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Universal Time Formulation of Special Relativity

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Universal Time Formulation of Special Relativity Indian Journal of Pure & Applied Physics Vol. 42, January 2004, pp 5-11 Universal time formulation of special relativity Vachaspati Physics Department, University of Allahabad, Allahabad 211 002 Received: 7 October 2003; accepted: 13 November 2003 Special Relativity is formulated with time, instead of the speed of light, treated as invariant under inertial coordinate transformations. Velocity of light in space is shown to be isotropic and the observed longer lifetimes of unstable particles in flight are explained. The slowing down of cesium clocks carried by Hafele and Keating in circumnavigating the earth, is discussed. An observational test to find out whether or not nature operates according to the universal time, is outlined. Keywords: Special relativity, Transformation equations, Universal time 1. Introduction alternative formulation of special relativity. An There are three landmark experiments that must observable consequence is described in section 7 that be explained by any alternative proposal seeking to may reveal whether nature operates according to replace the usual theory of special relativity; they are electromagnetic time or to universal time. But before 1. Michelson and Morley1 experiment showing that we do this, below is a rapid view of what has gone velocity of light is isotropic in space. The before. experiment has been repeated with increased The earlier attempts at alternative theories have 11 sophistication and result confirmed with high been described, among others, by W. Pauli and by C. 12 accuracy2,3; Møller . Ever since the publication of Einstein’s 13 4 paper , there has been controversy about his notion of 2. Rossi and Hall’s experiment with muons that 14 showed their lifetimes in flight were larger than time. H Dingle did not believe in the slowing down of clocks in motion and, in particular, in the asymmetric when they were at rest in accordance with the 15 theory. There are now numerous and precise aging of twins. C. G. Darwin showed that asymmetric aging occurs if the twins exchange electromagnetic confirmations supporting this conclusion, 16 especially from high energy particle accelerators signals to record the readings of their clocks. M. Sachs and storage rings5,6,7,8. objected to the idea that there is a one-to-one correspondence between an observer’s estimate of the 3. J. C. Hafele and R. Keating9,10 sought a direct time parameter in a frame of reference that moves confirmation of time dilation by taking very relative to him and a physical process, such as aging, accurate cesium clocks round the earth on that is going on in the moving frame. J. P. Hsu17 commercial airline in easterly and in westerly introduced a common time for all observers who may be directions, respectively. This is discussed later in in relative motion with respect to each other. He asserted section 6. that it is possible to synchronize clocks in relative It has been shown below that the above, and motion so as to indicate the common time. In his similar, experiments can all be understood if time is formulation speed of light is not the same in different taken as an invariant parameter and the speed of light coordinate systems; if it is c in one, it is c = (c – v)/√(1 – varies in different inertial coordinate systems that are v2/c2) in the other that is moving with velocity v. The not at rest with respect to each other. In other words, speed of light is thus anisotropic in the second system. there is no time dilation, but there is variation in the To overcome it, he argued that roundtrip measurements, speed of light; the theory can be considered as an such as those in Michelson and Morley’s experiment1, __________ * Present address: 32 Revere Road, Monmouth Junction, would cancel v, and c’, though different from c, would NJ 08852, USA) be isotropic in the mean. 6 INDIAN J PURE & APPL PHYS, VOL 42, JANUARY 2004 18 ’ ’ L. Brillouin has criticized relativity on several dx0 = u0 dtA , dx0 = u0 dtA. …(4) counts, one of them being that the unit of length is Notice that the time interval, dt , between two based on the wavelength of a spectral line of krypton- A events is the same, but the velocity, u , now takes the 86, and the unit of length is based on the frequency of 0 place of c and is taken different in the two cases. a spectral line of cesium. Hence, the