1

UG =−35

GM 2

RUG =−

35

GM 2

R

Relativistic Time Measurements and OtherExperiments Require Going Beyond Einstein

to the Yilmaz Theory of Gravity

Carroll Alley

Department of Physics

University of Maryland at College Park

Talk at the Science Session of Commission 31

XXVIth IAU General Assembly, Prague, Czech Republic

21 August 2006

2

UG =−35

GM 2

RUG =−

35

GM 2

R

“Deputy-General Relativity”* should now take command

*Professor John Wheeler’s name for the Yilmaz Theory,chosen to encourage its development and application

(Theme of the speaker’s talk at the “Wheeler Fest”In Princeton, Feburary 2006)

Th

3

“My field equations are like a house with two wings: The left-hand side is made of fine marble,

but the right-hand side of perishable wood.”

-- Albert Einstein

The Yilmaz formulation of relativistic gravity completes Einstein’s field equations, making the right-hand side also

of fine marble.

4

SOME OF THE STRENGTHS OF THE YILMAZ THEORY

(FULFILLING EINSTEIN’S HOPES FOR GENERAL RELATIVITY)

N-Body interactive solutions, exact in the low velocity limit

Exact multimode gravitational wave solutions of arbitrary strength

Compatibility with quantum theory

These, and many others, follow from the introduction of a true

tensor for the gravitational field stress energy instead of

the coordinate artifact (“pseudotensor”) of general relativity

5

OUTLINE

1. The speaker’s path to advocacy of the Yilmaz theory:

THE NEED TO COMPREHEND EXPERIMENTAL OBSERVATIONS

2. Introduction to the Yilmaz Theory from the Principle of Equivalence

3. Some experiments requiring the Yilmaz Theory:

Lunar Laser Ranging measurements showing that

gravitational binding energy gravitates

Proper Time measurements showing that clock rates are independent

of latitude on the rotating Earth

The measured isotropy of the local one-way speed of light on rotating

platforms (recent results of Professor Yan Hua Shih et al. at UMBC)

6

OUTLINE (continued)

4. Mathematical structure comparisons between GR and the Yilmaz Theory:

Consequences of the Freud decomposition and Identity

5. Some properties of the Yilmaz theory of importance for astrophysics:

Compact objects exist without event horizons - No “Black Holes”

Radiation can always escape into a narrow cone, extremely redshifted

No singularities - the Kretschmann invariants are zero, not infinite

Exact gravitatiional wave solutions exist transporting energy and

Momentum in accordance with the analog of the Poynting theorem

7

Key Postulate of Special Relativity

The speed of light is isotropic and has the value c

(c = 2.99792458 x 10^8 m/s)

whenever measured by an INERTIAL OBSERVER

This was given geometric expression with spacetime diagrams by the mathematician Herman Minkowski

[recall last talk of Einstein , Princeton, 1954]

The Yilmaz theory extends this property of the local speed of light to ACCELERATED OBSERVERS

and to OBSERVERS IN A GRAVITATIONAL FIELD

8

Minkowski spacetime diagramsand Einstein time comparison

Relativistic Doppler Factor,called k by Hermann Bondiin his “k-calculus”

t =t1 + 12 t3 −t1( ) = 1

2 t1 + t3( )

x= 12 t3 −t1( )c

k =1+v c

1−v2 c2=

1+v c1−v c

kAC =kBCkAB =kABkBC

9

Establishment of an axis ofsimultaneity using light pulses

“Slicing up” of spacetime(Minkowski, 1907)

Events 1 and 2 are simultaneousfor observer A

for observer B: 2 occurs before 1for observer C: 1 occurs before 2

10

Branches of the invariant Hyperbolas

c2 ′t 2 − ′x 2 =c2t2 −x2

=±1

Diagram for proving theInvariance of the interval

′t1 = kt1, t3 = k ′t3

11

′t1 = kt1, ′t3 =1

kt3

′t −′x

c= k t −

x

c⎛⎝⎜

⎞⎠⎟

′t +′x

c=

1

kt +

x

c⎛⎝⎜

⎞⎠⎟

′t 2 −′x 2

c2 = t 2 −x2

c2

c2 ′t 2 − ′x 2 = c2t 2 − x2 ≡ s2

k-calculus derivation of the invariant interval

Multiply equationsTo derive

Invariant interval

Invariant interval

add and subtract equations to derive

Lorentz transformation

12

Lorentz Transformations

The algebraic relations between the different coordinates (t, x) and (t’, x’) assigned to the same event by the two

different observers can be found by adding and subtracting

the earlier pair of boxed equations:

′t =1

2k +

1

k⎛⎝⎜

⎞⎠⎟t −

1

2k −

1

k⎛⎝⎜

⎞⎠⎟x

c=

1

1− v2 c2t −

vx

c2

⎛⎝⎜

⎞⎠⎟

′x =1

2k +

1

k⎛⎝⎜

⎞⎠⎟x −

c

2k −

1

k⎛⎝⎜

⎞⎠⎟t =

1

1− v2 c2x − vt( )

13

dτ 2 =dt2 −v2dt2

c2 = 1−v2 c2( )dt2

dτ = 1−v2 c2dt

Proper Time Increment Coordinate Time Increment

c2d ′t 2 −d ′x 2 =c2dt2 −dx2 =ds2 ("metric")d ′x =0, d ′t ≡dτ proper timeincrement

Differential form of the Invariant Intervaland the definition of Proper Time,

a clock’s own time (eigenzeit)

ds2 =c2dτ 2 =c2dt2 − dx2 +dy2 +dz2( )

14

“Route Dependence of Proper Time

τ A final −τ A initial = 1− vA2 c2dt

initial

final

∫

τ B final −τ B initial = 1− vB2 c2dt

initial

final

∫τ A final −τ A initial ≠ τ B final −τ B initial

The elapsed proper time of an ideal clock does not depend explicitly on itsacceleration, only on the above integrals. A real macroscopic physical clock will, of course, be affected by the stresses associated with its acceleration.

15

Principle of Equivalence

Rocket isAccelerating

upwards

16

Introduction of gravity by the Principle of Equivalence

17

k =1+

vc

1−v2 c2=

1+ tanhθ

1− tanhθ( )2=coshθ +sinhθ =eθ

kAC =kABkBC =eθABeθBC =eθAB+θBC

Exact Special Relativistic Implementation of the Principle of Equivalence

v

c=tanhθIntroduce velocity parameter θ:

⇔

18

The Principle of Equivalence predicts that relative clock rates are affected by the Newtonian Gravitational Potential

Treat the acceleration of “Aclab” in a succession of comoving inertial framesof reference matching its increasing velocity. In each of these

dv

c=d tanhθ( ) =

1coshθ( )2

θ=0

dθ =dθ

dθ =dvc=

gdtc

=gc

⎛⎝⎜

⎞⎠⎟

dxc

⎛⎝⎜

⎞⎠⎟=

gdxc2

θ(x) = dθlow

high

∫ =1c2 gdx

low

high

∫ =φ(x)c2

φ(x) isthepotential differenceof the force field sensedinAclab. ThisistobeidentifiedwiththeNewtoniongravitational potential differenceinGravlab.

19

Clocks in a Gravitational Field

Δτ ≠Δt

Δτ = k(x)Δt = eθ (x )Δt

Δτ = eφ(x )

c2Δt

This result suggests generalizing the invariant interval ofMinkowski from special relativity to curved spacetime

ds2 =c2dτ 2 =c2dt2 − dx2 +dy2 +dz2( )

⇒ g00c2dt2 + 2 g0icdtdxi

i=1

3

∑ + gikdxidxk

i,k=1

3

∑For the clocks at rest (no spatial differentials)

ds2 =c2dτ 2 =k2c2dt2 =e2φ(x)

c2 c2dt2 , g00 =e2φ(x)

c2

20

Analogy of the relation between proper time and coordinate timein curved spacetime to that between proper physical distance and the

coordinate difference of longitude on the curved spherical Earth

(Δs)2 =R2 (cosβ)2 (Δα)2 +R2 (Δβ)2

The proper physical distance for one degree difference inLongitude depends on the latitude

Metric coefficients

(β =45o)

(β =0o)

21

CURVED SPACETIME CHRONOMETRY

(J. L. Synge)

The use of Atomic Clocks and Laser Light Pulses in the examination of

gravitational fields may be compared to the use of

iron filings on paper over magnets and

horsehair cuttings on the surface of oil over charged electrodes

in the examination of magnetic and electric fields

(demonstrations by Michael Faraday in mid 19th century led him,

and from him, James Clerk Maxwell, to the field concept)

