General Relativity

Scalar-Tensor Theories

Hot “Theory” A

Talked About “Theory” B

Matter’s (Bodies’) Equation of Motion

Light’s Equation of Motion

Real Clock rates

Spatial lengths…….

Space-Time coordinates are not fundamental,but are intermediate tools and chosen for their convenience.

LLRSolar System Ranging to Planets and Probes

Binary Pulsar timing

CosmologyGalactic Structures and Dynamics

Neutron Star, Black Hole structures

Calculate experimental observablesand compare with data.

Non-Linear GravitySun

Earth

Moon

R(SE)

r(EM)

r(EE’)

2

2

( ) ( ) is Newtonian interaction

( )

( ) ( ) ( ) contributes to Einstein's classic planetary perihelion precession (in black).

( ) ( )

( ) ( ) ( ) is 3-body interaction contribu

( ) ( )

GM S m E

R SE

GM S m E GM S

R SE c R SE

Gm E m M GM S

r EM c R SE

2

ting to lunar orbit EP test (in blue).

( ) ( ') ( ) is 3-body interaction proportional to Earth's gravitational binding energy (in violet).

( ') ( )

This interaction is 50 times larger than th

Gm E m E GM S

r EE c R SE

2

2

-10 -6 15

e lunar orbit interaction, resulting in:

( ) ( ) (4 2 ) ( ) / =

( ) ( ) ( ) /

2nd post-Newtonian order 4-body interaction (in red) --- is a 5 10 x 4 10 = 2 10 EP test

M G M NG U G c

M I M NG U G c

3

4

3

4

6 10 15

For the Sun-Earth interaction this 2nd Post-Newtonian order

interaction leads to potential EP violation in LLR at the level:

4 10 5 10 2 10

i j k l

ijkl ij ik il

i j k l

ijkl ij jk kl

ij

m m m mG

r r rc

m m m mG

r r rc

x

4( ) ( ) / ...ij i j i jij

with G M G M G U U cR

i

j k l

i

j k

l

Gravitational Self Energy pullingon Gravitational Self Energy

Compactifying pairs of mass sources intotwo bodies as indicated below leads to:

2nd Post-Newtonian order Gravity Interactions

2 42

2 2 2

22

2 2

ˆ ˆ11

2 28 2 2

1 11 2 1 2

4 2

i j i j i ij ij ji ii

i ij ij

i j i j k

i jijkij ij ik

Internal Body Coupling Generalizations

Gm m v v v r r vv vL m c

rc c c

Gm m G m m mv v

r r rc c

becomes

2 42

2 2 2

2

2 2

ˆ ˆ1( ) 1

2 28 2 2

21 1* * 2 1

4 2

ij i j i ij ij ji ii

i ij ij

ij ij ijk

i jij ijkij ij ik

v v v r r vv vL M I c

rc c c

v vr r rc c

'2

24

2

2

1 2 2 1 2 ....

( )1 2 ....

( )| |

ij ij

ooij i j i ij

iab ab

i i

ii i

i i

Gg U U

r r r rc

MGg

rc

M GGwith U and r r r

rc

These indicated mass and mass-squared parameters will in general individually differ from body inertial mass energies by bodies’ gravitational binding energies.

These indicated PPN coefficients will in general be renormalized by matter surrounding Solar System … just as G is renormalized.

4 meters

6 centimeters

velocity V

Velocity V+u

Why is Earth contracted 6 cm. in diameter parallel to its motion? Because the electrodynamics,including magnetic forces, Lorentz-Contracts the atoms and molecules of the Earth body.

Why is Moon’s orbit contracted 4 meters in diameter parallel to its motion? Because the gravitationaldynamics , including gravitomagnetic forces, Lorentz contracts the orbit dynamics.

2 2 2 2 21 / .... 1 / ....M E E E M M S Ma g Order u u u u c g Order u c

Orbit Lorentz contraction, perigee, perihelion, geodetic precession+SO, M(G)/M(I)

moon

earthsun

V

2 2

2 2

2

2 2

/ /

/

/ 2

/

t r V c so r r r V c

moving COE about r V c

But v COE change is about r V r

which also moves COE about r V c

Photo by NASA

LLR as Gravity Theory Laboratory

Gravity’s Non-linearity (self coupling) --- the beta parameterFrontier measurement today and can remain so.Measuring 2PN order gravity theory structureM(I), M(G)… M(gamma), M(beta)……

Universality (composition independence) of Free-fallCompetitive with lab tests and can remain so.

dG/dt (and even superior spatial G gradient) constraintFrontier measurement today and can remain so.

