and we should cite criterion - completely aside from citing entire papers or books everyone should read in order to be swayed into a given philosophical camp (although I am guilty of advocating that myself, with Rosen's work - even here a cheap plug).
2. Let's next recognize that such deep questions are a discussion of world view and epistemology. Thus we need to know the epistemological criteria being used to defend one position or another.
3.We cannot reduce the key arguments to a positivist rejectability because world views are not rejectable in that way. They are the foundations of thought, not the rejectable outputs of thought. For example, I have eyes so I describe and explain my world in terms of what I see. I assume the visible and then propose theories in those terms. I can test the theory against other things I can see. If I were in the ultimate movie theatre of the future, however, I might never discover the flaw in my starting assumption, that the sights are real. And I might never know about the projector or the screen because what is stated in those terms, assumes those terms as reality. However, there are still some criteria, that if carefully applied, could tell me in this theater that something is wrong with my world view and I could begin to try other ideas about reality, testing them by these criteria.
4. The only criteria that we can't use in this test of world view, is the one we would like to use and are comfortable using when testing theories WITHIN a world view. We would like to compare predictions of the theory to actual data. But the meta-theory that reality is composed of the images I see in this hypothetical theater does not predict which images should appear next, it is the generator of theories about how those images are related to each other, assuming they are real. The derived theory then makes testable predictions and we can accept or reject the theories, but not the foundation on that evidence. So, what can be used?
5. There are epistemological requirements we more or less have agreed to in science and philosophy, even spirituality, when examined. I'll list these. There are six that I am aware of from the best known philosophers of science (which may not include a lot of Eastern philosophers, but so far I've found that at this level the criteria are in agreement). In no special order:
On Tuesday, May 23, 2017 at 12:10:27 AM UTC+5:30, marchal wrote:Hi John,
Apology for late answer.
On 15 May 2017, at 17:49, Edwards, Jonathan wrote:
On 15 May 2017, at 12:21, Bruno Marchal <mar...@ulb.ac.be> wrote:
The human explanation of general relativity requires chalk and blackboard, but relativity does not rely conceptually on chalk's and blackboard's existence. ---A good sign because chalk and blackboard are disappearing :)
Nobody said it did, did they? It requires observations, which are grounded in qualia.
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Siegfried Bleher
Dear Kashyap,
It seems I do not have the email message you are referring to—I don’t believe I am getting all of them—is it related to the question about measurement in QM you sent Serge on June 7th?
My thesis was on dynamics in nonlinear Hamiltonian systems, in 1989. Since then I have focused on quantized versions of classically nonlinear systems (and recently complex systems), and for 25 years now, yoga and yoga philosophy.
I recently made it back to UMD for a Dynamics Days conference—nice to see friends and advisors from long ago! Dieter Brill, Jogesh Pati and O. W. Greenberg were among the faculty I got to study with. Did you know them?
Best wishes,
Siegfried
From: online_sa...@googlegroups.com [mailto:online_sa...@googlegroups.com]
On Behalf Of Vasavada, Kashyap V
Sent: Sunday, June 11, 2017 9:02 PM
To: Online_Sa...@googlegroups.com
Subject: RE: C. S. Morrison and Siegfried Bleher, RE: [Sadhu Sanga] Physics and qualia
Dear Colin and Siegfried,
Siegfried: Thanks for exhibiting the actual equations, rather than stating in just words.
Colin : Sigfried’s calculation makes it clear that the smaller the period for acceleration and deceleration, the closer the final result will be to the SR formulas (2nd term in his equation). e.g
If accelerating and decelerating periods are say 0.05 instead of ¼ and constant velocity period is 0.9 (total time),
using Siegfried’s math I get for SR T= 1.67 T(0) and for mixed SR and accelerations T= 1.62 T(0). These are for v/c =0.8. For larger v/c time dilation will be much more. Also for extremely small periods of accelerations and decelerations, the answers will be essentially same as the SR results.
