On Einstein’s resolution of the twin clock paradox

von C. S. Unnikrishnan

On Einstein’s resolution of the twin clock paradox
C. S. Unnikrishnan

Gravitation Group, Tata Institute of Fundamental Research, Mumbai, India

In: Current Science, Vol. 89, No. 12, p. 2008 (2005)

Abstract: Einstein addressed the twin paradox in special relativity in a relatively unknown, unusual and rarely cited paper written in 1918, in the form of a dialogue between a critic and a relativist. Contrary to most textbook versions of the resolution,
Einstein admitted that the special relativistic time dilation was symmetric for the twins, and he had to invoke, asymmetrically, the general relativistic gravitational time dilation during the brief periods of acceleration to justify the asymmetrical aging. Notably, Einstein did not use any argument related to simultaneity or Doppler shift in his analysis. I discuss Einstein’s resolution and several conceptual issues that arise. It is concluded that Einstein’s resolution using gravitational time dilation suffers from logical and physical flaws, and gives incorrect answers in a general setting. The counter examples imply the need to reconsider many issues related to the comparison of transported clocks. The failure of the accepted views and resolutions is traced to the fact that the special relativity principle formulated originally for physics in empty space is not valid in the matter-filled universe.

… den ganzen Artikel lesen

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Siehe auch YouTube-Video vom Autor: Galileo, Time and Space: Gravity affecting clocks

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3 Antworten zu “On Einstein’s resolution of the twin clock paradox”

  1. Frank Wappler

    C.S. Unnikrishnan wrote („On Einstein’s resolution of the twin clock paradox“, Curr. Sci. 89, 2009; 2005):

    > […]
    > Counter-examples
    > […]
    > Consider an atomic clock B than is synchronized with another clock A during an initial encounter. Since the relative velocity between the clocks is known

    … provided it is specified and agreed how to determine values of „velocity between clocks“ in the first place (see below) …

    > and since the velocity of light is a constant […]

    … a statement for which the notion „velocity“ appears prerequisite, too.

    > […] Now consider the situation when A sends a signal at his time tA to B to freeze the clock and then decelerate, on receiving the signal. Since A can estimate the time at which the signal will reach B, he can also freeze his clock at that time […]

    Such an estimate would surely have to be consistent with the relation

    (1a)
    speed at which B moved away from A
    /
    speed of light
    ==
    duration of A from indicating the statement of the signal, „tA„, to indicating the freeze
    /
    duration of A from indicating the encounter with B to indicating the freeze;

    or

    (1b)
    1 –
    (speed at which B moved away from A
    /
    speed of light)
    ==
    duration of A from indicating the encounter with B to indicating the statement of the signal, „tA
    /
    duration of A from indicating the encounter with B to indicating the freeze.

    (Note that there is a particular relation between A’s indication of freeze and B’s indication of freeze which can be described in greater detail, involving the notion of „simultaneity“.)

    > The readings can be compared after B comes to rest relative to A

    … how this might be accomplished would first need to be specified and agreed, of course …

    > or after a round trip […]

    The „frozen readings“ of A and B may be compared when they meet again, too; just as directly as they were „synchronized“ at their initial encounter.

    Of course, comparing incidental „readings“ should be especially interesting if the corresponding durations could be compared as well, obtaining the value of ratio

    (2)
    duration of B from indicating the encounter with A to indicating the freeze
    /
    duration of A from indicating the encounter with B to indicating the freeze.

    > In fact, the possibility of switching off or freezing the clocks makes all the standard resolutions inadequate.

    This may well be debatable; but it surely doesn’t stand in the way of further analyzing the stated (counter-)example:

    As well as the ratio

    speed at which B moved away from A
    /
    speed of light

    from (1a) or (1b) above, the ratio

    speed at which A moved away from B
    /
    speed of light

    may be measured as well.

    If both were found equal throughout the trial under consideration, as may be assumed in the following, then this value is abbreviated as „\beta;“. Thus:

    (3)
    speed at which B moved away from A
    /
    speed of light
    ==
    speed at which A moved away from B
    /
    speed of light
    ==
    \beta;.

