Notion of Absolute Clock Synchronization in SR
in [Relativity]
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From: Tom Roberts on 31 Mar 2010 22:28 Paul B. Andersen wrote: > Be however aware that the ECI-frame - Earth Centred Inertial frame > is a misnomer in this context. It is no inertial frame because > space-time is curved, and the curvature is essential. So you cannot > use Einstein's synchronization method to define simultaneity > (except for between clocks on the same gravitational potential). Technically this is correct. But the GPS does use Einstein's synchronization method, for clocks on the ground and in orbit. It does so by deliberately using coordinate clocks that are compensated for their gravitational potential at their location. Real clocks that are moving (all of them except when at the poles) are also compensated for their motion. The ECI coordinates of the GPS are accurately those of an inertial frame of SR co-moving with the center of the earth; they are valid between the earth's surface and the GPS satellite orbits, plus some margin higher and lower. As the earth's center is not moving in a straight line, this puts a limit on the duration over which one can use these coordinates as inertial ones; fortunately that limit is considerably longer than the GPS needs. IOW: for short enough time periods, the ECI coordinates of the GPS are fully compensated for the curvature of spacetime, and they can be used as if SR is valid for them. For this to work, it is essential that the spacetime volume used be limited (in both space and time), and gravitation must be weak (near earth the gravitational potential is about 10^-6, which is indeed small [in GR it is unitless]). Tom Roberts
From: GSS on 2 Apr 2010 05:21 On Apr 1, 7:15 am, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote: > GSS wrote: >> On Mar 25, 10:42 pm, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote: >>> Clock synchronization is INHERENTLY frame dependent. That is, a given pair of >>> clocks can be synchronized in one AND ONLY one inertial frame; they are not >>> synchronized in any other inertial frame. > >> Agreed that a given pair of precision atomic clocks A and B (separated >> by constant distance S) can be synchronized in ONE AND ONLY ONE >> inertial reference frame. But is there any restriction on the choice >> of THAT inertial reference frame? > > No. Let us consider two clocks at points A and B fixed on earth's geoid (separated by constant distance S), and mutually synchronized to UTC through GPS. Let us assume the instantaneous velocity of both A and B is V, and their instantaneous positions P(A) and P(B), in the BCRF. Through established relations of relativity, we can transform the UTC time to BCRF time (say TDB). Since both points A and B are in motion in BCRF, their instantaneous times in BCRF will depend upon V, P(A), P(B) and the instantaneous orientation of line segment AB with velocity vector V. From such instantaneous computed BCRF times at point A and B, we can compute the instantaneous BCRF time difference between points A and B, which is the same as the synchronization offset (dT_b) between the two clocks A and B. But this instantaneous sync. offset cannot remain constant throughout the day due to the rotational motion of the earth about its axis. As the earth rotates, the instantaneous orientation of line segment AB with velocity vector V will keep changing, as sometime point A will be ahead of B in the direction of motion and after 12 hours, point B will be ahead of A in the direction of motion. As such the instantaneous synchronization offset (dT_b) will undergo a diurnal variation and oscillate between positive and negative values during the day cycle. Hence it is obvious that the synchronization of clocks A and B in BCRF done at one instant of time, will not remain valid at any other instant during the day cycle. This shows that two clocks A and B fixed on earth's geoid (separated by constant distance S), cannot be synchronized in BCRF because of 'their motion' in BCRF. Some reader had earlier pointed out that the two clocks A and B can be mutually synchronized only in an inertial (or practically regarded as inertial) reference frame in which they can be effectively *regarded at rest*. Do you agree? GSS > But it is most natural and easiest to synchronize them in the frame in which > they are at rest. Note that except for certain special cases the two clocks must > be at rest in THE SAME inertial frame. > > The rotating earth is such a special case, for synchronization > in the ECI frame. > >> Specifically, can we practically >> (not through gedankens) synchronize two clocks A and B in an inertial >> reference frame in which they are known to be in uniform MOTION (and >> not at rest)? > > Yes. The GPS does something similar. > >> [... synchronize in the BCRF] > > Synchronizing two earthbound clocks in the BCRF can be done in exactly the same > manner as they can be synchronized in the ECI. But there are difficulties: > A) There is no surface corresponding to earth's geoid, so you must > specify precisely where a standard clock displays the correct time. > Normally earth's geoid is used, even though it is moving in a rather > complex manner relative to the BCRF. > B) Gravitation of planets is important (the GPS can treat the sun, moon, > and planets as minuscule error terms; for the BCRF they are > considerably larger, and the fact that earth's geoid is rather low > in earth's gravity complicated matters). > C) There are no orbiting clocks to assist you (as the GPS satellites > do for the ECI). > D) The volume over which synchronization can be achieved is not obvious > to me; it surely extends to Pluto, and probably beyond, but cannot > possibly extend to the nearest star. > > Tom Roberts