Any EP-violating cosmic accelerations?

LLR data’s need for gravitomagnetic interaction.

Geodetic precessionGravitational Spin-Orbit Force

Inverse Square nature of gravity

Action-reaction equality in gravity

Lunar orbit’s solar tidal energy and mass equivalence?

LLR through the decadesPreviously200 meters

0 0 0

0

0

( ) ( ) .....

( ( ) ....) sin ( )

( ) ( ) ....

Amplitudes and frequencies and their

LLR data can be thought of as an empirical time seriesfor the round trip times.

t

n n nn

n n n

t t t

A A t t t dt

with t t t

time derivatives are then related to and fit by

free initial conditions plus physical theory parameters including gravity theory.

Locked toward Sun

. .

. .

Eccentricity20905 km

Evection3699 km

Variation2956 km

ParallacticInequality110 km

Four Key Lunar Orbit Perturbations or “Inequalities”

8.9 Years

191 Days

K. Nordtvedt, LLR, 25 July 2003

f A 2f D A

2f D f D

Earth’s Mass-Energy consists of nuclear chromodynamic, electromagnetic, weak, kinetic, and gravity contributions.

All of Physics as we know it!

Gravitational Binding Water Molecule Oxygen Nucleus

Proton

u u

d?

How do all these forms of energy contribute to Earth’s Inertial Mass and Gravitational Mass?

2

2

2

2

( ) ( ) (4 2) ( )( )

1 ( )

( )

( ) (4 2) ( ) ( ) ( ) ( )1

( ) ( ) ( )

( ) (4 2) ( ) (1 ) ( ) ( )1 with extend

( ) ( ) ( )

Consequences of Gravity's Non-Linearity

G M G M MG R G

G c R

M Ia c G

M I G

M G M M M G U G

M I M G M I c

M G M M I U G

M I M G M I c

2 2

2 10 2 13

ed Lorentz Invariance

( ) ( ) ( )ˆ ˆ1 (4 3 ) + 3 in lowest order

( ) ( ) ( )

For Earth, ( ) / 5 10 and ( ) / 4 10

, , ( ), ( ) 2

ab

M G U G T rotI pp

M I M I c M I c

U G Mc T rot Mc

M and M are all subject to nd and higher order structur

e of theory

2 4

2

1 1 1 1 1ˆ ˆ( ) 1 1

2 8 2 2 2

21 1 2....

4 2

Limits of co

Post-Newtonian Gravity for Gravitationally-Bound Bodies

ij

i i i i j i ij ij ji ij ij

ij ij ijk

i jij ijkij ij ik

L M I v v v v v r r vr

v vr r r

ij

2 2 2

2 ' 2

uplings , , as bodies become test bodies:

( )

( )

( )

( ')

( )

ij ijk ij

i j i j

ij i j i j

ijk jk i k j i i j k

ijk ij k i j k

i i

G M G m G m m

G M m G m m

G m G M m m G m m m

G m G M m m

M I m

00

Every non-linear order of the theory is "in" the , , ...... PPN coefficients

( , )

"weak field" versus "strong field" --- a bit of Red Herring?

SS

G

U r t solar system potential U surrounding matter potential

g

2

3

2

3

1

3

2 2

3

1 2( ) 2 ( ) 2 ( ) ...

[1 2 2 2 ] , ( , )

2 ( , ) [1 2 ] , ( , )

1( , ) [2 ( 1) ]

2

nSS SS n SS

n

nn

n

nSS n

n

nSS n

n

U U U U c U U more

U U c U Time renormalization t R T

U r t U nc U G renormalization G G R T

U r t n n c U renormalizati

2

2

, ( , )

But from the spatial metric non-linearity the proper length variable gets rescaled:

[1 ]

The metric field in the solar system is not perturbative from an empty universe;

SS SS nn

on R T

r U d U

it is perturbative from it's place in the surrounding actual universe.

The lowest order PPN coefficients obtain contributions from every non-linear order

2

2 42

2

From Landau and Lifshitz, the 1 / Lagrangian for a collection of charged particles is:

1

2 28

How does an Object's internal interaction energy get converted into Inertial Mass?

i ii

i

c

ev vL m c

c

2 2

2

ˆ ˆ1 .....

