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.

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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 ) 2 Transformation Equations To interpret β, we have to specify whether we 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 =

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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. u0’ - u’ = u0 - u = c …(14) We considered above displacements in one 4. Relation between A andE Variables direction; if we include all directions, We can find the relation between the velocity of Eq (14) would be a particle in this formulation and that in the usual 2 2 2 2 2 relativity by going back to relations (4) and (3) from u0’ - u’ = u0 - u = c …(15) which we see that 2 2 Hence u0 = √(u + c ) …(16) u0 dtA = c dtE so that which shows that the speed of light depends on dtA/dtE = c/u0 . the square of the source’s relative velocity, u, and not on its direction. This is in accordance with the Hence, a velocity uE = dx/dtE can be related to 1 observation of Michelson and Morley that the speed the velocity uA = dx/dtA by of light is isotropic in space. 2 2 uE = uA dtA/dtE = (c/u0) uA = cuA/√(c + uA ) …(20) Multiplying (13) by dtA throughout, we get (using equation (16)). dx ’ = γ dx + (V /c) dx, 0 A 0 A Squaring both sides and solving for uA we get uA 2 2 dx’ = γA dx + (VA/c) dx0 , = uE/√(1 – uE /c ), so that 2 2 whose finite form, analogous to Lorentz uA = uE/√(1 – uE /c ), …(21) transformation (8), is 2 2 2 2 2 2 1/ 2 and u0 = √(uA + c ) = [uE /(1 – uE /c ) + c ] x0’ = γA x0 + (VA/c) x, x’ = γA x + (VA/c) x0 …(17) = c/√(1 - u 2/c2) …(22) The difference from Lorentz transformation E arises from the way the velocity, V, is measured in the Notice that |uA | is always bigger than |uE | and two cases; in (8) it is measured with respect to tE and the speed of light, u0, is always bigger than c, or equal

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to it when uE = 0. Also, c is no more the maximum where wE0 = dx0/dτ, wEi = dxi/dτ (i = 1,2,3), and speed a particle can attain; it can have any velocity proper time τ is given by and there is no upper limit to its magnitude. dτ = dt √ (1 – u 2/c2). Moreover, c is now the least speed light can E E have in vacuum; it can have any greater value The equation of motion using the universal time, depending on the relative velocity of the source and tA, is the observer. ν m duAμ /dtA = (e/c) uAνF μ …(25) 5. Measuring Time-Intervals which is just the same as (24) if we identify τ There are usually two ways in which a time- with tA. It is therefore apparent that, whatever the interval is determined for moving objects; one is electromagnetic field, uAμ will come out same as wEμ. based on the frequency of electromagnetic waves, and the other consists in finding the velocity of an object Therefore we can write down traversing a known distance. These are described 2 2 u = w = u dt /dτ = u /√((1 - u /c ). briefly below. A E E E E E Frequency method. ⎯ If we solve the wave It follows that the time ΔtE in moving a distance equation, x will be x/uE according to E-clock, and ΔtA = x/uA = 2 2 2 2 (x/uE)√ (1 – uE /c ) according to the A-clock. We see (∂0 - ∇ )ψ = 0, again that, as in (23), we get waves of the form 2 2 ΔtA = √(1 – uE /c ) ΔtE . iϕ 2 ψ = ψ0 e , ϕ = k(x0 – n.x), n = 1, Universal time and proper time ⎯ The equality

where n is the direction of the wave propagation of tA and τ holds in case we are dealing with a single (with which we are not concerned here) and k is its particle. Proper time is employed for one world line;