From: Tom Roberts on 3 Apr 2010 12:12 GSS wrote: > This shows that two clocks A and B fixed on earth's geoid (separated > by constant distance S), cannot be synchronized in BCRF because of > 'their motion' in BCRF. Yes. For the approach of the GPS to work, each clock must have a constant speed in the ECI, and must have a constant gravitational potential. For the BCRF, clocks at rest on earth's geoid have neither constant speed wrt the BCRF nor constant gravitational potential (though in practice the latter could be dealt with as in the GPS). One could construct a device that when sitting at rest on earth's geoid always displays the current time of an imaginary coordinate clock of the BCRF at its current location. It would need to be able to observe its location and speed wrt the BCRF and adjust its displayed time accordingly. It would not qualify for the usual term "clock". > Some reader had earlier pointed out that the > two clocks A and B can be mutually synchronized only in an inertial > (or practically regarded as inertial) reference frame in which they > can be effectively *regarded at rest*. Do you agree? This depends on one's theoretical context. By saying "inertial frame" you imply the context is SR -- in SR one could set the offsets of two clocks such that they are synchronized in any inertial frame you choose. They would be synchronized with each other in that frame, but not with coordinate clocks of the selected frame (these two would "tick at a different rate" than those coordinate clocks). As soon as one introduces gravitation into the context, e.g. by considering the ECI or BCRF, then there are no inertial frames. There are only locally-inertial frames (e.g. the ECI). This complicates things considerably. Tom Roberts
From: Sue... on 3 Apr 2010 14:07 On Apr 3, 12:12 pm, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote: > GSS wrote: > > This shows that two clocks A and B fixed on earth's geoid (separated > > by constant distance S), cannot be synchronized in BCRF because of > > 'their motion' in BCRF. > > Yes. For the approach of the GPS to work, each clock must have a constant speed > in the ECI, and must have a constant gravitational potential. For the BCRF, > clocks at rest on earth's geoid have neither constant speed wrt the BCRF nor > constant gravitational potential (though in practice the latter could be dealt > with as in the GPS). > > One could construct a device that when sitting at rest on earth's geoid always > displays the current time of an imaginary coordinate clock of the BCRF at its > current location. It would need to be able to observe its location and speed wrt > the BCRF and adjust its displayed time accordingly. It would not qualify for the > usual term "clock". > > > Some reader had earlier pointed out that the > > two clocks A and B can be mutually synchronized only in an inertial > > (or practically regarded as inertial) reference frame in which they > > can be effectively *regarded at rest*. Do you agree? > > This depends on one's theoretical context. > ====================== > By saying "inertial frame" you imply the context is SR I did not infer that from the question. Can you offer an answer that does not assume Newton's light corpuscle moving under the influence of inertia? (if such thing existed energy density would not be necessary as a reference in GR) http://en.wikipedia.org/wiki/Einstein_synchronisation Common view over equal paths will certainly synchronise real clocks. Sue... -- in SR one could set > the offsets of two clocks such that they are synchronized in any inertial frame > you choose. They would be synchronized with each other in that frame, but not > with coordinate clocks of the selected frame (these two would "tick at a > different rate" than those coordinate clocks). > > As soon as one introduces gravitation into the context, e.g. by considering the > ECI or BCRF, then there are no inertial frames. There are only locally-inertial > frames (e.g. the ECI). This complicates things considerably. > > Tom Roberts
From: GSS on 5 Apr 2010 13:25
On Apr 3, 9:12 pm, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote: > GSS wrote: >> This shows that two clocks A and B fixed on earth's geoid (separated >> by constant distance S), cannot be synchronized in BCRF because of >> 'their motion' in BCRF. > > Yes. For the approach of the GPS to work, each clock must have a constant speed > in the ECI, and must have a constant gravitational potential. For the BCRF, > clocks at rest on earth's geoid have neither constant speed wrt the BCRF nor > constant gravitational potential (though in practice the latter could be dealt > with as in the GPS). > ..... >> Some reader had earlier pointed out that the >> two clocks A and B can be mutually synchronized only in an inertial >> (or practically regarded as inertial) reference frame in which they >> can be effectively *regarded at rest*. Do you agree? > > This depends on one's theoretical context. > > By saying "inertial frame" you imply the context is SR -- in SR one could set > the offsets of two clocks such that they are synchronized in any inertial frame > you choose. They would be synchronized with each other in that frame, but not > with coordinate clocks of the selected frame (these two would "tick at a > different rate" than those coordinate clocks). > You have made a very important statement which I would like to repeat with some emphasis. "In SR one could set the *offsets* of two clocks such that they are *synchronized* in *any* inertial frame you choose. They would be synchronized with each other in *that* frame." Let us extend the analogy of two clocks fixed on earth's geoid to a million (or more) clocks fixed on earth's geoid. Let us synchronize all these clocks in ECI frame by synchronizing their time to UTC by using GPS service. In this state, each and every adjoining pair of clocks can be considered as mutually synchronized with zero time offset between them. Let us *adjust* the offsets of all these clocks such that they are now synchronized in BCRF. A little reflection will show that the required offset will finally come out to be zero since each and every adjoining pair of clocks on earth's geoid will have to be mutually synchronized in BCRF. A further extension of this logic will show that when a group of million (or more) clocks fixed on earth's geoid, are synchronized to UTC through GPS, they can also be regarded as synchronized in BCRF or the Galactic reference frames. Then why can't we accept such a synchronization to be valid in an absolute or universal reference frame as well? For mutually synchronizing adjoining pairs of clocks on earth's geoid in a particular reference frame, we need not compare their time to the coordinate time of that frame. GSS |