22

Flying Atomic Clock Experiments carried out

during 1975-1977 by the Quantum Electronics Research Group

in the Department of Physics of the University of Maryland

in collaboration with

Leonard Cutler, Hewlett-Packard Company, chief designer of the cesium clocks

Gernot Winkler, U. S. Naval Observatory, Director of the Time Services Division

Ph. D. Theses:

Robert Reisse

Ralph Williams

23

Schematic diagram of the local flightsWith laser pulse time comparison

24

Radar tracking data converted to the expected gravitational andvelocity effects on time during the flight

ds2 =c2dτ 2 =e2φc2 c2dt2 − dx2 +dy2 +dz2( ) ≈ 1+

2φc2

⎛⎝⎜

⎞⎠⎟c2dt2 −v2dt2

⇒dτair −dτ ground

dτ ground

≈Δφc2 −

v2

2c2

25

Growth of the calculated proper time integrals during the flight andComparison of the predicted net effect with

the laser pulse time measurements

dτair −dτ ground( )∫

26

Flying atomic clock raterelative to ground atomicclock rate as an “altimeter”

27

How to determine the values of the spatial metric coefficients?

Physical conditions are imposed on the local speed of light:

that it propagate isotropically and have the local value c

Independent of the acceleration of the observer and independent of the strength of the local

gravitational field g = - gradient(φ).

These postulates of Yilmaz extend the key postulate of Einstein about the speed of light in special relativity.

28

In curved spacetime the d’Alembertian wave operator is:

W2ξ = 1

−g∂α −ggαβ∂βξ( ) =gαβ∂α∂βξ + 1

−g∂α −ggαβ( )∂βξ

Setting the underlined expression = 0 is the harmonic or DeDondergauge condition.

It eliminates the dependence on the gradient of thepotential (the operand ξ can be considered as ).

[ It is also necessary for compatibility with quantum theory,allowing the wave group velocity to match the Hamiltonian

particle velocity.]

=0g = det(gαβ )

29

Determination of the spatial metric coefficients

ds2 =e2φc2 c2dt2 − A2dx2 +B2dy2 +C2dz2( )

ds2 =e2φc2 c2dt2 −A2 dx2 +dy2 +dz2( )

−g=eφc2 A3

∂α −ggαβ( ) =0 ⇒ ∂i eφc2 A3 A−2( )

⎛

⎝⎜⎞

⎠⎟=∂i e

φc2 A

⎛

⎝⎜⎞

⎠⎟=0

eφc2 A=1, A=e

−φc2 , gii =−e

−2φc2

Isotropy requires A = B = C. We will also require in this limit that the metriccoefficients not be explicit functions of time and use the gauge condition

30

The exponential metric follows from compelling physicalArguments, one of which has just been presented

ds2 =c2dτ 2 =e2φc2dt2 −e−2φ dx2 +dy2 +dz2( )

The field equations are obtained by evaluating the Einstein-Hilbert curvature tensor for this metric.The gravitationsl field stress-energy

tensor appears added to the matter tensor

12 Gμ

ν = 4πGN

c4τ μ

ν + tμν( )

Einstein-Hilbertcurvature tensor

stress-energytensor of matter

stress-energy tensorof the gravitational field

In general relativity the stress-energy of the gravitationalfield is explicitly excluded as a source of spacetimecurvature. It is explicitly included as a source in the

Yilmaz theory: “gravity gravitates”

missing ingeneral relativity

31

The gravitational field stress-energy tensor in this low velocity case isthe standard expression for a scalar field:

tμν =−∂μφ∂

νφ+ 12 δμ

ν∂λφ∂λφ

The absence of this quantity in general relativity is responsiblefor the lack of interactive N-body solutions among massive bodies in that theory.

32

In the static or low velocity case which has been treated so farthe quantity is the Newtonian many-body gravitational potential:

φ xr

( ) =GNmA

xr

− xrA

+CA∑

The constant C can be chosen to make = 0 at the position of the observer giving the Minkowski valued metric (1, -1, -1, -1)

of special relativity.

