DERIVATION OF THE LORENTZ ETHER THEORYThis is a summary of the first talk I gave at the NPA meeting in College Station, Texas, May 1997, on the derivation of the Lorentz ether theory: ABSTRACT --- The Lorentz ether theory (LET) reproduces all the equations of special relativity (SR) and makes the same experimental predictions.[A. I. Miller, "Albert Einstein's Special Theory of Relativity," Addison-Wesley (1981).] The differences between it and SR are in the interpretations placed on those equations, and the starting assumptions used to derive them. This talk will review the derivation of the LET and compare it with SR. It will be shown that all of the counter-intuitive notions for which SR is famous can be understood as resulting from the adoption of mutable standards of measurement. The Lorentz transformation will be shown to follow from the contraction of the length unit, the dilation of the time unit, and the adoption of local time zones according to the Einstein synchronization convention. Relationships between the relative coordinates of SR and the absolute coordinates of the LET will be derived and the reason for the importance of the space-time interval will be explained. Expressions for absolute time, position and velocity will be presented and shown to be Lorentz invariant. ASSUMPTIONS 1. There exists ether, a physical substance that fills the entire universe and acts as the medium for the transmission of light waves. It will be assumed to be rigid. [Allowing deformability and differential motion in the ether leads to gravitational effects which will not be considered here.] In principle, the ether can be detected and absolute velocities determined with respect to it, but the means of accomplishing this has not yet been discovered. 2. There exist absolute standards of measurement for distance and direction. From this it can be proven that the geometry of space is Euclidean. [This proof was given in my geometry talk at a New Mexico Objectivist Club meeting in 1997 and will eventually be published somewhere.] 3. There exists an absolute standard of measurement for time. From this, and the fact that motion is relative, it can be proven that the kinematics of space-time is Galilean [not Lorentzian]. That is, absolute space and time coordinates in different frames of reference in relative motion must be related by the Galilean transformation equations (GTE's), not the Lorentz transformation equations (LTE's). [I will publish this proof somewhere eventually - but for now I will just take it to be an assumption.] By absolute space and time coordinates I mean coordinates measured with the absolute standards which are assumed to exist in principle but whose actual construction in terms of concrete materials is not yet specified. Absolute coordinates will be denoted by capital letters. Let the absolute coordinates measured in the frame of the ether be called X and T. Absolute coordinates measured in a frame moving with constant absolute velocity U in the X direction will be called X' and T'. They are related to X and T by the GTE's: X' = X - U T and T' = T (1) and the relationship between the absolute velocities measured in the two frames may be obtained by differentiating this, giving: V' = V - U (2) This also applies to the velocity of light, for which C' = C - U (3) Ordinary rulers and clocks are not absolute standards of measurement. They experience length contractions and time dilations as follows: 4. Length Contraction The length of a ruler varies as a function of its velocity relative to the ether according to the formula L = L0 sqrt(1-V^2/C^2) where L0 is the length of the ruler when it is at rest. This only applies when the ruler is aligned with the direction of motion. If it is perpendicular to this direction there is no effect. 