frequency. ψ remains unaltered if x0 → x0 + Δx0 such if there are several particles, a different proper time is that kΔx0 = 2π. Hence 2π/k can be used as a measure defined for each. The proper time formalism then for time change Δx0; in other words, changes of x0 are becomes unwieldy. On the other hand, the universal measured in terms of the unit 2π/k. The atomic clocks time, tA, will be the same for all particles. When we have a complicated system consisting of particles and measure Δx0 = cΔtE . fields, proper time cannot be used, but universal time If, on the other hand, we try to measure can be employed for them all. universal time, we shall have to write Δx0 = u0 ΔtA. If the signal emitting device is not moving, u0 = c; there Four-momentum ⎯ It is clear from Eq (25) that it is is then no difference between ΔtE and ΔtA. If it is appropriate to define the four-momentum of a particle moving, we can know ΔtA only indirectly by using by (22): pμ = muAμ . 2 2 μ 2 2 2 ΔtA = √(1 – uE /c ) ΔtE. …(23) Since uAμ uA = u0 - u = c [see (15)], it follows that Velocity method ⎯ If we know the velocity of a μ 2 2 2 2 2 particle and the distance it travels during an interval, pμ p = m c and p0 = √(p + m c ). we can find the time duration by dividing the latter by We can, as usual, identify cp0 with energy, E, the former. The velocity of a charged particle can be giving mc2 for rest energy. found by subjecting it to a magnetic field and using the appropriate equation of motion. 6. Lifetimes and Twin Paradox The usual relativistic equation of motion of a The experiments of Rossi and Hall4 and of J. 5,6 charged particle in an electromagnetic field Fμν with Bailey et al have given definite evidence that the F0i representing electric and Fij magnetic field (i,j = lifetime, ΔT, of mu mesons is greater in flight than 1,2,3) is when they are at rest. Denoting the rest system of a meson by S, which, when observed from S’ has m dwEμ /dτ = (e/c)wEνFμ …(24) velocity VE, we have

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2 2 ΔT = ΔT0/√(1 – VE /c ), …(26) equally; the difference will be in the speed of light which will not be c for them; it will be different from ΔT0 being the lifetime at rest. It should be noted c but same for both of them. Homer will experience that the times employed here are in accordance with the same speed of light coming from Roamer as E-clocks as explained in the previous section, so we Roamer will experience of light coming from Homer; can write Eq (26) as 2 2 namely, both will find it as u0 = c/√(1 – VE /c ). 2 2 [Δ x0]in flight = [Δ x0]at rest /√(1 – VE /c …(27) In the usual theory it is hard to explain why Let us see what Eq (27) implies when time is Homer cannot be regarded as moving with respect to measured by an A-clock. From Eq(4) we see that now Roamer, which would imply that, when they meet, Homer should be younger than Roamer. This presents [Δ x0]at rest = u0 Δ tA = c Δ tA (see remark after Eq. (11)) a paradox; indeed, if H ≥ R and R ≥ H, then H and R must be equal. and [Δ x0]in flight = u0’ Δ tA . Hafele and Keating10 carried four cesium clocks therefore Eq (27) becomes, on canceling Δ tA round the earth in jet flights, once eastward and once from both sides, westward, and recorded directionally dependent time u ’ = c/√(1 – V 2/c2). …(28) differences which were in good agreement with 0 E prediction of the conventional relativity theory. The This agrees with our previous result, Eq (22). flying clocks lost 59 ± 10.nanoseconds during east- When universal time is used, the lifetime remains ward trip and gained 273 ± 7 nanoseconds during same as before, namely Δ tA , but the speed of light westward trip. These results seemed to provide proof increases. of the velocity dependent time dilation effect of The experiments on mu meson lifetimes (Bailey special relativity theory. et al.5,6) are statistical; a stream of particles of known However, Richard Schlegel21 has shown that this velocity are sent into a storage ring and their number conclusion is not warranted because Sagnac22,23 effect is observed after they emerge. Denoting by N and 0 had not been taken into account. When this is done, N(t ) their initial and final numbers respectively, E the directional time loss (gain) for an eastward N(tE) = N0 exp(- tE/τ) (westward) circumpolar journey turns out as velocity independent and the east-west asymmetry disappears. where τ is the lifetime. For particles at rest, τ = 2 2 There would be no observed difference in time rate ΔT0; for moving particles, τ = ΔT = ΔT0 /√(1 – VE /c ). between clocks being flown eastward and those being On the other hand, we may not write τ as ΔT0 /√(1 – flown westward. This opens the question whether 2 2 ’ VE /c ) but as (u0 /c) ΔT0 as in (28); the result is the there is, or is not, the velocity dependent time dilation same as before. Thus either description, whether we as envisaged in special theory of relativity. regard changed τ as due to time dilation or due to increase in the velocity of light, gives the same 7. Persistence Effect and Observational Test number N(tE) of surviving muons. An explosive object, such as a bomb or a star, is However, with the time dilation explanation, we situated at x =0, and is viewed by observers (Fig. 1) N are faced with the question: why do the mesons last and F (for ‘Near’ and ‘Far’ respectively). The longer when in motion? Does some of their inner observers are at rest with respect to each other and mechanism gets altered? The mesons remain always with respect the to the object. The observer N is close in their own rest-frame and nothing changes inside to the object and F is far away at distance x. At time T them; then why do they start behaving differently in (since everything is at rest, TE = TA = T) the object flight from when at rest? Or is their longer lifetime an explodes and its fragments fly at high speed, uA (we illusion? These questions become more acute when are dealing with a one dimensional case). The near we consider the twin paradox next. observer, N, will see the unexploded object until the If one of the twins (call him Homer) stays at moment T and then no more. However, the fragments home and the other goes on a round trip (call him are high speed and light from them will travel at 2 2 Roamer), Roamer is supposed to have aged less than speed u0 = √(c + uA ). Therefore the light starting at T Homer because of the time dilation effect (26) on E- from the fragments will reach the observer F earlier clocks. But, according to A-clocks, both would age than the light starting at the same time T from the