The velocity of light has the value c locally.

ds2 =gμνdxμdxν =e2φc2dt2 −e−2φ dx2 +dy2 +dz2( )

φ=0⏐ →⏐ ⏐ c2dt2 − dx2 +dy2 +dz2( ) =0 for light

There is no need to transform to the Riemannian normalcoordinate system where the first derivatives on the metriccoefficients vanish to achieve the local Minkowski values

33

In the general case all known exact solutions have metric coefficients which are exponential functionals of a gauge field

gμν = ηe4 φ̂−12φ1̂( )

( )μν

whereη istheMinkowski metric

φμν =φ̂=

φ00 φ0

1 φ02 φ0

3

φ10 φ1

1 φ12 φ1

3

φ20 φ2

1 φ22 φ2

3

φ30 φ3

1 φ32 φ3

3

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

φ=trφ̂Under a gauge condition similar to the Feynman gauge of electrodynamics,

the gauge potential is a solution of the curved spacetime d’Alembert equation

W2φμ

ν =τ μν ⇒ σuμu

νvc→ 0⏐ →⏐ ⏐ ∇2φ=−σ

34

Important exact solutions in the general case are those fortransverse-traceless plane gravity waves of arbitrary strength

φ̂ =

. . . .

. φ11 φ1

2 .. φ2

1 φ22 .

. . . .

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

φ11 = −φ2

2 = 12 h+ (t, z)∑

φ12 = φ2

1 = 12 h× (t, z)∑

−g11 = e2 h+ (t , z )∑

−g22 = e−2 h+ (t , z )∑

−g11 = −g22 = cosh 2 h× (t, z)∑−g12 = −g21 = sinh 2 h×∑ (t, z)

These are exact multimode solutions which carry energy and momentumand there is an analog of the Poynting theorem for EM waves.

Solutions with such properties do not exist in general relativity

GR requires the weak field approximation and cannot handle strong fields

gμν ≈ημν +hμν +higher order terms hμν = 1

35

36

Laser Ranging to Retro-Reflectors on the Moon

37

• The Most Significant Physics Results of nearly 37 years of• continuous Lunar Laser Ranging are the following:

– The massive bodies Earth and Moon fall to the Sun with the same acceleration. That is, gravitational binding energy

gravitates like the mass equivalent of other energy “Strong Principle of Equivalence”

“Gravity gravitates”

– The Brans-Dicke Scalar-Tensor Theory fails since it predicts different accelerations for the Earth and the Moon

toward the Sun

– General Relativity can not account for the measurements since it lacks both N-Body interactive solutions and the

localization of gravitational field energy needed to calculate binding energy

– The gravitational theory of Yilmaz is required to describe the measurements

38

39

In Newtonian theory the gravitational binding energy of a spherically symmetric solid body can be found by calculating the work needed to pull the body apart, spherical shell by spherical

shell, and taking the parts to an infinite distanceOne finds

UG =−35

GNM 2

R

The ratio of the fractional binding energies , Earth to Moon, is

UG

Mc2

⎛⎝⎜

⎞⎠⎟Earth

UG

Mc2⎛⎝⎜

⎞⎠⎟Moon

=−4.18×10−10

−1.95×10−11 =21.4

In the Brans-Dicke theory the value of GN is a field quantity and depends on the distance of Earth and Moon from the Sun.

They would fall to the Sun with different accelerations

40

Relation between gravitational binding energy and the t00

component of the gravitational field stress energy tensor

t00 =−

18π

(∇ )2

UG = t00dv

0

R

∫ + t00dv

R

∞

∫ =−110

GM 2

R−12

GM 2

R

=−35

GM 2

R

This calculation cannot be done in general relativity because there Is no localizible gravitational stress-energy tensor

41

The general relativity theorists who analyze the lunar laser ranging data assume that is the many body Newtonian

potential. However this requires the Yilmaz theory!

42

Note N-body potential for the Yilmaz theory!

Harmonic Gauge

∂ν −ggμν( ) =0

⇒ δ =2αγ −β =1

43

Comment on the “Nordtvedt Effect” Nordtvedt studied in the context of the PPN formalism

whether the gravitational binding energy could affect thegravitational mass

MG =M 1+ηUG

Mc2

⎛⎝⎜

⎞⎠⎟

Laborious calculations give

η =4β − γ − 3

which vanishes for the general relativity values β =γ =1

The Yilmaz theory addresses the issue very simply since theequations of motion include the field stress-energy tensor

dpμ

dτ= 1

2 ∂μgαβ τ αβ + tαβ( ) ≡ 1−g∂ν ( −gtμ

ν )

44

For the exponential metric of the low-velocity limit, one finds

dpk

dτ=σ ∂kφ+∂kφ t0

0 −tii( ) =∂kφσ 1+

t00

σt0

0 −tii

t00

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥

In terms of densities, the corresponding Nordtvedt parameter is

η =t0

0 − tii

t00

Explicit evaluation yields

t00 −ti

i( ) = 12 ∇φ( )2 −1

2 ∇φ( )2( ) =0

There is no anomalous influence of the gravitationalbinding energy.

