Time Varying Chameleon Fields, Propagating Scalar Waves and Advanced Propulsion T. Marshall Eubanks Space Initiatives, Inc., E-mail:
[email protected]
Submitted to the 2018 Estes Park Advanced Propulsion Workshop The discovery of the accelerating expansion of the universe has led to a renewed interest in modifications of General Relativity. In particular, the addition of a scalar field (or fields) with energies comparable to the cosmological constant could potentially provide a cosmologically significant “dark energy” component to gravity (1, 2). However, massless (Jordan-Brans-Dicke) scalar fields are subject to stringent constraints from laboratory and solar system tests of gravity and of the principle of equivalence (3). These tests require massless scalar fields to be much weaker than gravity, and thus unable to provide cosmologically significant corrections to standard gravity on any size scale. Chameleon fields are massive self-interacting scalar fields whose effective mass depends on the density of the surrounding normal matter; for sufficiently large and dense bodies in vacuum chameleon fields are restricted to a thin surface layer, greatly reducing the chameleon forces from such objects and allowing chameleon fields to evade the stringent constraints from laboratory and solar system tests while remaining dynamically important on galactic and intergalactic scales (2, 4, 5). For a laboratory sized or larger baryonic object in vacuum the thin chameleon surface layer will act in many ways like a surface electrical charge on a conductor (6). A time-varying chameleon field follows a Klein-Gordon equation for a massive scalar field with the resulting group velocity, vc , being (7) 1 vg (1) =q m2 c 1 + kef2 f in natural units, where mef f is the effective chameleon particle mass and the Planck constant and the speed of light are set to unity. Define Ref f to be the Compton radius of mef f ; this parameter sets the scale constant of the chameleon Yukawa potential and is constrained experimentally to be . 1 µm in baryonic matter (8). At the surface of a laboratory-sized mass of dimension R therefore vg Ref f ∝ ≪ 1. c R
(2)
while in vacuum Equation 2 is inappropriate and vg ∼ c. Vibrations of a surface with a thin chameleon field in vacuum will generate propagating scalar waves, similar to the generation of waves from a moving surface charge in electromagnetism, although scalar waves will also support monopole radiation. The coupling between chameleon and matter fields thus implies that a non-relativistically vibrating mass in vacuum can be much more efficient at generating scalar radiation than at generating tensor gravitational waves; scalar wave generation will be most efficient when the size of the object is comparable to the crossing time of the scalar waves. A laboratory mass with R ∼ 0.1 m will have a crossing time, R / vg ∼ R2 / (c Ref f ), . 30 µs, and thus would generate scalar radiation most effectively at frequencies of order 30 kHz. Scalar radiation can transfer energy and momentum through vacuum, and thus could be used for propulsion in deep space. The effects described in this paper would not be evident in the numerous static or very low frequency 1
tests for a fifth force, and there are at present very few limits on frequency-dependent violations of the weak Equivalence Principle. However, it would be straightforward to search for this radiation with existing laboratory techniques, and it is possible it has been detected in work aimed at verifying Mach effect thrusters.
References 1. P. J. Peebles and B. Ratra. The cosmological constant and dark energy. Reviews of Modern Physics, 75:559– 606, April 2003. 2. J. Khoury and A. Weltman. Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space. Phys. Rev. Lett., 93(17):171104, October 2004. 3. C. M. Will. The Confrontation between General Relativity and Experiment. Living Reviews in Relativity, 17:4, June 2014. 4. J. Khoury and A. Weltman. Chameleon cosmology. Phys. Rev. D, 69(4):044026, February 2004. 5. C. Burrage and J. Sakstein. Tests of Chameleon Gravity. ArXiv e-prints, September 2017. 6. C. Burrage, E. J. Copeland, A. Moss, and J. A. Stevenson. The shape dependence of chameleon screening. J. Cosmology and Astroparticle Physics, 1:056, January 2018. 7. J. Ø. Lindroos, C. Llinares, and D. F. Mota. Wave propagation in modified gravity. Phys. Rev. D, 93(4):044050, February 2016. 8. J. Sakstein. Tests of gravity with future space-based experiments. Phys. Rev. D, 97(6):064028, March 2018.
2