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unexploded object. Light from the unexploded bomb It is likely that not all fragments of the object will reach F at time T(bomb) = T + x/c; light from the will fly at the same speed. Therefore the observer F fragments will reach F at time T(fragments) = T + x/u0 . will not see all fragments at once, but will go on Since u0 > c, it follows that T(fragments) < T(bomb) and F seeing increasingly more as time goes on. This will be will conclude that the object did not explode at time T interpreted by F as meaning that the object continued , but a little earlier than T; the object did not explode exploding over long time starting from T - ΔT, even all at once, but melted away slowly. though the explosion happened instantaneously at The overlap time when the two, the object and time T. It is also possible that some of the earlier its fragments, will appear together is fragments are slower than the later ones, in which 2 2 case their time-ordering can be reversed for F, the ΔT = x/c - x/u0 = [1 - c/√(c + uA )](x/c). later ones will be seen by F sooner than the earlier

For illustration, if uA = c (this corresponds to uE fragments. = c/√2 ), If observations of astronomers confirm this persistence effect, it would appear that nature operates ΔT = (1 – 1/√2) x/c ≈ (0.4/1.4)x/c ≈ 0.3 x/c. according to universal time. Perhaps the experiment x/c is the time that light from a stationery object could be carried out in the laboratory by shattering an takes to reach F. Thus ΔT would be about three-tenths object into fragments and making observations from of that time. If x/c = 1 million years, ΔT would be far and near. about 300,000 years, and the observer F will think 8. Concluding Remarks that the explosion started 300,000 years earlier than T, and the object took 300,000 years to vanish It has been shown above that it is possible to completely. formulate relativity with time treated as universal. The universal time is invariant against uniform translatory motion of coordinates, but the speed of light, now denoted by u0, changes according to rule (13). It depends on the square of the relative velocity, u, of the source and the observer, and is thus isotropic in space [eq. (16)], agreeing with the conclusion of Michelson and Morley1. Universal time is equivalent to the proper time in case of a particle. Proper time, however, is used for a single particle; when there are several particles, a different proper time is employed for each. On the other hand, universal time can be used for any number of particles or fields. There is no problem in defining simultaneity when universal time is used. It makes sense to speak of “this instant tA” anywhere and the duration of an event will likewise be the same everywhere. There is no twin paradox; both, the stay-at-home Homer and the traveler Roamer age equally. The lifetimes of unstable particles in motion remain the same in flight Fig. 1 ⎯ An object is viewed by two observers, N and F, both at also, the observed increases are ascribed to the rest with respect to each other and to the object. N is near and F is changed speed of light. far from the object. The object suddenly explodes at time T and its fragments fly apart at high speed. N will see the unexploded In the usual formulation of relativity, one is left object until time T and then no more. If light travels at speed c, F to wonder, why, while all other velocities are affected will also get the same picture. However, if speed of light is uo = by the relative motion of the observer and the 2 2 1/2 (c + uA ) , light from fragments will travel faster than light from observed, is the velocity of light an exception? The the object did; this will make F believe that (1) the object did not explode at time T, but earlier, and (2) the object did not explode burden of keeping the velocity of light so sacrosanct suddenly but exploded slowly. is then borne by time. This affects our perception of