“Gravity gravitates” like other forms of energy

45

Global Clock Transport Flights summer solstice 1977

Thule

Andrews AFB

Christchurch, NZ

Andrews AFB

46Air Force C-141 and the “Two-Ton Timex”

47

The Earth as an elevator falling toward the Sun

“Floor” = Northern Hemisphere: Washington, USA (low clock)“Ceiling” = Southern Hemisphere: Christchurch, NZ (high clock)

Does the “Ceiling Clock” run fast with respect to the“Floor Clock”because of the difference in the Sun’s gravitational potential?

Before the clock flights and measurements were carried out in 1997,no clear answer could be obtained from the general relativity community.

The experimental answer is NO, to first order. The explanation lies in the Principle of Equivalence

48

Clocks on the rotating Earth

He stated that a clock at the equator would run slow withrespect to one at the pole because of its velocity

In fact, the change in gravitational potential from the equator to the pole on the oblate Earth just compensates the change in velocity.

(equatorial radius - polar radius = 21 km)

Identical clocks on the geoid run at the same rate, independent of theirlatitude.

The effect of gravitational potential on clocks was discovered byEinstein only in 1907!

Einstein’s “mistake” in his 1905 paper on special relativity

49

50

Experimental Results from the Clock Transport to Thule

Einstein 1905 Prediction (ECI Coordinate Time):

Andrews: Latitude 38.82 degrees velocity = 360 meters/s Thule: Latitude 76.53 degrees velocity = 110 meters/s Thule Clocks - Andrews Clocks: [-5.5 - (-61.7)] ns/day = 56.2 ns/day 4 days 224.8 ns

Experimental Measurement:Thule Clocks - Andrews Clocks after trip = 38 5 nsCalculated Effect of transport and dwell time = 35 2 ns

Prediction of Einstein in 1905 is clearly wrong!

dτ /dt= 1−v2 c2 ≈1−v2 2c2

dτ /dt−1=−7.2×10−13

=−61.7ns/daydτ /dt−1=−6.7×10−14

=−5.5ns/day

51

The change of Gravitational Potential on the oblate Earth from the equator to the pole compensates for the change in rotational velocity

The combinationφ−v2

2= geopotential = const.

where is the multimode Newtonian gravitational potential andv is the surface velocity of the rotating Earth. The geopotential is

constant along the geoid, which roughly coincides with mean sea level.

Such exact multimode solutions are not available in generalrelativity because of the absence of N-body interactive solutions.

They are available exactly in the Yilmaz theory.

Including the monopole and quadrupole terms, where R is the equatorial radius of the oblate Earth and β is the geocentric latitude

φ=−Gdm

′r∫ = −GM

r1+ 1

2 J2

R

r⎛⎝⎜

⎞⎠⎟

2

1− 3sin2 β( )⎡

⎣⎢

⎤

⎦⎥

52

One-way Speed of Lightin a Rotating Frame

Let r = radius = angular velocity

Is the locally measured speed of light in the rotating frame

c or r c ?

Most treatments of the ring laser gyroscope use r c. For r large, r = constant,

this is inconsistent with special relativity.

Yilmaz predicted c in his Boston Studies in the Philosophy of Science lecture of 1969 and in his MG4 paper of 1986.

53

There is great confusion in the literature about the Sagnac EffectM. G. Sagnac, Comptes Rendu, Acac. Sci. 157 209 (1913)

For example the paper

The Sagnac Effect: Correct and Incorrect ExplanationsBy G. B. Malykin

Physics-Uspekhi 43 (12) 1229-1252 (2000)

Contains 290 references to published papers!