    Further, trying to obtain relations corresponding to (1a) or (1b), but involving duruations of B, one may arrive at

    (4a)
    (speed at which B moved away from A
    /
    speed of light)
    ==
    duration of B from some particular indication, „tB„, to indicating the freeze
    /
    duration of B from indicating the encounter with A to indication „tB„;

    or

    (4b)
    1 +
    (speed at which B moved away from A
    /
    speed of light)
    ==
    duration of B from indicating the encounter with A to indicating the freeze
    /
    duration of B from indicating the encounter with B to indication „tB„.

    There, the indication „tB“ of clock B was surely (in B’s judgement) after B’s indication of the encounter with A, and before B’s freeze.

    Also, surely, this indication „tB“ of clock B should have some particular relation to A’s indication „tA“ of stating the signal.

    This relation can be described in greater detail, involving the notion of „simultaneity“; thereby also motivating the following relation (5) between the durations appearing in (1b) and (4b):

    (5)
    duration of A from indicating the encounter with B to indicating the statement of the signal, „tA
    /
    duration of B from indicating the encounter with B to indication „tB
    ==
    duration of B from indicating the encounter with A to indicating the freeze
    /
    duration of A from indicating the encounter with B to indicating the freeze.

    (Short of spelling out the motivation of (5) completely, note that defining its denominators involved explicit appeal to the notion of „simultaneity“; while defining its numerators did not.)

    Finally inserting (1b) and (4b), with abbreviation (3):

    (6a)
    duration of B from indicating the encounter with A to indicating the freeze
    /
    duration of A from indicating the encounter with B to indicating the freeze
    ==
    (
    (1 – β) *
    duration of A from indicating the encounter with B to indicating the freeze
    )
    /
    (
    duration of B from indicating the encounter with A to indicating the freeze
    /
    (1 + β)
    );

    or

    (6b)
    duration of B from indicating the encounter with A to indicating the freeze
    /
    duration of A from indicating the encounter with B to indicating the freeze
    ==
    Sqrt[ 1 – β^2 ]
    ==
    duration of A from indicating the encounter with B to indicating the statement of the signal, „tA
    /
    duration of B from indicating the encounter with B to indication „tB„.

    If comparison of the „readings“ mentioned above corresponds to this ratio of durations then it is said that the two clocks had kept (sufficiently) equal „design“ throughout the trial; or otherwise not.

    Also note that the quantity

    (7)
    (1 – Sqrt[ 1 – β^2 ]) *
    duration of A from indicating the encounter with B to indicating the freeze

    may by some be referred to as „net proper time difference“.

  2. Frank Wappler

    p.s.
    Besides my reply above (#1, 29. Januar 2013 um 02:28) there is already a certain thread of correspondence directly arising from C.S. Unnikrishnan’s article on record in Current Science, and publicly available from the same website; by

    Ø.G. Grøn („Relativistic resolutions of the twin paradox“, Curr. Sci. 92, 416; 2007),

    K. Hazra („On the resolutions of the twin paradox“, Curr. Sci. 95, 706; 2008), and

    C.S. Unnikrishnan („Response“, Curr. Sci. 95, 707; 2008).

    In my humble opinion, together they exhibit all-too-common flaws:
    Grøn being not (quite) pedagogical enough to appear relevant;
    Hazra and Unnikrishnan being not (quite) inquisitive enough to be conclusive.

  3. Mark Lofts

    Der Schwerpunkt auf der Auflösung der Zwillingsparadox ist vom Einsteins Aufsatz, um die Simpel zu verwirren und Kritiker mit pedantischen unwichtigen bezüglich einiger ‘durch Gravitation’ Beschleunigung Einzelheiten zu verblüffen, abgelenkt und gedeckt. Warum unwichtig? Wegen eine entscheidende Tatsache. Die auf gleichbleibender Geschwindigkeit Reise bloß in einen STR-Effekt sich verwickelt, im Gegensatz zu übrigen der Fahrt Teilen, innerhalb welcher außerdem auch ein von Beschleunigung und Langsamerwerden mit Umkehrung bzw. ‘Gravitation’ ATR-Effekt erzeugt wäre.

    Also die beide Effekte sind unabhängig voneinander, weil die auf gleichbleibender Geschwindigkeit Entfernung von Gravitation oder Trägheit nicht unterworfen sein kann, außer wenn die Simpel von irreführender mathematischer Argumentation, woran die Zahlen sich aufgehoben werden, irreführt sind und dementsprechend beschloßen!