New Experiments to Constrain the Coefficients of the 

Robertson Transformations for Inertial Systems 

Contract NAG8-047 

Final Report for the Period September 30, 1985, to September 30, 1986 

(November, 1986 1 

Prepared By 

Douglas G. Torr 

Principal Investigator 

Larry L. Smaley 

Co-Principal Investigator 

School of Science 

Huntsville, Alabama 35899 

The University of Alabama in Huntsville 

Prepared For 

National Aeronautics and Space Administration 

Marshall Space Flight Center, Alabama 35812 

George C. Marshall Space Flight Center 

(NASA-CR-179908) NEW EXPEI3IMEhTS TO Nd7- 12320 

CCNSTHBIN THE COEFFICIEhlS CY ?BE ROO~R?s~~ 

PRANSFO€IFATICES FCR Ih’EETIAL SYSTEMS kirtai 

iidt&ort, 30 Scp. 3985 - 30 SeF, 1386 [hlahama Unclas 

Univ,, Huntsville.) 36 p CSCL 2uc G3/70 44925 
t 

Final Report on NASA/MSFC Grant NAGS-047, entitled: 

"New Experiments to Constrain the Coefficients of the 

Robertson Transformations for Inertial Systems" 

SUMMARY 

The work proposed under NASA/MSFC Grant Number NAG8-047 was 

successfully completed. Under this contract, a feasibility study was 

conducted to evaluate a measurement of the variability in the one-way speed 

of light by a direct-time-of-flight approach. 'Ihe proposed experiment 

design was successfully completed and the initial tests indicated that the-. 

approach is viable. The experiment is now being carried out under new 

funding . 

1. Introduction 

The purpose of Grant NAGS-047 was to provide for the design and 

commencement of a laboratory version of an experiment to measure variations 

in the one-way velocity of light, a topic in fundamental physics that is 

receiving increasing attention in the literature. The experiment had 

previously been performed at Utah State University (USU), in Logan, Utah. 

* 

* 

G. Spavieri, Phys. Rev. A - 34, 1708 (1986) and "Accuracy of "me 

Transfer in Satellite Systems", Report by the ?:atfoilal Academy of Science 

(National Academy Press, Washington, D.C., 1986). 
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Re USU version of the experiment lacked the rigid controls that can be 

achieved in a laboratory environment. A new version of the experiment was 

designed with the collaboration of Mr. Dave Gagnon (Naval Weapons Center, 

China Lake, California). Work on the actual construction of the 

experimental setup was started. 'he effort made under this project 

continues at present under Grant NAG8-067. 

2. ExDerimental SetuD 

Numerous discussions were conducted among members of the group until 

the experimental setup of Figure 1 was finally adopted. 'he essence of the 

experiment consists in comparing the output of two cesium time standards 

after the signal of one of them goes through a ceramic waveguide with high 

refractive index. Phase locked oscillators were included in the system to 

operate at higher energies than those provided by the standards. 'his 

feature was expected to improve the signal to noise ratio. All this was 

mounted on a rotating table together with data acquisition and telemetry 

systems. Data were transmitted to external data reduction and monitoring 

systems through a fiber optic link. 

3. Tasks Performed 

3.1. Cesium Standards 

Cesliiiii standards ("closks") with high performance tubes were 

rented from Hewlet-Packard under a Lease-Purchase agreement. The 

frequency stabil-ity of the clocks was characterized at 100 XHz 

and 5 MHz. 
3.2. Rotating Table 

'Ihe desired characteristics of the rotating system were 

determined. An air turbine was chosen to perform the rotation of 

the table. Electric motors were discarded as they produce 

undesirable electromagnetic noise. The construction of the 

rotating mechanics was subcontracted with Reisz Engineering, 

Huntsville, Alabama. 

3.3. Waveguide Sys tem 

A microwave guide was designed by Mr. Gagnon. The 

dielectric substance chosen was barium titanate because of its 

extremely high dielectric constant. A company in Lambertville, 

New Jersy, was contacted for the construction of the wave guide. 

RFD, Ltd., of Tampa, Florida, provided us with high reliability 

phase locked frequency multipliers (oscillators). 5 MHz power 

attenuators were constructed to match the power requirements of 

the PLO's with the power output from the clocks. 
3.4. Data Acquisition 

A data acquisition system was designed. The search started 

for a system that would have the specifications of the design. 