The speaker renewed his study of the Yilmaz theory around 1981when he was perplexed whether to use c or r c

in the last leg of the planned LASSO (Laser Synchronization fromSstationary Orbit), an ESA/French experiment which never came

o full fruition

54

One-way speed of light experiment carried out by the Quantum Electronics GroupOf the Department of Physics on the University of Maryland at College Park

Ph. D. Thesis of Robert Nelson

WashingtonBeltway

55

• First Direct Comparison of the One-Way Speed of Light between the East-West and West-East Directions of the Rotating Earth

– Pulses of laser light were sent from the Goddard Space FlightCenter to the U. S. Naval Observatory utilizing a mirror on topof the Washington National Cathedral and sent back by aretro-reflector array. The sending and receiving timeswere recorded by a hydrogen maser atomic clock.

– A second hydrogen maser atomic clock was carefullytransported from the GSFC to the USNO and the timeof arrival (and reflection by an adjacent retro-reflector)of the laser pulse recorded by the transported clock.

– Nine repetitions showed the difference between the EW and WE transit times to be no greater than 100 ps, the limitation being systematic errors of unknown origin. Unfortunately this is the amount to be expected if the speed of light is affected by the surface speed of ~350 m/s.

– The isotropy of the speed of light was not established by these experiments. It is hoped to redo them with higher precision and reduced systematic errors in the in the near future.

56

Δt =1

2(t2 − t1) − (t3 − t2 )[ ] = t2 −

1

2t1 + t3( )

57

If speed of light =r ±c, Δt wouldbe80psIf speedof light=±c, Δt wouldbe0ps

Experimental resultswereinconclusive

58

We hope to repeat this experiment during the coming year with the following improvements:

The air path for the laser light pulses will be replaced with dark optical fiber between the College Park and Baltimore County campuses of the University of Maryland (critical Δt = 144 ps)

Better timing electronics

Better environmental isolation during the transp;ort of the hydrogen maser atomic clocks

Definitive results are expected

59

Light Propagation in aRotating Reference

System

Heyi Zhang, Xuehua HeAnd Yanhua Shih

Department of PhysicsUniversity of Maryland

Baltimore County

Unpublished work

(presented here with permission of the

authors)

60

61

Increase in frequency isaccompanied by decreasein wavelength

Decrease in frequency isaccompanied by increasein wavelength

λν = cas locally measured on the

rotating disk for each of the cw and ccw propagating waves.

The locally measured phasevelocity is c, not r c.

This agrees with the predictionof Yilmaz in 1969

62

Mathematical structure comparisons between General

Relativity and the Yilmaz Gauge Field Theory of gravity

A mathematical identity exists in Riemannian geometry,

the Freud identity, whose full implications are ignored in GR.

This has the following consequences:

Absence of interactions between massive bodies for solutions of the Einstein field equations

Replacement of the correct energy-momentum

conservation law of special relativity by the

incorrect law of conservation of rest-mass

63

Einstein’s Introduction of Gravitational Field Stress-energy

Dν ( 12 Gμν ) = 1

−g∂ν ( −g( 12 Gμ

ν ))−12 (∂μgαβ )( 12 Gαβ ) ≡0

12 (∂μgαβ )( 12 Gαβ ) =− 1

−g∂ν ( −guμ

ν )

Dν ( 12 Gμν ) = 1

−g∂ν ( −gτ μ

ν + −guμν ) ≡0

Bianchi Identity:

“Obstreperous term”(Schrödinger’s name)

Einstein transformed the obstreperous term into a plain divergence:

One can then write, putting the matter tensor

as required by Gauss’ Theorem for a conservation law.

τ μν = 1

2 Gμν

64

In 1918 Schrödinger evaluated the field stress-energy for the Schwarzschild metric in rectangular coordinates and found

every componentto be zero

He was very surprised and wrote a paper about it. Einstein replied that he was as surprised as Schrödinger, but that the

reason was that the Schwarzschild solution was only for one body. All would be well with the two body solution.

No such solution ever found in GR until recently.(Two-disk solution, but there is no force between them,

contrary to fact.)

Also in 1918, Bauer evaluated uμν for the Minkowski metric where

there is no gravitational field. Using polar coordinaates he found

uμν is a coordinate artifact, not a tensor. Einstein was severely

criticized by the leading theorists and mathematicians for using it !

uμν =0

uμν ≠0

65

Freud Decomposition and Freud Identity (1939)

Uμν = 1

2 Gμν +uμ

ν

∂ν ( −gUμν ) ≡ 0

∂ν ( −gτ μν + −guμ

ν ) ≡ 0

Freud identified a quantity

which satisfies the identity

He regarded the identity as a proof of Einstein’s conservation law:

However the Freud identity is an independent indicial identitydistinct from the Bianchi identity.

