Weak Equivalence Principle

# The Equivalence Principle

There is no way of distinguishing between the effects on an observer of a uniform gravitational field and of constant acceleration.

The Principle of the Equivalence of Gravitation and Inertia provides rests on the equality of gravitational and inertial mass, demonstrated by Galileo, Huygens, Newton, Bessel, and Eotvos. Einstein concluded that one could not detect the difference between a uniform static gravitational field and uniform acceleration. Einstein elevated this concept to become the Principle of Equivalence which is the foundation of the General Theory of Relativity.

The equivalence principle can be stated as "At every spacetime point in an arbitrary gravitational field, it is possible to chose a locally inertial coordinate system such that, within a sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems.

# The Weak Equivalence Principle

A key stone to Einstein's theory of General Relativity Theory is the principle of the uniqueness of free fall.

## The Principle of the Uniqueness of Free Fall

The world line of a freely falling test body is independent of its composition or structure.
The uniqueness of free fall trajectories allows one to regard spacetime as filled as filled with a set of curves, the test body trajectories, which are unique aside from parameterization.

When translated into Newtonian language, the uniqueness of free fall states than any two test bodies must fall with the same acceleration in a given external gravitational field.

A short form of the two principles of equivalence is:

The Strong Principle of Equivalence states all the laws of nature are the same in a uniform static gravitational field and the equivalent accelerated reference frame.

The Weak Principle of Equivalence states all the laws of motion for freely falling particles are the same as in an unaccelerated reference frame.

That is the Weak Equivalence Principle is a restatement of the equality of gravitational and inertial mass.

# Neutrino Test of The Equivalence Principle

If a Gamma Ray Burster occurs at a distance of about 100 MPc (326 million light years), observation of both the time of arrival of the gamma rays (via a GRB monitoring satellite such as currently operating GRO) and detection of ultra-high energy neutrinos (e.g PeV energy) would allow a test of the Weak Equivalence principle to about the 10^{-11} level on cosmic scale in a direct test of the uniqueness of free fall.

### References:

hep-ph/9612391

"An Investigation of Equivalence Principle Violations Using Solar Neutrino Oscillations in a Constant Gravitational Potential"

Authors: J. R. Mureika (University of Southern California)
Comments: 12pp, LaTeX; 12 figures (bitmapped postscript);
Submitted to Phys Rev D

Neutrino oscillations induced by a flavor-dependent violation of the Einstein Equivalence Principle (VEP) have been recently considered as a suitable explanation of the solar electron-neutrino deficiency. Unlike the MSW oscillation mechanism, the VEP mechanism is dependent on a coupling to the local background gravitational potential $\Phi$. We investigate the differences which arise by considering three-flavor VEP neutrinos oscillating against fixed background potentials, and against the radially-dependent solar potential. This can help determine the sensitivity of the gravitationally-induced oscillations to both constancy and size (order of magnitude) of $\Phi$. In particular, we consider the potential of the local superculster, $|\Phi|=3\times 10^{-5}$, in light of recent work suggesting that the varying solar potential has no effect on the oscillations. The possibility for arbitrarily large background potentials in different cosmologies is discussed, and the effects of one such potential ($\Phi = 10^{-3}$) are considered.

hep-ph/9507355

"Gravitationally--Induced Three--Flavor Neutrino Oscillations as a Possible Solution to the Solar Neutrino Problem"

Authors: J. R. Mureika, R. B. Mann
Comments: 5 pages, latex, epsf, 2 figures, talk given by J.R. Mureika at 6th Canadian Conference on General Relativity and Relativistic Astrophysics; to appear in Proceedings of 6th CCGRRA.

Neutrinos can undergo flavor--oscillations if they possess flavor--dependent couplings to the surrounding gravitational field (the VEP mechanism). The neutrino fields can be massless, in accord with the Minimal Standard Model, but at the expense of the Einstein Equivalence Principle. We show that it is possible to explain the observed Solar Neutrino data from the various experiments using the VEP solution in a realistic three--generation framework, and further note how the three--flavor model can offer larger allowed regions of parameter space over the two--flavor models.

On the Smoot Group page you can read about the various research projects including a detector to use neutrinos from Gamma Ray Bursters for a test of the Weak Equivalence Principle.
Revised 29 May 1996; smoot@cosmos.lbl.gov