Special Relativity - the Twin problem

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provo Visible Light Wave USA 515 Posts	Posted - 02/22/2003 : 04:14:38 The so called Twin paradox of SR has made a numbetr of debuts on these forums - the most common explanation of the age difference is based on the turn-around acceleration forces that take the problem out of SR - but consider the consequences if the traveling twin rides aboard a rocket ship that is aimed precisely close to a distant planet so that it will be slingshot-ed back toward earth by the gravitational field thereof (like a highly elliptical orbit of a comet) - here the traveling twin is in free-fall during the entire journey (while there is some change in velocity, it is small compared to stopping and turning around)- the point of difference lies in the fact that as the traveling twin approaches the gravitatinal field of the distant planet - he feels no acceleration - while he is nearly captured thereby, he is redirected back toward the earth with just enough velocity to complete the journey. When he arrives at the earth - he is younger that his stay at home sibling. Does gravitational redirection change anything?? Alert Mentor now
emu Radio Wave Canada 87 Posts	Posted - 02/22/2003 : 04:27:48 Since they both take off and return to Earth, this is your reference frame. Calculate relative velocities for each twin relative to earth. Where's the beef? Alert Mentor now
provo Visible Light Wave USA 515 Posts	Posted - 02/22/2003 : 08:24:35 If the stay at home twin doesn't move with respect to the earth - and the traveling twin is always in a zero G field - one cannot use the traditional rationale that the actual age difference is the result of turn-around acceleration. We can simplify the situation even more by considering one twin traveling an ellipical orbit and the other twin remains at the aphelion point - the twins set their clocks as the traveling twin passes by - when the traveling twin returns to the same point - do the clocks read the same time? - if no - then is time dilation real or apparent (the traveling twin is in a zero G field during the entire orbit)? Alert Mentor now
Hurkyl Visible Light Wave USA 723 Posts	Posted - 02/22/2003 : 16:22:59 Of course, the paradox part of the twin paradox relies on the use of the special relativistic time dilation formulas to produce a contradiction, those formulas are no longer valid in general relativity! I'm not as familiar with GR as I am with SR, but I'll take a crack at it. There is now a clear asymmetry in the problem; I believe it's possible for each twin to identify that the space-bound twin is the one that is in the gravity well of the distant planet, so it's doubtful that you could construct a paradox. So without any acceleration, the problem is whether the spacebound twin's time in the distant planet's gravitational well slows his clock more or less than the earth's gravitatinal well slows the earthbound; i.e. no simple answer. Hurkyl Alert Mentor now
Erich Schoedl Infrared Wave USA 222 Posts	Posted - 02/22/2003 : 20:04:51 The time dilation of the spacebound clock would be actual, not apparent. The space-bound clock would slow by a velocity relative to the earth bound clock (assuming small movement of earth over the span of the trip). The clock would also slow by GR as it were closer to the source of gravity (ie the sun). Using simple Schwarzschild solution, and SR, you could find out exactly how much slower a clock would be on a particular orbit around the sun and back to Earth. In that case the speeds and gravity would be small enough to make the first approximations of simple gravitational redshift dependent on radius from source. That is a good example of how proper acceleration can't thwart the twin paradox (of course it cheats by using GR). It is the average speed relative to a given reference frame that matters. In addition to the redshift induced by GR. Most real experiments would have to account for some GR effects like these, and current theory should give the right answers without any paradox. Alert Mentor now
provo Visible Light Wave USA 515 Posts	Posted - 02/23/2003 : 03:59:08 Erich - Hurkyl ... You are double teaming me again - but your posts are quite welcome. I don't know why SR problems keep poppin in my noggen - but It is instructive to get your feedback. Relativity is a sensitive subject - there are many who resent hypos posed by an upstart crow with the temerity to interogate what is considered to be irrefutable - so much for that - let me address your posts...as best I can after a little libation. Hurkyl - I don't know if I am absolutely correct on this, but from other postings I have read it seems that there is not a dual (double) effect due to time dilation gravity well plus orbital velocity - from what I have read the dilation is one or the other - in other words, in a centrifuge the amount of time dilation can be calculated from the formulas of SR (i.e., the velocity difference) or they can be calculated from the effective acceleration that would correspond to