same physical We call the system of space coordinates (x1, x2 , phenomenon, a spectral line, is used for two ’ ’ ’ definitions, length and time, so that the velocity of x3) as S and of (x1 , x2 , x3 ) as S’. light remains undefined and looks arbitrary. From (3) and (4) it follows that 19 In a very interesting paper, F. Selleri has ’ u0 = c dtE/dtA , u0’ = c dtE /dtA . …(5) proposed a theory with a privileged frame of ether which enables him to introduce relative time and The relation (1) is satisfied if absolute simultaneity. He suggests using two clocks, ’ 2 one giving the “natural time” and the other giving dx0 = (dx0 + β dx)/√(1 - β ), ’ 2 “Einstein time”. He did not make clear how the clock dx = (dx + β dx0)/√(1 - β ) …(6) showing “natural time” can be set up. where β is a constant. From (6) follows that A comprehensive review of the conventionality ’ 2 of synchronization and test theories of relativity is x0 = (x0 + βx)/√(1 - β ), given by R. Anderson, L. Vetharaniam and G. E. ’ 2 Stedman20. x = (x + β x0 )/√(1 - β ) To interpret β, we have to specify whether we 2 Transformation Equations use (3) or (4). We start with the line-element, First we use (3). Then (6) gives 2 2 2 ds = dxo - dx , ’ 2 cdtE = (cdtE + βE dx)/√(1 - βE ), and the problem is to find transformations, (x0 , 2 ′ dx’ = (dx + βE cdtE)/√(1 - βE ) x)→ (x0 , x′), that leave this line-element invariant; i.e., such that and ’2 ’2 2 2 ’ dxo - dx = dxo - dx . [dx’/dtE ]dx=0 = βEc …(7) I shall consider the one dimensional case so that x = (x, 0, 0), x’ = (x’, 0, 0), and The subscript E on β has been put to distinguish it from the next case when I use time tA.. ’2 ’2 2 2 … dxo - dx = dxo - dx ; (1) From Eq. (7) we see that βEc is the velocity with ’ dxo and dxo are proportional to infinitesimal which a fixed point in S appears to move to an time intervals in the two coordinate systems and one observer in S’, and is therefore the velocity, VE , with usually writes which S is moving with respect to S’. When we write V /c instead of β, we get the famous Lorentz dx = c dt, dx ’ = c dt’. …(2) E 0 0 transformation, However, we shall use two kinds of clocks, E ct ’ = (ct + V x/c)/√(1 – V 2/c2), (for Electromagnetic) and A (for Absolute or E E E E 2 2 universal), to measure time. When measuring time x’ = (x + VE tE)/√(1 – VE /c ) ...(8) with an E- clock, I write t or t ’ ; when measuring it E E Let us next repeat the above process using the with an A-clock, I write tA. Since universal time is ’ connection (4) (6) gives (I put suffix A over β invariant, tA = tA. Eq. (2) refers to time measured with an E-clock; now) 2 therefore I rewrite it as u0’ dtA = (u0 dtA + βA dx)/√(1 - βA ), dx = c dt , dx ’ = c dt ’ …(3) 2 0 E 0 E dx’ = (dx + βA u0 dtA)/√(1 - βA ), Notice that c is a constant velocity that remains and, on dividing both sides by dtA , we find the same in the two coordinate systems. If time is 2 measured by an A-clock, u0’ = (u0 + βAdx/dtA)/√(1 - βA ), dx’/dtA = VACHASPATI : UNIVERSAL TIME OF SPECIAL RELATIVITY 7 2 (dx/dtA + βAu0)/√(1 - βA ) …(9) is therefore called VE ; in (17) it is with respect to tA and is therefore called V . When dx = 0, i.e., when x is a fixed point in S, A we get 3 Velocity Addition 2 From Eq (13) we can find the way velocities are [u0’]dx=0 = [u0 ]dx=0 /√(1 - βA ), added in this scheme. It can be easily seen that if we 2 [dx’/dtA]dx=0 = βA[u0]dx=0 /√(1 - βA ) …(10) introduce another coordinate system S’’ such that As before, [dx’/dt ] is the velocity with ’’ ’ ’ ’ ’ ’ ’ ’ A dx=0 u0 = γA u0 + (VA /c)u , u’’ = γA u’ + (VA /c)u0 which S is moving with respect to S’; call it VA : ’2 2 γA’ = √(1 + VA /c ), [dx’/dtA]dx=0 = VA … (11) we find that Since all points in S are stationery (dx = 0), the ’’ ’’ ’’ ’’ speed of light in S is the observed speed, c, i.e., u0 = γA u0 + (VA /c)u, u’’ = γA’’ u + (VA /c)u0 [u0]dx=0 = c. From the second equation of (10), γ ’’ = √(1 + V ’’ 2/c2), therefore, A A 2 where VA = βAc/√(1 - βA ), ’’ ’ ’ from which VA = γA VA + γA VA …(18) 2 2 βA = (VA/c)/√(1 + VA /c ), Eq (18) tells the way velocities are combined. u’’ 2 2 2 gives the velocity of the particle and u ’’ gives the 1/√(1 - βA ) = √(1 + VA /c ) ≡ γA …(12) 0 ’ velocity of light as observed by S’’ from a source Writing u = dx/dtA, u = dx’/dtA, (9) becomes stationery in S. The rule (18) corresponds to ’ u0 = γA u0 + (VA/c)u, u’ = γA u + (VA/c)u0 …(13) ’’ ’ ’ 2 VE = (VE + VE )/(1 + VE VE /c ) …(19) One sees easily that 2 2 2 2 2 of the usual relativity.
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