Appropriate microprocessor controller and 12 bit digital to 

analog converter were ordered. 

3.5. Telemetrv Svstem 

Bidirectional optical compilers were constructed. An 

RS-232/optical 

system was designed. 

coupler circuit diagram for the interface control 

3.6. Data Storage 

A microcomputer with appropriate interfacing and data 

storage requirements was selected. 

3.7. Signal Processine 

Method and Software packsges for data analyses were 

selected. 

4. Data Taking 

Clock data were taken (not the actual data €or the present experiment) 

in order to IiiiprOve the analysis of the above mentioned expertnent at 
Page 5 

Utah State University. Basically, the noise of the clocks was restudied 

and used as input for some refinements in the analysis of that previous 

experiment. 

5. Theoretical Work 

Further theoretical work was done which resulted in a paper which is 

presently under consideration for publication by the journal Foundations of 

Physics (See attached copy). 

6. Budget Information 

A greater part of the budget than had been proposed was used for 

paying the rental of the clocks until a new grant from this NASA center was 

obtained to provide for the bulk of the hardware of the experiment. 
TENTATIVE VERIFICATION OF ROBERTSON'S PRIVILEGED 

"REST FRAME" HYPOTHESIS 

D. G. Torr and Tsao Chang' 

Department of Physics 

The University of Alabama in Huntsville 

Huntsville, Alabama 35899 

'On leave from Shanghai University of Science and Technology, 

Shanghai, China. 
Abstract 

Experimental results are reported in a companion paperI3 which, 

although marginal, suggests a possible detection of anisotropies in 

the one-way velocity of light which are consistent withthe motionof 

the solar system inferred from measurements of Smoot et al., of the 

2.X cosmic background radiation. We use Robertson's approach to 

theoreticallyinterprettheseobservationsusingMaxwell's equations 

for anisotropic space. We find that the results are consistent with 

the null results of other experiments, if the metric is velocity 

dependent. The physical cause of the anisotropy appears to be 

related to the Fresnel convection of light in a rotating system. We 

argue briefly that the results may provide experimental evidence for 

ten-dimensional space. 

1. Introduction 

Robertson' has devised a theoretical framework which has been 

generally used in analyzing experimental tests of Special Relativity 

(SR). His approach is based on the assumption that there exists 2 

priori a reference frame S(xg) which intrinsically possesses the 

properties of any "rest frame" of special relativity. In this frame 

light propagates isotropically at velocity c in vacuo. Space is 

assumed to be homogeneous so that the coordinate transformation 

equations between S(xP) and any arbitrary inertial frame S'(xt') 

1 
moving with relative velocity y with respect to S are given by 

Robertson' then reduces the 16 unknown coefficients a,b of (1) to 3 in 

the following way: (1) by the introduction of additional symmetry 

arguments; (2) bysuitablechoicesof originsandcoordinateaxes; (3) 

by the adoption of Einstein's clock-synchronization in S and SI. 

Robertson' then shows that if the Michelson-Morley (MM) , KennedyThorndike (KT), experiments yield exactly null results, and IvesStilwell (IS) the second order relativistic doppler effect, then the 

Lorentz-Einstein transformations are derived. 

Vargas2 s3 expressed the transformation equations in the most 

general formallowedbyobservationalconstraints taking intoaccount 

errors of measurement, namely 

x' = a(x - vt) 

t' = hx + jt 

y' = ey 

z' = ez 

where the coefficients a, e, h and j form a four parameter family of 

Robertson-Vargas (RV) transformations. Preciseknowledge of a, e, h 

and j determines the mechanics, and electrodynamics associated with 

the transformations, since each family of coefficients uniquely 

defines an associated physical theory, each of which must closely 

2 
resemble SR. Lorentz4 derived two sets of experimentally equivalent 

transformations referred to by Rindler5 as LTA and LTB. LTB are the 

Lorentz-Einstein transformations. The values of the coefficients 

for LTA6s7 and LTB are given in Table 1. 