a equivalent gravitation well - but not both. This has led to a lot of speculation that time dilation in SR and GR are due in some manner to the same phenomena - e.g., motion with respect to space .. The ether types are quick to conclude that since the G field corresponds to an escape velocity at any particular orbit of interest, then space effectively flows inwardly toward masses. While the numbers correlate, I am not endorsing the theory - I site it only as example in connection with the fact that the gravitation potential translates to a corresponding escape velocity - which is what we measure due to gravity. So, without reaching any conclusions - it would seem that we can consider (maybe) only the velocity attribute of the travling twin in determining his clock rate and consequently determine the amount of dilation therefrom . Erich - if the actual dilation is actual (as you have said) and as I believe it could be - how do we reconcile this with the symetrical result demanded by SR for relatively moving reference frames - if we look at how the GR time dilation formula is derived - it depends upon both the receiver observer and the source transmitter being in the same acceleration frame. Here the stay at home twin, albeit subject to the earths local G field, is not in the accelerating frame of the elliptical orbit twin - so the derivation upon which the GR formula depends is inapplicable - and since there is no local G field which the travelling twin can detect - is time dilation inilateral? Alert Mentor now
Hurkyl Visible Light Wave USA 723 Posts	Posted - 02/23/2003 : 17:33:21 quote: there are many who resent hypos posed by an upstart crow with the temerity to interogate what is considered to be irrefutable You merely interrogate; you aren't trying to dictate with an iron fist that relativity is wrong, so we don't mind (and even encourage) your questions. Calculations of time dilation boil down to computing the derivative: d/dt where measures proper time and t measures coordinate-time. Casting everything into GR's perspective, the SR formula for time dilation is simply the flat space solution to that derivative. Time dilation, technically, isn't a composition of multiple dilation factors. Specifically, the SR formula follows from the flat metric: c² d² = c² dt² - dx² - dy² - dz² If you divide through by dt², you get: c² d² / dt² = c² - dx² / dt² - dy² / dt² - dz² / dt² And by the definition of classical velocity, this is the same as: c² d² / dt² = c² - v² And then solving yields: d / dt = sqrt(1 - (v / c)²) Different metrics (such as a metric where the acceleration in a centrifuge is gravitational) would simply yield different equations that have different solutions, there's no mathematical reason to think that SR time dilation should be a factor. However! In the "weak field", (i.e. GR looks like Newtonion Gravity) there is a coordinate chart where the metric is given by: c² d² = (1 + 2U / c²) (c² dt² - dx² - dy² - dz²) Where U is the Newtonian gravitational potential energy. If you do the same work you do above, you get: c² d² / dt² = (1 + 2U / c²) (c² - dx² / dt² - dy² / dt² - dz² / dt²) d² / dt² = (1 + 2U / c²) (1 - v² / c²) d / dt = sqrt(1 + 2U / c²) * sqrt(1 - (v / c)²) d / dt = (1 + U / c²) * sqrt(1 - (v / c)²) (This last step is done by noting that U is small WRT c, so this approximation for square roots can be used) So in this weak field approximation, we see that time dilation is governed by two factors; one that is equivalent to SR time dilation, and one generated by gravitational potential (note that U is always negative). One usually models something like the earth system or the solar system by weak field models, so it makes sense to say that the net time dilation is the product of the individual factors... but in general we cannot make this decomposition. Hurkyl Alert Mentor now Edited by - Hurkyl on 02/23/2003 17:53:14
provo Visible Light Wave USA 515 Posts	Posted - 02/24/2003 : 10:57:46 Hurkyl - Yes - I agree that the two aspects can be treated from an overall energy perspective - which may be the key to understanding the underlying nature of both SR and GR time dilation. In most practical problems as you know, the two factors are treated separately and algebrically combined - for example in the GPS satellites the gravitational potential is reduced by the altitude (radius 26.6 Km) so that the primary correction to the clocks is due to GR (+45,900 ns/day) and the velocity correction is -7200 ns/day. If there is no correction due to acceleration as experiments seem to confirm (only gravitational potential and velocity) then one could introduce an on board clock with the appropriate GPS corrections for a circular orbit referenced to an earth centered system and the difference between the on board GPS clock and an on board uncorrected clock which ran at the rate of an earth clock could be used to measure actual time dilation on the spacecraft - taking it a step further - if the GPS clock is set to reflect the average speed of a long journey (ignoring the gravity effect at the start) then