5. Time Dilation The time interval between ticks of a clock varies according to the formula T = T0 / sqrt(1-V^2/C^2) where T0 is the time interval when the clock is at rest. [This assumption can be omitted if one is willing to accept the notion that all clocks are equivalent to a "light clock" which consists of a light pulse perpetually bouncing between two mirrors attached to the ends of a ruler.] Space and time coordinates measured with these mutable standards are called relative coordinates and will be denoted by lower case letters: x and t in the ether frame, and x' and t' in the moving frame. Since there is no length contraction or time dilation in the ether frame we must have x = X and t = T (4) and also for velocities, v = V, u = U and c = C. (5) The relative coordinates in the moving frame are related to the relative coordinates in the ether frame by the LTE's, which we can now derive with the help of one more assumption: 6. Local Time Synchronization Clocks at different locations are to be "synchronized" using the Einstein Synchronization Convention (ESC) as follows: Let one particular master clock emit a light pulse when it reads t=0. All other clocks will be set to d/c when they receive the pulse, where d is their distance from the master clock measured with relative units (ie. actual rulers). Notice that this procedure will produce absolute synchronization if the master clock (and the rulers) is at rest in the ether, but not otherwise. The meaning of local time is the same as for local time zones on the earth. Instead of synchronizing clocks to each other, they are synchronized to some locally observable objective event. For example, "set your clock to noon when the sun is directly overhead" roughly gives the ordinary kind of local time. Clocks that are 15 degrees of longitude apart will read an hour time difference. They are synchronized to the sun, not to each other. With the ESC, clocks are "synchronized to light" rather than to each other. This is the best that can be done if you are unable to detect the ether. We will see later that absolute synchronization is possible in principle, but it requires knowledge of the absolute velocity of the clocks, which we do not yet know how to measure. DERIVATION OF THE LORENTZ TRANSFORMATION EQUATIONS The LTE's relate x' and t' to x and t. Since the relative length unit in the moving frame is contracted by a factor of sqrt(1-U^2/C^2), the relative coordinate will be expanded by the same factor. The product of the coordinate times the unit is equal to the objective length, which must be independent of the chosen units. Thus x' L = X' L0 (6) so we must have x' = (L0/L) X' = G X' (7) where G = 1/sqrt(1-U^2/C^2) > 1. In other words, if the rulers are shorter, then it takes more of them to span the same objective distance. Applying (1), (4) and (5) to (7) then gives: x' = g (x - u t) (8) where g = 1/sqrt(1-u^2/c^2). This is the first LTE. In an exactly similar manner we may also find that t' = T'/G = t/g (9) but this is not yet the second LTE. Since this equation has no dependence on position, moving clocks at different locations will read the same time. But this is not right. By assumption 6, the clocks must be offset to different local time zones by using the ESC. We must add an offset: t' = t/g + o(x',u) (10) where o(x',u) depends only on the location of the clock (x') and on the velocity u, but not on the time (t') so that rate will not be altered. The ESC is applied as follows: Let the clock at the origin of the moving frame be the master clock. It emits a light pulse at t'=0, which is later received by another clock at x'. The absolute time of reception TR' is the absolute distance X' divided by the absolute velocity of light C', and using (7) and (5) this becomes X' x'/ g x' c + u TR' = -- = ----- = --- sqrt( ----- ) (11) C' c - u c c - u and TR' = TR = tr. The ESC says that at this time, the clock at x' must be set to x'/c, so applying (10) we may write: tr/g + o(x',u) = x'/c and then using (11) we can solve for the offset and find: o(x',u) = - u x'/ c^2 So the second LTE becomes t' = t/g - u x'/ c^2 (12) In this form [(8) and (12)] the meaning of each term in the LTE's is clear. The g factor in (8) is from length contraction, the 1/g factor in (12) is from time dilation, and the -ux'/c^2 term in (12) is the amount by which each clock must be set back so that it will be "synchronized to light" rather than to the other clocks. Equation (12) can be put into a more familiar form by using (8) to replace the x' term and then simplifying, to get: t' = g ( t - u x/ c^2 ) (13) The symmetrical appearance of (8) and (13) is related to an important property of the LTE's called the group property. If the LTE's are applied twice, to go from frame 1 to frame 2 and then to frame 3, the net result can be expressed as a single LT directly from 1 to 3. LT's can thus be "multiplied" as if they were numbers, and they have algebraic properties. What this means is that even though in the derivation of the LTE's, the starting frame had to be the ether; in the application of the LTE's the ether does not have to appear at all. They can be used to connect any two frames and can be used even if you can't detect the ether (which we can't). However, it needs to be emphasized that while the ether is not required in order to use the LTE's, it IS required in order to understand what they mean. The transformation for velocities can be found by differentiating (8) and (13) and is given by: dx' g ( dx - u dt ) v - u v' = --- = ---------------------- = ------------- (14) dt' g ( dt - u dx / c^2 ) 1 - u v / c^2 whete v = dx/dt. Note that c - u c - u c' = ------------- = c ----- = c (15) 1 - u c / c^2 c - u so that the speed of light in relative units is invariant. AN ALTERNATIVE DERIVATION Without the offset term the LTE's would not have the group property. An alternative derivation (perhaps the one that Lorentz used) would be to start with equations (8) and (10) and ask what offset would be required in order to make the transformations form a group. This can easily be solved and will be left as an exercise for the reader. [Hint: Write the equations in matrix form and set the inverse of L(u) equal to L(-u), where L(u) is the transformation matrix.] RELATIONSHIP BETWEEN ABSOLUTE AND RELATIVE COORDINATES Given an arbitrary frame with relative coordinates x and t, and absolute coordinates X and T, it is easy to find the formula that relates them if we assume that the ether can be measured and has some velocity ve or VE. Use the LTE's to transform the relative coordinates and velocity to the ether frame: x0 = ge ( x - ve t ) (16) t0 = ge ( t - ve x / c^2 ) (17) v0 = ( v - ve ) / ( 1 - ve v / c^2) (18) where ge = 1/sqrt(1-ve^2/c^2). The appendix "0" denotes that the attached quantity belongs to the ether frame. Use the GTE's to transform the absolute coordinates and velocity to the ether frame: X0 = X - VE T (19) T0 = T (20) V0 = V - VE (21) Then use the fact that in the ether frame, relative and absolute units are the same, so we can equate x0 = X0, t0 = T0 and v0 = V0 to get the following three equations: X - VE T = ge ( x - ve t ) (22) T = ge ( t - ve x / c^2 ) (23) V - VE = ( v - ve ) / ( 1 - ve v / c^2 ) (24) Applying (24) to the starting frame itself, which has v = V = 0, we immediately find that VE = ve. It does not matter whether the ether velocity is measured in relative or absolute units, it must have the same numerical value eigther way. Using this fact, (22) and (24) can be simplified to the following: X = x / ge (25) which just indicates that the absolute and relative position coordinates are related by the length contraction factor as in equation (7), and c^2 - ve^2 V = v ------------ (26) c^2 - ve v Applying (26) to the velocity of light we find C = c + ve (27) as one would expect. INVARIANTS The quantities in equations (16) through (21) are all invariants. They are the same for all observers in different frames of reference, moving with respect to each other; and they are even the same for observers who use units of measurement which are relative or absolute. The formulas depend on which type of unit is being used, but the numerical value does not. Equations (16) through (18) give the expressions for the invariant position, time and velocity in terms of relative coordinates and they are all Lorentz invariant. Equations (19) through (21) give the expressions in terms of absolute coordinates and they are all Galilean invariant. Example 1. Eq. (19) is a Galilean invariant expression for the absolute position. Proof: Starting with X0 = X - VE T use the GTE's to transform to another frame moving with velocity U: X' = X - U T T' = T VE' = VE - U Therefore: X0' = X' - VE' T' = X - U T - ( VE - U ) T = X - VE T = X0 Q.E.D. Example 2. Eq. (17) is a Lorentz invariant expression for the absolute