TABLE 1 

Coefficient LTA LTB (SRL 

a 7* 7 

h 0 - +7v/c2 

j 7-' 7 

e 1 1 

*Where 7 = (1 - v2/c 2 ) -1/2 

The presence of terms in LTA4 which violate the symmetries 

associated with LTB became the primaryfactor whichdrove thephysics 

of the early twentieth century towards LTB. 

properties of LTB had become apparent, as had theapparent absenceof . 

groupstructure inLTA. ThusalongwithLTBaroseacertainformalism 

which, it was thoughta, would be seriously disturbed by the 

anisotropies of LTA, however small they might be. 

At that time the group 

While the RV transformations do not possess group properties in 

four dimensions, they do in seven dimensional space, which is the 

carrier space for the transformations. Vargas' finds that both LTA 

and LTB are isomorphic in the RV group, and that rather than 

3 
constrainingthephysics, the RVgroup structureexpands the possible 

physical effects to be observed, and might be important for other 

areas in physics. For example, of particular significance is the 

fact that according to Vargasg, space-time is at least sevendimensional. Vargas finds that a passive rotation of any arbitrary 

coordinate system is not the same as an active rotation of a physical 

system, and this adds three more dimensions raising the total 

dimensionality of space to ten. Thus any experimental test which 

confirms the existence of Robertson's hypothesized rest frame, also 

Implies a space-time of dimensionality equal to at least ten. 

Because it is impractical to consider testing all possible values of 

theRobertson-Vargas familyofcoefficientsa, e, hand j, werestrict 

ourselves, at this stage, to testing the LTA transformations. 

The LTA transformations which are derived from the coefficients 

given above in vector form are 

Here (R,T) arespaceand time coordinates in Robertson's rest frame S; 

(r,t) pertain to any other arbitrary inertial frame E; ,v is the 

velocity of F with respect to S. In order to avoid confusion with 

superscript notation, we do not use primes here and in Section 2. 

4 
Chang'' has derived Maxwell's equations for anisotropic space 

from Equations (3). Below we show that potentially these equations 

predict a detectable variation in the one-way velocity of light as a 

functionof the orientationof the propagation path, for propagation 

under non-vacuum conditions. 

gclv = 

2. Theoretical Analysis of the Propagation of EM Waves 

p1 0 0 

0 1 0 

0 0 1 

In terms of Equations (3), the expression for four-line element 

ds2 in F is 

Equation (4) can be rewritten in tensor form: 

herexp= (x,y,z,ct)andglrv isthe covariantmetric tensor. 

contravariant metric tensor is given as: 

Thenthe 

5 
For an electromagnetic (EM) wave in free space, the EM wave equation 

will be 

where A is a component of electric field Lor magnetic field 51. 

Substituting (6) into (7), we get 

This is anexplicit expressionof the EMwaveequation inanisotropic 

free space. 

Chang" has proposed two theories for deriving the wave 

equation. Althoughthetheorypublished in1983 is not rigorous, it 

gives asimplederivationof the EMwave equation inan infinitemedium 

without sources. Chang will present a rigorous treatment of the 

derivation in a separate paper. 

In terms of (4), a notation xo can be introduced: 

6 
whichdenotes the zerothspace coordinate in four dimensional space. 

Then (4) can be rewritten in a form which has four dimensional 

symmetry : 

ds2 = (ax")' - [(dx')' + + (dx3)I2 

Here xo is not equivalent to the time coordinate in theF frame (i.e., 

except for the privileged frame). The significance of this 

formulation is that physical laws have the same form in any inertial 

frame if xo is used as the zeroth spacecomponent. However, thesame 

physical laws will have different forms in different frames if xo is 

given by the quantity [ct - (,v r)/c]. The four dimensional symmetry 

is broken when we use (r,t) as variables. We recognize that if no 

physical consequence arises from this approach, this will be 

equivalent to a resynchronization. 

We need to derive the wave equation for propagation in a 

homogeneous medium of refractive index v. Following chang'sl' 1983 

illustrative approach, equations (9) and (10) yield the following 

relations for the differential operators: 

7 
The notations (a/ax0,V3) in (11) have the same definition as in 

special relativity, buttheymust bechanged into the operatorsshown 

on the right hand sides of (11). Following the above rule, one may 

write the wave equation in the form: 

d2A 

V,2A - q2 =o 

(dx0l2 

for anEMwavepropagating inauniformmediumwithouta source, where 

qistherefractiveindex. Nowwereplacethenotations (v3,a/ax0)by 

using (ll), and (12) becomes: .. 