2 2

Using equations of motion from least action

Inertial mass of system of interacting charges accelerating as a whole is:

1

i j i j i ij ij j

ij ij

i i

ii

e v v v r r v

r c c

d L L

dt v r

vM m

2 2 2 3

1 1 1

22 2

The inertia of the electric interaction energy results from the field lines of

mutually accelerated charges developing curvature wh

i j i j

i i i ij iji ij i ijij ij

e e e em v v r r

rc c c r

ich then yields net forces

proportional to that acceleration-driven field line curvature .

Same thing happens with nuclear forces, QCD forces, gravity forces.....

Accelerating sources

2

2

4 2

2

( ) ( ) ( )(4 3 ) valid in perturbative metric gravity

( )

(4 3 ) is another interpretation of ( ) / ( ) 1

2 10 ( ) / from LLR

/ /(4 3 ) (4 3 )

LLR constra

G r G U r

G c

G UM G M I

G c

Gg Sun c

G

dG dt dU dtH

G c

2

3

ins / via frequency shifts in lunar motion:

( ) ( ) {1 2 / ( )}

( ) ( ) ( ) ( ) / ( )

( ) ( ) {1 / ( ) / 2}

/ 10 from LLR

o o

o o o o

o o

dG dt

f t f t dG Gdt t t

t f t t t f t dG Gdt t t

r t r t dG Gdt t t

dG Hdt

Cosmic EP-violating acceleration field?Source? Dark Matter? Dark Energy? 72 – 24 – 4 !

14 2 7

3 2

2 8.9( )

| | 10 / 2 10

ar L A

yearsL L A

a cm s cH

a

15 2

2

3cos ( ) 2 /(8.9 )

2 ( )

is sidereal lunar frequency, is anomalistic frequency

For 1 a 4 10 / is detectable

But for ( ) /

Cosmic (Sidereal) EP-violating Acceleration?

c oo

o

ax t t years

x mm a cm s

U Mc

3

2

33 1 5

10 difference between Earth and Moon

this allows detection of cosmic gradient of = / at level

4 10 6 10 1/( )

This tests whether dark matter to ordinary matter interaction

fulfills the

U

e c

cm cT

-5EP? Recall ripples in cosmos are at the 10 level.

132 3

( ) 8 10 ( )GM

a ru V g Sunc R

Gravitational Spin-Orbit Force

A spinning body does not move on same gravitational trajectory as a test body.

A body’s angular momentum couples to the Newtonian gravitational gradient.

The orbitting Moon can be viewed as a spinning body.

r

u

V

2H O C. of M.

2

2

2

2

4

13

sin

2.3 10 sin 0.3

3 10

H OGA GAISO

GP GP EH O E

H O

E

GA GA

GP GPH O E

MM Ma C Lat g

M M M

MLat

M

M M

M M

If the gravitational forces betweenEarth’s water and core are not equaland opposite, Earth experiences a

self-acceleration along polar axis, andconsequently perturbs lunar orbit.

2 14

3

1Strength of Newtonian Solar Tidal Energy / 7 10

2

ˆ ˆ3 .......

Lunar orbit's Newto

Does Earth's ( )/ ( ) ratio become anomalous at the level of lunar orbit's Solar Tidal Energy?

S

r Q r c

GMQ I RR

R

M G M I

21 1nian level energy function ( )

2 2 is a constant of the motion in the limit the rotation

of the Sun's Newtonian tidal tensor can be neglected.

( ) / is the earth's Newtonian poten

e v u r r Q r

Q

u r Gm r

2

tial.

But a solar tidal energy contribution to orbit energy , in addition to constant part, oscillates

1at twice the synodic frequency, so therefore the "local" energy ( ) does as well.

2Lunar o

e

v u r

rbit's solar tidal energy does not "fall" with the Moon as do the "local" contributions

to lunar orbit energy. So exactly how this tidal energy affects moon's effective ( ) / ( ) ratio

requires sepa

M G M I

rate investigation in Post-Newtonian Metric Gravity.

LLR is close approaching the level of precision needed to measure this solar tidal contribution.