66

−gUμν = ∂α H μ

να

Hμνα =

δμν δμ

α δμσ

−ggνρ −ggαρ −ggσρ

Γνρσ Γα

ρσ Γσρσ

=−Hμαν

∂ν ( −gUμν ) = ∂ν ∂α H μ

να ≡ 0

Structure of the Freud Quantity and theIndicial Nature of the Identity

67

Correct Way to Write the Freud Decomposition (Huseyin Yilmaz 1988)

12 Gμ

ν =Uμν −uμ

ν =(τ μν + zμ

ν )−(−tμν + zμ

ν )

=τ μν + tμ

ν .

Express the true tensor as the difference between the non-tensors.Each non-tensor must therefore have a true tensor part added to

the same non-tensor part which cancels

In general relativity the gravitational field stress energy is suppressed from the field equatiions. Therefore

uμν =zμ

ν

The results of Schrodinger and Bauer are now understandable.

A non-tensor

zμν

68

The Non-tensor for the Standard Schwarzschild Metric

−gzμν =

1

2

−sinθ 0 0 0

0 sinθ 0 0

0 − M − r( )cosθ −sinθ 0

0 0 0 −sinθ

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

ds2 = 1−2Mr

⎛⎝⎜

⎞⎠⎟dt2 − 1−

2Mr

⎛⎝⎜

⎞⎠⎟

−1

dr2 −r2 dθ 2 + sin2θd 2( )

∂ν −gzμν

( ) = ∂r −gzθr

( ) + ∂θ −gzθθ

( ) (other terms = 0)

= ∂r − M − r( )cosθ⎡⎣ ⎤⎦+ ∂θ −sinθ( )

= cosθ − cosθ ≡ 0

The density divergence vanishes identically (factor 1/2 dropped)

69

Consequences of the Freud and Bianchi Identities

∂ν ( −gUμν ) = ∂ν ( −g(τ μ

ν + zμν )) = ∂ν ( −gτ μ

ν ) ≡ 0

Dν ( 12 Gμν ) = 1

−g∂ν ( −gτ μ

ν )−12 (∂μgαβ )τ

αβ

+ 1−g∂ν ( −gtμ

ν )−12 (∂μgαβ )t

αβ ≡0

−12 (∂μgαβ )(τ αβ + tαβ ) + 1

−g∂ν ( −gtμ

ν ) ≡ 0.

12 (∂μgαβ )τ

αβ ≡0 if tμν =0asinGR.

The vanishing of the density divergence is a general property

∂ν ( −gzμν ) = 0

The first term on the right vanishes by the Freud identity.Grouping terms,

Freud:

Bianchi:

Therefore,

70

The Equations of Motion dpμ

dτ= 1

2 ∂μgαβ τ αβ + tαβ( ) ≡ 1−g∂ν ( −gtμ

ν )

The interaction is carried by thegravitational field stress energy tensor!

This is analogous to electrodynamics where the Lorentz force is the divergence of the Faraday/Maxwell electromagnetic field

stress energy tensor

In the absence of , as is the case for general relativity,there is no interaction

tμν

Exact two-disk solutions have been found which show explicitlythe absence of interaction (Alley, Burrows and Yilmaz 1994).

Exact solutions of GR must exhibit DIVISORS OF ZERO!12 (∂μgαβ )τ

αβ ≡0

71

One-dimensional problem: two thin disks and a guard ringto suppress edge effects (like a parallel plate capacitor).

The potential is a function only of the coordinate z.

ExactTwo-diskSolution

72

ds2 =e−2φdt2 −e2φ(1+ε )(dx2 +dy2 )−e2φ(1+2ε )dz2

12 Gμ

ν =

1+ε 0 0 00 ε 2 0 00 0 ε 2 00 0 0 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

∇2φ+(1−ε 2 )tμν =τ μ

ν +(1−ε 2 )tμν

Einstein: ε 2 =1 Yilmaz: ε 2 =0

τ αβ =gαμτ μβ =

e2φ(1+ε)−e−2φ(1+ε ) ε

2

−e−2φ(1+ε ) ε2

0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

∇2φ≠0

Metric Solution for the One-dimensional Problem

73

12 ∂μgαβ =

−e−2φ

−e2φ(1+ε )(1+ε)−e2φ(1+ε )(1+ε)