Equation (13) can be rewritten as: 

Equation (14) is a fundamental equation for an EMwave propagatingin 

an infinite homogeneous medium in anisotropicspace. We assume that 

8 
. 

the solution for a plane wave in orm: 

Substituting (15) into (14), we get: 

aw ao a0 

-' ak, -1 ak, 

We define the group velocity as: 

- 

Equation (16) gives the solution: 

C 

7) + v cos e/c 

Here 8 is the angle between I! and the direction of the EM wave. If Q 

= 1, ug is the expression forthe one-wayvelocity of light invacuum. 

According to (17), it is easy to see that the roundtrip time interval 

foraplaneEMwaveinaninfinitestationarymedium (orinavacuum) is 

direction independent: 

9 
where L is the propagation distance. Therefore, the wave equations 

for propagationina homogeneous infinite space (which correspondsto 

the situation for most optical experiments such as those of Brillet 

and Hall", and Trimmer et al.I2, to cite two examples) yield null 

results. Equation (18) also gives a method to measure the index of 

refraction 7). The expression for the group velocity of light for 

propagationinamovingmediumintheS frame is derivedin Appendix I. 

' 

3. Experimental Arrangement and Measurements 

- 

Detailsoftheexperimental arrangement are givenin acompanion 

paper by Kolen and Torr' 3. Only a summary description is given here. 

Measurements of theone-way time-of-flight of EMwaves weremade 

by propagating a 5 MHz wave generated by a stable Cesium atomic 

frequency standard with high performance beam tubes. The 5 MHz 

signal was injected intoa 1km coaxial cable filledwith drynitrogen 

maintainedatapressureof 2psiaboveambient. Thecenter conductor 

was supported by a polyurethane spiral helical structure. The 

refractive index was 1.04. The roundtrip time-of-flight was also 

measured simultaneously by reflecting back part of the signal by an 

impedance mismatch at the end of the cable. The phase of the source 

and reflected signals was compared using a linear phase comparator 

10 
. I 

which yielded an equivalent detection threshold for the time-offlight of -2 x lo"* seconds, 

To make the one-way measurement, a second identical frequency 

standard was located at the west end of the cable, which was oriented 

in the east-west direction. The arrival time of a point of constant 

phase was compared at the west end with the phase of an identical 

signal generated by the west-end clock. Ideally, the difference in 

arrival times between the phase points at the phase comparator 

represents an arbitrary "turn-on" phase difference between the two 

unsynchronized standards at t= 0 plus anadditional time difference 

directly dependent on the one-way time-of-flight of the phase point 

propagated through the cable. A detailed technical description of 

the experiment has been given elsewhere' 9 ' 4. 

Laboratory calibration at zero separation yeilded a mean value 

of Av/v = 2.5 x The 

clock housings were isolated from external electrical and magnetic 

influences and the temperature was actively controlled. Active 

modulation of the thermal environment was introduced via random 

cycling of the heating and cooling periods. Clock and cable 

temperatureswere monitoredevery lominutes. The thermallyinduced 

variations in frequency were computed using a predetermined 

functional relationship. Analysis of the thermally induced 

variations confirmed that over a period of approximately one month, 

frequency stability over a diurnal cycle. 

the randomization procedure reduced the average value to below 

11 
detection threshold (Av/v = 3.5 x lomt5 for one month's 

integration). Data were taken for 150 days over 1 1/2 years. The 

detection threshold was estimated to be AV/V 1 x Data were 

averaged over 149 sidereal cycles. 

4. Tentative Analysis of the One-way Time-of-Fliqht Results 

Results for the one-way time-of-flight are shown in Figure 1. 

Theexperimentyieldedadeparture from aconstant value for thetimeof-flight parameterized by (see Figure 1) 6t = 80 x lomt2 sin(wt) 

seconds forL=lkm,whereherewis theangular velocityof theearth. 

From the perspective of an observer in the Robertson frame, the 

oscillator frequencies of the clocks are retarded according to (3), 

and following the sameprocedure usedin specialrelativitytoderive 

the doppler shifted received frequency, v,, we find it is given by 

(Appendix 11) 

Where Err ICs, and 0 are the wave vectors and angular frequency 

respectively observed from the S frame. The square root factor in 

(19) comes from the effect of time dilation due to the motion of the 

source and receiver. And we have 

12 
,I v =v+gs -r v=v+llr 

where (20) effectively defines the relative velocities g, and gr, 

between source, receiver and the laboratory measured In the S frame 

respectively. We also note that (20) is the appropriate velocity 

addition law In the Robertson rest frame. 