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time profoundly. The matter was much debated until 3 Lamoureaux S K, Jacobs J P,Raab F J & Forster E N, Phys. about thirty years ago, but the controversy has died Rev. Letters, 57, (1986) 3125-8. down in recent years. However, it continues to be a 4 Rossi and B Hall D B, Phys. Rev., 59, (1941) 223. 5 Bailey J, Bartl W, von Bochmann G,et al., Il Nuovo Cimento, matter of unease. Vol. 9A, (1972369-432). The situation is remedied here by using 6 Bailey J, Borer K, Combley F et al., Nature, 268, (1977) universal time. Like all velocities, light’s is also 301-304. subject to similar transformation law. An 7 MacArthur D W, Phys. Rev. A, 33, (1986) 1. observational test is proposed in section 7. 8 MacArthur D W,,Butterfield K B, Clark D A et al, Phys. Rev. Letters, 56, (1986) 282-285. One might ask, how do we measure the 9 Hafele J C, Am J. Phys., 40, (1972) 81. universal time, tA? We have electromagnetic devices 10 Hafele J C & Richard R. Keating, Science, 177, (1972) 166 to measure tE , but we are not aware of any that and 168. measure tA directly. We can, of course, find tA by 11 Pauli W, Theory of Relativity, Second edition, B.I. Publications, Bombay, India, and Pergamon Press, Oxford, converting tE to tA. However, this does not mean that there is no universal time. That we are not able to England, pp. 1-9 (1965). 12 Møller C, The Theory of Relativity, Second edition, Oxford measure something directly, does not imply that it University Press Delhi, pp. 40-43, 544-550 (1972). does not exist. We have learnt from quantum 13 Einstein A, Ann. Phys., 17, (1905) 891. mechanics that, while we work with wave functions, 14 Dingle H, Nature, 179, (1957) 866 and 1242. the measurable things are the expectation values, and 15 Darwin C G, Nature, 179, (1957) 976-977. that nature works on a deeper level than what appears 16 Sachs M, Physics Today, 25, (1972) 9. on the surface. 17 Hsu J P, Il Nuovo Cimento, 74 B (1983) 67. 24 18 Brillouin L, Relativity Re-examined, Academic Press, p. 5 It may be mentioned that E. J. Post in 1967 (1970). showed that Lorentz transformations are not 19 Selleri F, Found. Phys., 27, (1997) 1527. appropriate for rotating bodies. A. K. J. Maciel and J. 20 Anderson R, I. Vetharaniam, G. E. Stedman, Physics Tiomno25 are of a similar view. Robert D. Klauber26 Reports, 295, numbers 3 and 4, 93-180 (1998). has discussed the problem of rotating disc at length 21 Richard Schlegel, Amer. J. Phys. 42, (1974) 183. and come to the same conclusion. A clock using a 22 Sagnac G, Compt. Rend., 157, (1913) 708, 1410; rotating disc should be able to display universal time 23 Sagnac G, J. Phys. Radium, Fifth series, 4, (1914) 177. directly; our earth’s surface seems satisfactory. 24 Post E J, Rev. Mod. Phys., 39, 475 (1967). References 25 Maciel A K A & Tiomno J, Phys. Rev. Letters, 55, (1985) 143-146. 1 Michelson A A & Morley E W, Am. J. Sci., 34, (1887) 333. 26 Robert D. Klauber, Found. Phys. Letters, 11 (1998), 405- 2 Brillet A & Hall J L, Phys. Rev. Letters, 42, (1979) 549. 443.