there is a ready made test of the actuality of time dilation which doesn't depend upon a return - or if as proposed there is no time dilation assocated with acceleration, but only velocity, then one could conceivably carryout the twin experiment with an actual day to day reading being transmitted back to earth as to the time lost by the GPS clock in comparison with the on board uncorrected clock - and of course if a turn around is carried out at the same velocity as the crusing speed - the rate of the clocks upon executing this manuver should not be affected. This method avoids the often cited objection that each twin sees the other siblings clock running slow - but those comments arise from the fact that the usual experiment is carried out using signals transmitted from twins which are receding from one another - i.e., they are in reality only detecting Doppler info and not actual time dilation - whereas in the on board experiment with two clocks the transmitter is functioning in an entirely different capacity - informing the earth receiver what it is witnessing on board. Alert Mentor now
Hurkyl Visible Light Wave USA 723 Posts	Posted - 02/24/2003 : 11:50:57 One usually assumes in all of these theoretical arguments that the doppler effect is corrected for before everyone jumps into their calculations, so the actual time dilation is what they compute. Hurkyl Alert Mentor now
provo Visible Light Wave USA 515 Posts	Posted - 02/25/2003 : 03:59:42 It should be treated separately - but in usual zeal to explain time dilation by using signals transmitted between moving systems - it is often overlooked - and in fact the entire analysis gets confused by considering what the receiver observes given that the transmitter is clock synchronized to send pulses at equal intervals according to proper time increments measured by the local clock at rest relative to the frame of the transmitter - Hurkyl - your analysis based upon energy poses the question as to the mechanism that ties general relativistic time dilation to special relativistic time dilation (example - the GR slowing of a clock in elliptical orbit as it reaches perigee is equal to the SR slowing at the same point due to the deeper gravitational well - the two dilations are equal and additive). If both depend upon Potential energy rather than kinetic as being foundational - (as I previously mentioned, if SR time dilation can be equated to velocity with respect to space then the theories that attempt to explain the tie between SR and GR time dilation devolve into the spatial inrush hypothesis) - then a new clue is provided for a more plausible alternative to inrush. If a clock rate gets determined by its potential in GR - maybe it gets determined in SR by its KE... or by how much its PE has been converted into motion . I wonder if every object can be considered as having an intrinsic clock rate that grinds to zero at the two extremes - 1) when positioned at the event horizon of a BH - 2) when all of its negative PE plus Mc^2 rest energy exists in the form of c velocity KE - Alert Mentor now
Erich Schoedl Infrared Wave USA 222 Posts	Posted - 03/04/2003 : 03:49:32 Provo "Erich - if the actual dilation is actual (as you have said) and as I believe it could be - how do we reconcile this with the symetrical result demanded by SR for relatively moving reference frames - if we look at how the GR time dilation formula is derived - it depends upon both the receiver observer and the source transmitter being in the same acceleration frame. Here the stay at home twin, albeit subject to the earths local G field, is not in the accelerating frame of the elliptical orbit twin - so the derivation upon which the GR formula depends is inapplicable - and since there is no local G field which the travelling twin can detect - is time dilation inilateral?" Think of it this way instead. A traveler orbits a very powerful black hole at very near the event horizon. He feels no acceleration if the tidal forces are small and the diameter big. But he is aging slower than a far away observer because of the gravitational redshift. His high velocity only adds to this time dilation. Even though he doesn't feel accleration, he is in a very different GR frame. The same is for the clock in the eliptic orbit around the sun in your example. Alert Mentor now
flamethrower Infrared Wave USA 263 Posts	Posted - 03/04/2003 : 17:01:27 Provo the point of difference lies in the fact that as the traveling twin approaches the gravitatinal field of the distant planet - he feels no acceleration - while he is nearly captured thereby, he is redirected back toward the earth with just enough velocity to complete the journey To sling-shot around the planet IS acceleration. Centripetal force causes the ship to accelerate towards the center of the planet � the same as if you spun a rock on a string. As well, the ship is now moving on another worldline - the same worldline as if it had stopped, turned around and went back to earth. Yelling at referees helps your team win games. 'Cause they really like that. Ya Think? Alert Mentor now