Intermsof (19), (20)and (111.4) inAppendixII1, the frequency 

difference recordedbythe receiver, i.e. calculated for the receiver 

frame is given to second order in v/c by 

(Ur - u,) 'V+U, 2 -us 2 

+ 

C2 2c2 

where c* = c/q. 

Note that we introduce here yr and y,, the velocities 

respectively of the medium near the receiver and source with respect 

to the S frame, e is the unit vector in the direction of propagation. 

13 
I 

Here the fourthand fifthtermsontheright handsideof (21) come from 

theexpansionofthesquareroot portionof (19). Tothe first order, 

theabove four equations arevalid forawavepropagating ina co-axial 

cable. From (3) and (17) we derive the following relations to the 

first order in v/c: 

and from (21) and (22), 

Suppose the medium is at rest in the laboratory system, i.e. 

TJ, - TJ, = v. Then (23) reduces to null result: 

- 

14 
The reason for this is given below. In deriving (17) for the 

group velocity, we used (14) where the metric isdetermined solelyby 

the uniform Robertson velocity of the laboratory. When the medium 

actually moves with source and receiver, the motion within the 

laboratoryframeaffectsthe metriclocally. In this case themetric 

depends on l-7, and E,, i.e., on E, and us. Since Chang's'' metric 

yields a null result, we suggest that the group velocity of light 

propagatingin themovingmediumdepends onlypartiallyonthe earth's 

rotation. The physical implication would be that light is only 

partially Fresnel convected in the case of rotational motion. In 

this case y, and v, can be generalized as 

Inorder torepresent thepartial Fresnelconvection, the coefficient 

ais introducedwhichhasarangeo I as I. It is obvious that if a 10, 

E, - 1, = v, which is the null result case. On the other hand, if a = 1, 

i.e., if there isno differencebetween theconvection androtational 

motion, va = YS, v, - xrr and (23) becomes 

- 

- 

15 
Since (gtr - SI,) 0 E= -wLv cos u't', we obtain 

where L is the distance between the receiver and source. Then 

For 6t1 = 80 95 ps, we find y in the equatorial plane to be less than 

163 kms-' with the vector pointing to z11:30 hours R.A. If 0 I a I 1, we 

have partialFresne1 convectionof light and v> 163km. The valueof 

a must be determined by experiment. In terms of (25) the following 

16 
, # 

result is derived for the fractional frequency variation predicted: 

which Is a generalized form of (27). We also have 

6t' = - LV (q2 - 1) a sin ultl/c2 

If we assume v = 390 km/s measured by Smoot, et al. ' 5, Equation (30) 

yields dt' = 80 sinu'tl psfor a= 0.46. Thephysical implicationis 

that the rotational motion of the earth results in only partial 

-- 

Fresnel convection of the EM wave. As mentioned above, this 

sensitivity of the metric to angular motion means that space-time 

exhibits velocity dependent properties in rotating homogeneous 

media. This result suggests the interesting conclusion: that 

within the context of the Robertson preferred frame hypothesis, a 

Riemannianmetric is consistent with the null predictions of special 

relativity, whereas if space-time is velocity dependent, the 

associated velocity dependent metric allows for the non-null case. 

It shouldbe noted that thisspace-time isa moregeneral depictionof 

the usual Riemannian case. Equation (30) which is an experimental 

result, if confirmed, will define the form ofthe metric. The reason 

17 
the dependence on rotational motion only shows up in the one-way 

results is because, in this case, we rely on the rotational motion to 

reversetheorientationof theexperiment. These resultsare ingood 

agreement in phase and amplitude with the results of Smoot, et al. l5 

In the case of the rocket-borne hydrogen maser experiment of Vessot 

and Levinel', and others where = 1, null results are derived. 

5. Discussion 

The theoretical significance of these results lies in the 

interpretation of what is thequantitythatclocks actuallymeasure. 

