General Relativity Scalar-Tensor Theories Hot “Theory” A Talked About “Theory” B Matter’s...
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Transcript of General Relativity Scalar-Tensor Theories Hot “Theory” A Talked About “Theory” B Matter’s...
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