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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Generalized Nordtvedt effect and tests of equivalence
principle for rotating bodies
Yu. V. Baryshev
Institute of Astronomy, St.Petersburg University, St.Petersburg, Russia
The most recent results of the lunar laser ranging (LLR) data analysis [1] gives
restriction on Einstein Equivalence Principle (EEP) violation due to Nordtvedt
effect ( proposed in [2]) at the level j LLR = (\Gamma0:7 \Sigma 1:5) \Theta 10 \Gamma13 , where j =
mG=m I \Gamma 1. It means that gravitational binding energy (j bind = E bind =Mc 2 =
4:63 \Theta 10 \Gamma10 for the Earth) gives the same contribution to both inertial and
gravitational masses with accuracy about 0.03%.
Here it is proposed a new application of LLR results for testing EEP violation
due to rotational energy of the Earth which corresponds to j rot = E rot =Mc 2 =
3:97 \Theta 10 \Gamma13 . In general relativity (GR) according to EEP a rotating body falls
down as non­rotating one (up to the tidal effects, which may be neglected at
present accuracy). To study theoretical predictions for possible violation of EEP
the Thirring­Feynman field gravity theory (FGT) is considered [3], [4]. It is based
on the principle of universality of gravitational interaction (UGI) which states
that the interaction Lagrangian has the form \Lambda (int)
= \Gamma 1
c 2
/ ik T ik where / ik is the
tensor potential of gravity field and T ik is the energy momentum tensor of any
kind of fields. It means that UGI states that gravity is universal force in the sense
that it ``sees'' any material field through its energy momentum tensor (including
gravity field itself).
Using Lagrangian formalism of the relativistic field theory the post­Newtonian
equation of translational motion for rotating bodies is derived in the frame of
FGT. The PN­3­acceleration of the rotating body in FGT has the form:
d ~ V
dt = \Gamma
`
1 + V 2
c 2
+ 4 'N
c 2
+ I! 2
Mc 2
'
~
r'N + 4
~
V
c
` ~
V
c \Delta ~
r'N
'
+ 3
Mc 2
Z
[~!~r]([~!~r] \Delta ~
r'N )dm
Following the method by Landau & Lifshitz ([5],p.331) for a post--Newtonian
derivation of rotational effects in relativistic gravity, we used for the velocity ~v
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of an element dm the form ~v = ~
V + [~!~r], where ~
V is the translational velocity
of the body, ~! is the angular velocity, ~r is the radius vector of an element dm
relative to its center of inertia, so that R
~rdm = 0, M = R
dm is the total mass
of the body, I is its moment of inertia.
The equation of motion of a rotating body shows that the orbital velocity of
the center of mass of the body will have additional perturbations due to rotation
of the body. The last term in this equation depends on the direction and value
of the angular velocity ~!. It has an order of magnitude v 2
rot =c 2 and may be
measured in laboratory experiments and astronomical observations. Note that
the binding energy does not enter into the equation of motion so that both FGT
and GR classical Nordtvedt effect are absent. A generalized Nordtvedt effect
includes possible rotational effects derived in FGT from UGI. The additional
perturbation terms in the equation of motion of rotating body are determined by
the direction and value of angular velocity: ffi g = (j 0 + j 1
cos\Omega t)g N . Taking into
account the inclination of the rotation axis of the Earth on the orbit around the
Sun and using the momentum of inertia of the Earth I = 0:33 MR 2 we get for
the constant perturbation the value j 0
FGT = +0:32 \Theta 10 \Gamma13 , and for the variable
part the amplitude j 1
FGT = 0:96 \Theta 10 \Gamma13 with the period P = 0:5P orb , where P orb
is orbital period of the Earth. These values are very close to the modern accuracy
of LLR and show the importance of further improvement of the LLR experiment
to test EEP violation for rotating bodies.
References
1. Anderson J., Williams J. Long­range tests of the equivalence principle. Class.
Quant. Grav., 2001, 18 (in press).
2. Nordtvedt K. Testing relativity with laser ranging to the Moon. Phys.Rev.,
1968, 170, 1186.
3. Thirring W. Field--theoretical approach to gravitation. Ann. Phys., 1961, 16,
96.
4. Feynman R., Morinigo F., Wagner W. Feynman lectures on gravitation. Cal­
tech, Addison--Wesley Publ. Company, 1995.
5. Landau L. D., Lifshitz E. M. The classical theory of fields. Pergamon Press,
N.Y., 1971.
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