−e2φ(1+2ε )(1+ 2ε)

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

∂zφ≠0

τ αβ =

e2φ(1+ε)−e−2φ(1+ε ) ε

2

−e−2φ(1+ε ) ε2

0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

∇2φ≠0

12 ∂μgαβ τ

αβ =−(1+ε)+ 2(1+ε) ε2=−(1−ε 2 )∇2φ∂zφ=0 for ε 2 =1

These exact two body solutions (ε = 1 and ε = -1) for the Einstein field equation exhibit divisors of zero so that the right hand side

of the geodesic equations of motion vanishes! This is a property of all solutions of the Einstein field equations.

It is a consequence of the Bianchi and Freud identities.

74

Remark about the Energy-momentum Conservation Law

This is expressed by the Freud Identity:

∂ν ( −gUμν ) = ∂ν ( −g(τ μ

ν + zμν )) = ∂ν ( −gτ μ

ν ) ≡ 0

The Freud quantity is equivalent to the gauge d’Alembertianfor the gauge field of the Yilmaz theory

∂ν ( −gτ μν ) ≡ 0

Uμ

ν = W2φμν − 1

−g∂α −ggρν∂ρφμ

α( )

when one considers the diferential form of the relation

gμν = ηe2 φ1̂−2φ̂( )( )

μν

75

How General Relativity derives the Geodesic Equations from theField Equations by violating energy-momentum conservation

12 Gμ

ν =τ μν =σuμu

ν

Dν (σuμuν )= 1

−g∂ν ( −gσuμu

ν )−12 (∂μgαβ )(σuαuβ ) ≡0

1−g∂ν ( −gσuμu

ν ) = 1−g∂ν ( −gσuν )uμ +σuν∂νuμ

Put ∂ν ( −gσuν ) =0 Conservationof rest mass!

Wrong! Energy−momentum −gσuμuνisconserved!

σuν∂νuμ =σ∂xν

∂τ∂∂xν uμ =σ

duμ

dτ

σduμ

dτ= 1

2 (∂μgαβ )(σuαuβ ) Geodesic Equations

76

Non-uniqueness of the General Relativity Central Body Metrics

ds2 = 1+2εM

r⎛⎝⎜

⎞⎠⎟

−1ε− 1+

2εMr

⎛⎝⎜

⎞⎠⎟

1εdr2 − 1+

2εMr

⎛⎝⎜

⎞⎠⎟

1+1εr2 dθ 2 + sin2θd 2( )

ds2 = 1−2Mr

⎛⎝⎜

⎞⎠⎟dt2 − 1−

2Mr

⎛⎝⎜

⎞⎠⎟

−1

dr2 −r2 dθ 2 + sin2θd 2( ) (ε =−1)

ds2 = 1+2Mr

⎛⎝⎜

⎞⎠⎟

−1

1+2Mr

⎛⎝⎜

⎞⎠⎟dr2 − 1+

2Mr

⎛⎝⎜

⎞⎠⎟

2

r2 dθ 2 + sin2θd 2( ) (ε =+1)

ds2 =e−2Mr dt2 −e

+2Mr r2dθ 2 + r2 sin2θd 2 +dr2( ) (ε =0)

ε = -1 Schwarzschildε = +1 Droste (r r - 2M) No Event Horizonε = 0 Yilmaz

77

Einstein-Hilbert Curvature Tensor for the Central Body Metrics(M/r = )

t r θ

12 Gμ

ν =

1+ε 0 0 00 0 0 00 0 ε 2 00 0 0 ε 2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

∇2φ+(1−ε 2 )tμν =τ μ

ν +(1−ε 2 )tμν

12 G0

0 =012 G0

0 =2∇2φ12 G0

0 =∇2φ+ t00

=τ 00 −1

2 (∇ )2

ε = -1 Schwarzschild Unphysical

ε = +1 Droste Unphysical

ε = 0 Yilmaz Correct NewtonianCorrespondence toPoisson equation and field energy

78

The Yilmaz theory completes Einstein’s theory, achieving Einstein’s wishes with exact solutions

for important physical systems.

The gravitational field stress-energy is a true tensor.

The successes of General Relativity use approximations to the Yilmaz theory.

There should be explicit recognition of this

paradigm shift to the new theory!