So the lesson is that the results of your imaginary experiment with A,B,C will depend on the actual data, accelerating and decelerating time for each , fraction of the total time for constant velocity, and the numerical values of velocities. Then we can put in numbers. If you double the speed of C so that he comes back in the same time to earth as B, C’s time dilation will be much more. If you give them the same velocities then taus will be different for the calculation.
Sigfried, good to hear from a fellow Maryland graduate. My guess is that you are much younger than me. I got my Ph.D in prehistoric time, 1964 in theoretical high energy physics.!! When did you graduate and what was your field of Ph.D. thesis?
In a previous e-mail, I asked you the same question about quantum mechanics as I asked Serge. Because of tons of e-mails, you may have missed it. If you cannot find it, I will send it again.
With Best Regards
Kashyap
From:
online_sa...@googlegroups.com [mailto:online_sa...@googlegroups.com]
On Behalf Of Siegfried Bleher
Sent: Sunday, June 11, 2017 1:32 PM
To: Online_Sa...@googlegroups.com
Subject: RE: C. S. Morrison, RE: [Sadhu Sanga] Physics and qualia
Dear Colin and Kashyap,
I would add the following small modification to Kashyap’s answer regarding the twin paradox and the issue of acceleration. This is, in particular, in response to Colin’s intuition, given his example of the triplets, that the relative velocity
between earth and the rocket is the source of the main part of the difference in the twin’s ages, notwithstanding the influence of the rocket’s acceleration and deceleration. The way to account for the difference in age of each member of the triplet is to
integrate the proper time
for each of them, where
is the spacetime metric tensor. If we consider only the rocket (no large
masses around), then the metric tensor is the Minkowski metric
for flat spacetime. If we let
t be the time of the trip as measured by someone on earth, and
is the time as measured by the traveling twin (or triplet), then we have
(from the expression for proper time above)
This includes acceleration and deceleration phases of the rocket, as well as any phase of constant velocity. To make things easy to calculate, suppose the rocket attains a fraction
of the speed of light
during a constant acceleration phase which, as observed from the rocket,
occurs for the time
Assume the deceleration phase of the trip is also constant, and takes
the same amount of time, so the total outward-bound time from rocket’s frame would be
where
is the time spent at constant velocity
Integrating the equation above gives the outward-bound time as
The total trip time is
(in the earth frame of reference). Smaller acceleration implies longer
(and shorter
), larger acceleration implies shorter
(and longer
). For example, suppose the rocket accelerates during ¼ of the outbound
trip, travels at constant velocity equal to
for ½, and decelerates for the remaining ¼ of the outbound trip. Then
, and
So the total trip time from the earth’s perspective would take 2.826
times the time in rocket frame. My apologies for including the math explicitly, but I want to point out there is time dilation during each phase of trip, whether the rocket is accelerating, decelerating or moving with constant velocity. There is, as Kashyap
points out, acceleration to distinguish between the two frames of reference of earth and rocket. But this would not account for the (presumably) longer time interval of the trip during which the rocket is traveling at constant velocity and accumulating the
larger part of its time dilation. The thing that resolves the twin paradox in my mind is how each measures distance. Distance as measured in the earth frame (i.e. the proper length) is measured between the earth and the distant destination, say planet B,
when these are assumed stationary relative to each other and their locations are measured simultaneously. On the other hand, the rocket is moving away from earth and toward planet B. So, whereas the two reference points are stationary in the earth frame,
they are not stationary in the rocket frame, so in the rocket’s point of view the distance to the destination is contracted (Lorentz contraction), and it takes him far less time to travel there than his twin on earth thinks. In fact, he (in the rocket) does
think earth’s clocks are running slow compared with his, but the reason he thinks so is that he has traveled much farther to earth’s perspective in between clock ticks on earth than he measures in his own frame. Now, if we want to include the influence of
earth’s gravitational pull, or other large bodies between us and planet B, we would have to integrate the proper time equation with an appropriate function for the metric tensor that accounts for the presence of these masses. The influence of other forces
is also calculable, in principle, from Einstein’s equation, if we include these other forces on the energy-momentum side of the equation. We would then have to derive the appropriate metric tensor from Einstein’s equation, and proceed to the proper time equation
above.
.
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