time. Proof: Starting with t0 = ge ( t - ve x / c^2 ) use the LTE's to transform to another frame moving with velocity u: x' = gu ( x - u t ) [where gu = 1/sqrt(1 - u^2/c^2)] t' = gu ( t - u x / c^2 ) ve' = ( ve - u ) / ( 1 - u ve / c^2 ) c' = c [ see eq. (15) ] ge' = 1 / sqrt( 1 - ve'^2 / c'^2 ) 1 = --------------------------------------- ve - u sqrt( 1 - ( -------------- )^2 / c^2 ) 1 - u ve / c^2 ( 1 - u ve / c^2 ) = ------------------------------------------------- sqrt{ ( 1 - u ve / c^2 )^2 - ( ve - u )^2 / c^2 } ( 1 - u ve / c^2 ) = --------------------------------------------------- sqrt( 1 + u^2 ve^2 / c^4 - ve^2 / c^2 - u^2 / c^2 ) ( 1 - u ve / c^2 ) = -------------------------------------------- sqrt{ ( 1 - u^2 / c^2 ) ( 1 - ve^2 / c^2 ) } = gu ge ( 1 - u ve / c^2 ) Therefore: t0' = ge' ( t' - ve' x' / c'^2 ) ve - u = gu ge ( 1 - u ve / c^2) ( t' - -------------- x' / c^2 ) 1 - u ve / c^2 = gu ge { ( 1 - u ve / c^2 ) t' - ( ve - u ) x' / c^2 } = gu^2 ge { (1 - u ve/c^2)(t - u x/c^2) - (ve - u)(x - u t)/c^2 } = gu^2 ge ( t + u^2 ve x / c^4 - ve x / c^2 - u^2 t / c^2 ) = gu^2 ge ( 1 - u^2 / c^2 ) ( t - ve x / c^2 ) = ge ( t - ve x / c^2 ) = t0 Q.E.D. The four remaining proofs will be left as an exercise for the reader. A special case of the invariant velocities is the invariant velocity of light. Using (18), the Lorentz invariant absolute velocity of light is c0 = (c - ve)/(1 - ve/c) = c (28) And using (21), the Galilean invariant absolute velocity of light is C0 = C - VE (29) These must, of course, be equal: c0 = C0. THE SPACE-TIME INTERVAL The Lorentz invariant absolute position, time and velocity all depend explicitly on the ether velocity. Thus, they cannot be evaluated until a method of measuring the ether velocity is discovered. However, any combination of these invariants will also be an invariant, and we may look for some special combination in which the ether velocity cancels out. This special combination is called the space-time interval and is given in terms of relative coordinates by: I = x0^2 - c0^2 t0^2 = ge^2 ( x - ve t )^2 - c^2 ge^2 ( t - ve x / c^2 )^2 = ge^2 ( x^2 + ve^2 t^2 - c^2 t^2 - ve^2 x^2 / c^2 ) = ge^2 ( x^2 - c^2 t^2 ) ( 1 - ve^2 / c^2 ) = x^2 - c^2 t^2 The invariance of this quadratic form in the relative coordinates is what makes possible the four-dimensional geometry of Minkowski space-time. It allows us to do calculations and solve problems without referring to the ether. It even allows the philosophically naive to pretend that there is no ether. But it must be remembered that relative coordinates are only a mathematical device. They are not real distances and times because they are not based on immutable standards of measurement. The interval may also be expressed in terms of absolute coordinates as I = X0^2 - C0^2 T0^2 = ( X - VE T )^2 - ( C - VE )^2 T^2 but the dependence on VE does not cancel out in this case, so this expression isn't particularly useful. EXTENSIONS TO THE LET The LET can be extended to include gravitational effects by allowing the ether to move differentially. The first such theory was by R. L. Kirkwood [Phys. Rev., vol. 92, pp. 1557 (1953), and vol. 95, pp. 1051 (1954)]. The second talk I gave at the NPA meeting in May 1997 was on the Kirkwood ether theory, but I am not going to bother writing up a summary of it because I don't believe the theory is correct - although it is quite interesting and does illustrate some of the features that a correct theory would have to have. His theory gives the ether a velocity but no density or pressure and it does not allow the conservation of ether. A better theory has been found recently by Ilja Schmelzer. He calls it the General Ether Theory. It is mathematically equivalent to general relativity but uses Euclidean space and absolute time. He gives the ether a density, velocity and pressure tensor and satifies all the appropriate conservation equations, but there are still many unresolved issues concerning the proper way to interpret what his theory means. I will not discuss his theory further here, but those who are interested can get more information, including copies of his papers, from his web site: http://get.ilja-schmelzer.net/ Return to home page