russ_watters Radio Wave USA 53 Posts	Posted - 03/04/2003 : 20:50:02 quote: consider the consequences if the traveling twin rides aboard a rocket ship that is aimed precisely close to a distant planet so that it will be slingshot-ed back toward earth by the gravitational field thereof (like a highly elliptical orbit of a comet) - here the traveling twin is in free-fall during the entire journey (while there is some change in velocity, it is small compared to stopping and turning around)- the point of difference lies in the fact that as the traveling twin approaches the gravitatinal field of the distant planet - he feels no acceleration - while he is nearly captured thereby, he is redirected back toward the earth with just enough velocity to complete the journey. When he arrives at the earth - he is younger that his stay at home sibling. Does gravitational redirection change anything?? No, gravitational redirection does NOT change anything. The acceleration of the spacecraft is EXACTLY identcal as stated by conservation of momentum. HOW the velocity is reached is immaterial. The average acceleration is simply (V1-V2)/T. Same V1 and V2 means same acceleration. The astronaut doesn't FEEL the acceleration because of the way it is imparted, but nevertheless it is still there. Alert Mentor now
Hurkyl Visible Light Wave USA 723 Posts	Posted - 03/04/2003 : 22:26:23 quote: To sling-shot around the planet IS acceleration. Centripetal force causes the ship to accelerate towards the center of the planet � the same as if you spun a rock on a string. Not quite... When you're in free-fall around a planet, you remain on a geodesic path. In fact, there exists a (locally) inertial reference frame where your coordinate position remains constant for your entire journey. When you spin a rock on a string, you apply a force that causes it to deviate from a geodesic path, there is no (locally) inertial reference frame where the rock is stationary. Hurkyl Alert Mentor now
flamethrower Infrared Wave USA 263 Posts	Posted - 03/05/2003 : 08:06:42 Hurkyl If there is a change in speed or direction of motion, then there is a change in velocity, hence acceleration. Yelling at referees helps your team win games. 'Cause they really like that. Ya Think? Alert Mentor now
Hurkyl Visible Light Wave USA 723 Posts	Posted - 03/05/2003 : 11:52:20 quote: If there is a change in speed or direction of motion, then there is a change in velocity, hence acceleration. (According to GR) There are inertial frames where the slingshotting ship experiences no change in speed or direction of motion; it is stationary in this frame at all times, thus zero acceleration. No such inertial frame exists for a rock spinning around on a string. Hurkyl Alert Mentor now
flamethrower Infrared Wave USA 263 Posts	Posted - 03/05/2003 : 17:21:51 The slingshot maneuver is an orbital mechanics problem using Newtonian physics. In order for the ship to move in a curved path around the planet, there must be acceleration towards the center of the planet supplied by the gravitational pull of the planet. In the case of the slingshot effect, a change in speed and direction of motion is present. This is centripetal force. Yelling at referees helps your team win games. 'Cause they really like that. Ya Think? Alert Mentor now
Hurkyl Visible Light Wave USA 723 Posts	Posted - 03/05/2003 : 22:48:30 Newtonian physics isn't General Relativity. Hurkyl Alert Mentor now
flamethrower Infrared Wave USA 263 Posts	Posted - 03/05/2003 : 23:40:46 Newtonian physics isn't General Relativity. True. But Orbital Mechanics is not based on GR. Gravity assist orbits are problems in Orbital Mechanics � ie. Hohmann transfer orbits. http://hypertextbook.com/physics/mechanics/orbital-mechanics-2/ http://www.go.ednet.ns.ca/~larry/orbits/gravasst/gravasst.html http://scienceworld.wolfram.com/physics/HohmannTransferOrbit.html Yelling at referees helps your team win games. 'Cause they really like that. Ya Think? Alert Mentor now
Hurkyl Visible Light Wave USA 723 Posts	Posted - 03/06/2003 : 01:31:50 Just because orbital mechanics doesn't tend to use the full functionality of GR's equations doesn't mean that there isn't a fundamental difference in GR between a slingshotting rocket and a rock on a string, the difference being that the slingshotting rocket has an inertial reference frame and the rock doesn't. You don't use the rocket's inertial frame in orbital mechanics because the equations would be very complicated. [:P] Hurkyl Alert Mentor now
flamethrower Infrared Wave USA 263 Posts	Posted - 03/06/2003 : 05:49:08 If you twirl a rock around your head on a string the centripetal force acts on the rock and the outward force acts on your hand. Similarly, the centripetal force (gravity) acting on the ship causing it to slingshot around the planet is paired with outward force acting on the planet, not the ship! There is no fundamental difference. Yelling at referees helps your team win games. 'Cause they really like that. Ya Think? Alert Mentor now

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