|“The habit has developed of assuming that a physical theory is necessarily sound if its mathematics are impeccable: the question of whether there is anything in nature corresponding to the impeccable mathematics is not regarded as a question; it is taken for granted.” Dingle|
Chambers Encyclopaedia – One kind of application of Special Relativity is to the fundamental concepts of physics, such as mass, energy, force etc. If the laws of conservation of mass and of linear momentum are to be valid in all inertial frames of uniform motion the mass of a particle is defined as m(1 − v²/c²)½ and its energy as mc²(1 − v²/c²)½. Here m is the mass the particle would have in an inertial frame in which it was at rest (its proper mass) and v is its mass relative to the frame. It then follows that the laws of conservation of mass and energy are identical if we assume that the proper mass of a particle is itself equivalent to the amount of energy mc². These definitions of mass and energy imply that v must be less than c and it is for this reason that Special Relativity restricts itself to relative velocities not greater than c.
The two definitions have played a capital part in modern atomic physics. The equivalence of mass and energy helps to explain the phenomenon of radioactivity and the explosions of atomic bombs. It also accounts for the production of the vast quantities of radiation emitted by the stars without an apparent cooling of these bodies, the necessary energy being found in the conversion of a small fraction of their masses into radiant energy.
McCrea rather protests too much in his attempts to refute Dingle’s arguments, although he concedes that Dingle “has not made any mistakes in his algebra.” Instead, McCrea believes that Dingle “deals with objects to which the [SR] theory explicitly denies a meaning.” This, of course, begs the question: if the theory denies a meaning to a physical model, is the theory coherent? He claims that Dingle’s assertion is false because “he is not talking about the same thing, but two different things... equations (3) and (4) concern two different sets of events, and so they cannot contradict each other.” McCrea adamantly refuses to acknowledge the validity of coordinate transformation systems and his refusal to treat both events in a common system is contrary to both the letter and spirit of special relativity theory.
McCrea repeatedly asserts, in the summary in Special Relativity (2) and elsewhere, that “Because A is never at E1, the phrase is meaningless and so Dingle’s (3) is meaningless. Correspondingly his (4) is meaningless.” That he should attempt to use “A is never at E1” as valid criticism was regarded by Dingle as “indefensible.” Dingle had explicitly defined A as a clock distant from H. Clock A is never at E1; it merely records the event.
To better illustrate the clock paradox Dingle developed his thesis further. A basic example of his argument is provided by considering only the x′-axis and x-axis in two coordinate systems k and K. The coordinate system k is moving at uniform velocity v relative to K.
|N||B||»»»»»»» v »»»»»»»|
x-axis of K, clock readings denoted by t, below
A and H are two regularly running clocks at a fixed distance from each other in coordinate system K. N and B are another pair of regularly running clocks at a fixed distance from each other in the moving coordinate system k. (The distances NB and AH are independent and arbitrary.)
The readings of N and B with respect to the coordinate system k are denoted by t′ and the readings of A and H, in the coordinate system K, by t. (For simplicity we regard K as stationary.) We now examine three successive events as k moves with uniform velocity v relative to K.
At event E0, B is adjacent to A and both are observed to read zero. We also synchronize N and H to read zero at this moment.
|t′ = 0|
|t = 0|
At event E1, B is adjacent to H; B is observed to read t′1 and H is observed to read t1. Clock A is distant from clock H but reads t1 when H reads t1.
|t′ = t′1|
|t = t1|
At event E2, N is adjacent to A; N is observed to read t′2 and A is observed to read t2. This is all independent of the theory – it is merely a description of a physical process. A theory is required when we wish to determine two independent things: (1) the values of t′1, t′2 for given values of t1, t2 or vice versa and (2) the relative values of A and B when these values apply. Clock B is distant from clock N, but reads t′2 when N reads t′2.
|t′ = t′2|
|t = t2|
We now apply Einstein’s theory, supposing that A is fixed at the origin of the K coordinate system and B is fixed at the origin of the k system.
(i) t and t′ are related by the Lorentz transformation so that:–
|Between events E0 – E1||t′1 = at1||(1)|
t1 in the K coordinate system is corrected by a (as A is offset) to read at1
|Between events E1 – E2||t2 = at′2||(2)|
t′2 in the k coordinate system is corrected by a (as B is offset) to read at′2
|and||a = (1 − v²/c²)½|
(ii) The relative velocity a (the ratio of distance vs. time, sometimes denoted by µ) of k in relation to K is determined by choosing a pair of events and comparing the intervals between the readings of A and B at these events.
Einstein chose events E0 and E1. At these events A reads 0 and t1 respectively and B reads 0 and t′1 respectively. The reason why A must be held to read t1 at E1 is that H reads t1 at this event, and by application of the Lorentz transformation the process by which A is set in relation to H synchronizes it with H.
Shortly however we reach a contradiction. Between events E0 and E1, A advances by t′1 = t1 and B by t′1 = at1 by (1). Therefore:
|rate of A||=||t1||=||1||(3)|
|rate of B||at1||a|
But now choose events E0 and E2. At these events A reads zero and t2 respectively and B reads zero and t′2 respectively. The reason why B must be held t′2 at E2 is that N reads t′2 at this event and the process by which B is set in relation to N synchronizes it with N.
However between events E0 and E2, B advances by t2 = t′2 and A by t2 = at′2, by (2). Then:
|rate of A||=||at′2||=||a||(4)|
|rate of B||t′2|
Equations (3) and (4) are contradictory; one result is the reciprocal of the other. Further, since v is non-zero, a is never 1 and it is obvious nonsense to derive (rate A)/(rate B) > 1 and (rate A)/(rate B) < 1 in the same system. Hence the theory from which they are derived must be invalid. Einstein in his theory gave only (3) and accepted it as the unique value of the rate-ratio; he did not verify the result by checking the interval between E0 and E2. Had he done so he would have seen that his conclusion was untenable.
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