2 42

2 2 2

2 42

2 2

2

ˆ ˆ11

2 28 2 2

ˆ ˆ11

2 28 2

1/ Electrodynamics versus PPN Metric Gravity

i j i j i ij ij ji ii

i ij ij

i j i j i ij ij ji ii

i ij ij

e e v v v r r vv vL m c

rc c c

versus

Gm m v v v r r vv vL m c

rc c

c

2

22

2 2

22 2 2

2

2

1 11 2 1 2

4 2

1 1 1( ) 1 1 2

2 2 2

1 1ˆ ˆ

2

i j i j k

i jijkij ij ik

i j

i i i i ii ij iij

i j

i i i ij iji ij ij

c

Gm m G m m mv v

r r rc c

Gm mM I m v m r Q r

rc c c

Gm mm v v r r

rc

Locked toward Sun

. .

. .

Eccentricity20905 km

Evection3699 km

Variation2956 km

ParallacticInequality110 km

Four Key Lunar Orbit Perturbations or “Inequalities”

8.9 Years

191 Days

K. Nordtvedt, LLR, 25 July 2003

f A 2f D A

2f D f D

3

2

ˆ ˆ[ ] [ ] 3

yields Newton's Variational Orbit with cos2D perturbation

In limit Sun's Tidal Tensor [ ] is not rotating

there a lunar orbit energy constant of the motion:

1

2

M E S M S

S

M

GMa g Q r Q I RR

R

Q

e v

2 2 14

2

2

2 2

2

2 2

1( ) [ ] / 7 10

2

Does ( ) 1 / ? How about ( )?

12 4

with 1 / 02 2

E M M S M M

i jiCoE i i i j

i ij ij

i jii CoE

i ij ij

U r r Q r Q r c

M I m e c M G

m mv GE R m r r r

rc c

m mv GE m dR dt

rc c

2 2

Sun's metric field expansion contains gravitational mass ( ) but also

additional mass parameters entering various metric potentials

( ), ( ), .

[ ( ) / ( )] 1 (4 / 1 [ / ]earth earth

M G

M M etc

M G M I U Mc U Mc

U bein

2 2

's

1 / 2 ( 1/ 2) [ ( ) / ( )] [ ( ) / ( )]

( ) / ( ) 1 / ( ) / ( ) 1 ' /

Sun's gravit t

' 2

a

sun sun

g a body internal gravitational binding energy

But M M G and M M G

with M M G U Mc a

and are combinations of nd PPN order theory parameters

nd M M G U Mc

2 6

6

ional binding is [ / ] 4 10

So experiments which reach precision level 4 10 in measuring and

begin measuring 2nd PPN order and '

Can LLR reach this level?

What viable theory types can deviat

sunU Mc

e at 2nd PPN but not 1st PPN Order?

2 2

2 4 / 3

2

Space and Time Variation of Fundamental 'Constants'

( ) / ( ) '

[ ( ) / ] /

( ) / /

, [ ( ) / ]

f U U being electric energy content of body s matter

Then a U Mc c

U M scales as Z A among the element nuclei

From LLR and the U Mc dif

3

10 2

5

10

/ 1.6 10 ( ) /

Spectral lines from very distant objects in the universe suggest a bound on

1time dependence of 10

Strongest space-t

ference of between Earth and Moon

g sun c

dHubble Rate

dt

ime constraint on fundamental constant variation is from LLR?

(although / comparison will be theory-dependent)versus d dt

0 0 0

0 0

( )

Integrate left hand side by parts for time interval T

( )

If right hand side d

T T T

i i i ij i i ii ij

TT

i i i ij i i i i i ii ij i i

NewtonianVirials

dt m a r dt f r dt f ex r

dt m v v f r f ex r m v r

oes not grow linear in T --- equilibrium --- then

left hand side integrand is on average zero. That is the spatial

tensor virial with its auxiliary scalar virial. For

1

2

ij ji

i i i iji ij

f f

m v v f r

( )

For isolated body right hand side vanishes

ij i ii

f ex r

The eccentric and evective motion of the Moon's orbital motion produces oscillations of the

Virial 2 about its average value of zero. The vanishing

Virial Theorem and Lunar Orbit's Eccentric Energy

T V of the scalar and tensor Virials

are required for inertial and gravitational mass equivilents of this orbital energy being equal.

Eccentric energy is of order / with evective energy somewhat smallerem r

2 2

. Because eccentric

energy oscillates at sidereal frequency, possible range perturbations result at annual frequency

and twice sidereal minus annual frequency. Characteristic size is / .5 .

SmaSe g r c mm

ller evective motion contributions are at slightly different frequencies.

These "perturbations" are probably coordinate effects; the proper distance coordinate in

the variable Solar Potential differs

2 2 2

from the isotropic coordinate as

1 / / 2

r

r g r c g r c