# Gravity Probing

This post lists the experiments which are used to test gravity theories carried out on the earth.

The test of gravity theories can be viewed as test of the fundations of gravity theories and the the theories themselves, say test of equivalent principle and general relativity or f(R) gravity theory. Thus we should break down general relativity theory into several stages. Here, we use the following table to do so.

• Physical Fundations: Hyperthesis:
Theory Mach WEP EEP SEP GC Notes
GR Partial Y Y Y Y
• Mathematical Description:
Theory Topoplogy Manifold Connection Metric
GR     No torsion Non-metricity tensor vanishes
• Theoretical Implifications:
Theory Gravitational Waves Newtonian Limit GR Limit Notes
GR

Most items in mathematics are the same in different theories.

## Hyperthesis

• WEP: weak equivalence principle
• EEP: Einstein equivalence principle
• SEP: strong equivalence principle
• GC, General Covariance
• Mach Principle: gravity coupled to matter

## Experiments

### Eotvos Torsion Balance

#### How

• Inertial mass $m_I$
• Gravitational mass $m_G$

In Newtonian system, the acceleration of an object will be $\vec a \propto \frac{\vec F}{m_I}$

In a static and uniform gravitation field, the gravity force is $\vec G = - g m_G \hat r$

Thus the acceleration in this case should be $\vec a \propto -\hat r g \frac{m_G}{m_I}$

When $m_G/m_I$ is constant, the falling accerelation are the same for different objects with same mass. However, if $m_G/m_I$ is not a constant, say $m_G\ne m_I$, different objects would fall at different acceleration.

Now if we put two ball with different mass on the Eotvos torsion balance, the balance would rotate and we can measure it.

#### Results

Detection of $R^k_{0l0}=(1/c^2)\partial^2\Phi/\partial x^k\partial x^l \sim 10^{-32} \text{cm}^{-2}$.

### Hughes-Drevershiy Experiment, etc

Anisotropy of gravitation/electromagnetism is not proved in our galaxy.

Similar to Eddington and Dyson’s bending light observation, radio signals serve as a more precise experiment to test Einstein’s theory. And these experiments are against scalar tensor theories because scalar tensor theories give a smaller bending angle (1.66 second of arc less than the observations).

## Summary Table

Tables constructed according to arXiv:1106.2476v3.

Test of fundamental principles

Experiment Results Note
WEP Eotvos torsion balance $\eta = (0.3 \pm 1.8) \times 10^{-13}$ More precise in space exp. [[^1]a] [1] [2]
Gravitational redshift of light   [3]
EEP Hughes-Drever Experiment $n \le 10^{-27}$ [4] [5]

Test of GR:

Experiment Results Note
Null geodesics test photon trajectory, spatial deflection $\theta = (0.99992\pm 0.00023)\times 1.75’’$, where 1.75 is the theoretical value Achieved through observing star position, etc [6]
Shapiro time-delay effect $\Delta t = (1.00001\pm 0.00001)\Delta t_{GR}$ [7] [8]
Time like geodesics Anomalous perihelion precession   Just use the PPN formalism [9] [10] [11]
Nordtvedt effect $\eta = (-1.0 \pm 1.4) \times 10^{-*13}$ [12] [13]
Spinning objects obiting   [14] [15]
Small-range Potential probing   [16] [17]
Radiation Speed of gravitational waves
Polarity of gravitational radiation
Dynamics of source objects

## Footnote

1. Eotvos experiment: using torsion balance to test the equality of gravitational mass and inertial mass. Wikipedia has a photo of how this works.

2. $\eta=2\frac{ABS(a1-a2)}{ABS(a1+a2)}$. $a1$ and $a2$ are the accelerations of the two bodies in Eotvos torsion balance. Thus $\eta$ is the accleration difference of the two objects.

3. References: R. W. P. Drever. A search for anisotropy of inertial mass using a free precession technique. Philosophical Magazin, 6:683-687, May 1961. ; V. W. Hughes, H. G. Robinson, and V. Beltran-Lopez. Upper Limit for the Anisotropy of Inertial Mass from Nuclear Resonance Experiments. Physical Review Letters, 4:342-344, Apr. 1960. ; S. K. Lamoreaux, J. P. Jacobs, B. R. Heckel, F. J. Raab, and E. N. Fortson. New limits on spatial anisotropy from optically-pumped 201 Hg and 199 Hg. Physical Review Letters, 57:3125–3128, Dec. 1986. ; T. E. Chupp, R. J. Hoare, R. A. Loveman, E. R. Oteiza, J. M. Richardson, M. E. Wagshul, and A. K. Thompson. Results of a new test of local Lorentz invariance: A search for mass anisotropy in 21 Ne. Physical Review Letters, 63:1541–1545, Oct. 1989.

4. Hughes-Drever Experiment: test the isotropy of mass and space through the NMR spectrum, or the mono-metric spacetime.

5. S. S. Shapiro, J. L. Davis, D. E. Lebach, and J. S. Gregory. Measurement of the Solar Gravitational Deflection of Radio Waves using Geodetic Very-Long-Baseline Interferometry Data, 1979 1999. Physical Review Letters, 92(12):121101, Mar. 2004.

6. References, I. I. Shapiro. Fourth Test of General Relativity. Physical Review Letters, 13:789–791, Dec. 1964. ; B. Bertotti, L. Iess, and P. Tortora. A test of general relativity using radio links with the Cassini spacecraft. Nature, 425:374–376, Sept. 2003.

7. Shapiro time-delay effect: time delay when light travels through a massive object.

8. Observational data for the value of perihelion precession of Mercury are summarized in E. V. Pitjeva. Modern Numerical Ephemerides of Planets and the Importance of Ranging Observations for Their Creation. Celestial Mechanics and Dynamical Astronomy, 80:249–271, July 2001.

9. PPN formalism is the lowest order of GR.

10. Anomalous precession:

11. K. Nordtvedt. Equivalence Principle for Massive Bodies. I. Phenomenology. Physical Review, 169:1014–1016, May 1968. ; J. G. Williams, S. G. Turyshev, and D. H. Boggs. Progress in Lunar Laser Ranging Tests of Relativistic Gravity. Physical Review Letters, 93(26):261101, Dec. 2004, arXiv:gr-qc/0411113.

12. Nordtvedt effect: massive objects in Eotvos torsion balance experiments. We can use the whole Earth-Moon system to test this effect.

13. There is a Lense Thirring effect here. GPB has done this.

14. GR can be reduced to Newtonian potential at small range.

15. Currently, most of the modification has a Yukawa potential form.