Since no clock synchronization was used in the experiments reported 

here, the results serve to determine the appropriate temporal 

coordinate to be identified with the measurement of time i.e., the 

results, if confirmed, showthatxo=ct- (l/c) 1 * rand notct. 

actual character of space-time, however, remains unchanged, even 

though the appearance of the metric becomes frame dependent. The 

invariance of the line element is preserved, although we must expect 

small new terms to appear in the equations of mechanics, 

electrodynamics, gravitation, particle physics, quantum mechanics, 

and other areas of Physics which, nevertheless, closely resemble 

relativity in numerical values. Vargas and Vargas and Torr' 

discusses the group properties of the LTA group in the context of the 

larger RV group and the impact on other areas of physics in detail. 

The 

18 
Acknowledqements 

Thisworkwas supported byNASA GrantNAGW-119 andNAG8-047. We 

thank M. R. Torr for laboratory support, Dr. V. Gordon Lind for 

departmental support at Utah State University, Y. Clifton and J. G. 

Vargas for useful discussions. 

19 
APPENDIX I 

The Velocitv of Light in a Moving Medium 

According to Equation (14), the four-line element in a mediumrest frame F (g,t) is 

ds2 = gpvd x hdxv = (dg)2 - 1/Q2(cdt - * dy/C)2 (1.1) 

Using the transformation Equation (3) 

We obtain 

Where - R,t are the space and time coordinate in S frame. For the 

propagation of light, let ds2 = 0, Equation (1.2) yields 

20 
I I' : '1' 1/11') I' u, = c 1 + - y'(1 - 

TI 

1 

where the upper case U, refers to the velocity of light in S. 

Here e is the unit vector of the velocity of light. 

To the first order of (v/c), (1.3) reduces to ._ 

which is the same as Fresnel's formula for the convection of light. 

The vector form of (1.4) is 

21 
I 

APPENDIX I1 

The Transformation of the Frequency and Wave Vector 

Consider that the phase angle is an invariant 

(11.1) 

Where fl and care the angular frequency and wave vector in Robertson 

rest frame S, and w and Lare in frame F. Using the transformation 

Equation (3), we obtain 

1 - 7-1 

- 

(11.2) 

(11.3) 

Assume a source with frequency v, moves withvelocityy, withrespect 

to S, Equation (11.2) becomes 

(11.4) 

Assume the receiver moves with velocity y, with respect to S, 

Equation (11.2) gives 

the 

22 
2nVr = YrfI(1 - v, * - Kr/Q) 

Equation (11.5) divided by (11.4) yeilds 

(11.5) 

(11.6) 

This is the Doppler formula [Equation (19)]. 

23 
APPENDIX I11 

The Relationship Between Land g 

We start with (16) 

+- (1 &)w - k2 = 0 

Replace 0, kby using (11.2) and (11.3). We obtain 

.- 

(111.1) 

In terms of the definition of group velocity, 

where the upper case tJr represents the velocity of light in the s 

frame. We have 

24 
I 

(111.2) 

Solving for WI), to first order, 

.- 

Substituting (1.5) into (111.3), andevaluatingK/natthelocationof 

the receiver and the source, we get 

wherev,,,arethevelocitiesofthemediumwithrespect to the S frame 

and c = c/q. Substituting (111.1) into (19), we obtain * 

25 
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Relativity, Found. Of Phys., 14, 625, 1984. 

'H. A. Lorentz, The Theory of Electrons, Dover Publications, Inc., New 

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5W. Rindler, Am. J. Phys., 38, 111, 1970. 

'H. A. Lorentz, Amst. Proc., g, 809, 1904. 

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"A. Brillet and J. L. Hall, Phys. Rev. Lett., 42, 549, 1979. 

'*W. S. N. Trimmer, R. F. Baierlein, J. E. Faller, andH. A. Hill, Phys. 

I3P. T. Kolen and D. G. Torr, Test of the Dimensionality of Space by 

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issue, Phys. Rev., 1986. 

I4D. G. Torr and P. Kolen, Found. of Phys., 12, 265, 1982. 

Rev. D, 88 3321, 1973. 

15G. F. Smoot, et al., Phvs. Rev. Let., 39, 898, 1977. 

26 
16R. F. C. Vessot and M. W. Levine, General Relativity and 

Gravitation, 10, 181, 1979. 

27 
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