Historical research guide · 0655 · 2016
Gedanken Experiment (Thought Experiment) about Gravo-Electric and Gravo-Magnetic Fields, and the Link to Gravitons and Gravitational Waves in the Early Universe
Andrew Walcott Beckwith
In one minute
Andrew Beckwith runs a thought experiment on a piece of textbook relativity most readers never meet: gravity has its own electric and magnetic fields. A mass produces a gravo-electric field, a spinning mass produces a gravo-magnetic one, and together they set an angular frequency — Beckwith works from an exercise in Padmanabhan’s Gravitation. The innovation is what he feeds into it. In place of a bulk mass he substitutes a swarm of gravitons: about 10³⁷ of them, each weighing about 10⁻⁶² grams, turning at a radius near the Planck length. Running that through the gravo-electromagnetic frequency gives relic gravitational waves whose wavelength was about 10⁻²⁹ to 10⁻³⁰ metres at birth and about 10⁹ to 10¹⁰ metres now — roughly 10³⁰ hertz in the early universe, stretched to 10⁻¹⁰ hertz today. He is candid that these are order-of-magnitude numbers, and he closes on the sharper question: whether an interferometer ever responds to a third, scalar polarisation, which would put a scalar–tensor theory in place of general relativity.
Connection
This is the electromagnetic-gravitational crossing point stated in textbook form — a gravo-magnetic vector potential and a gravo-magnetic field, produced by spin, in the same shape as the ordinary magnetic ones — which is the structure chapters 10 and 4 build on. Its numerical output is a specific high-frequency band to hunt in, the same band the radio-telescope detector proposals in this library are designed to reach.
From the source
A source with an arbitrary density distribution produces, at lowest order in the metric perturbation, a gravo-electric potential going as minus G times the mass over distance and a gravo-magnetic potential going as the angular momentum crossed into position divided by the speed of light squared and the cube of the distance — the direct gravitational counterparts of the electric and magnetic potentials.Section 1, Equation (3), following Padmanabhan, Gravitation, exercise 6.15, pp. 278–279
The gravo-electric and gravo-magnetic fields of a spinning source set an angular frequency whose square goes as G times the mass over the cube of the radius, corrected by a term of order twice G times the spin divided by the speed of light squared and the fourth power of the radius.Section 1, Equations (4) and (5)
Replacing the total mass by a graviton count times a graviton rest mass of about 10⁻⁶² grams, and the spin by that same count of gravitons turning at a radius of Planck length times a power of ten, converts the gravo-electromagnetic frequency into an early-universe prediction driven by the number of initial gravitons, taken as about 10³⁷.Sections 2 and 3, Equations (6) and (7)
The relic gravitational radiation so produced has a wavelength of about 10⁻²⁹ to 10⁻³⁰ metres when the source radius is well under a metre and about 10⁹ to 10¹⁰ metres at the present radius of 4.4 × 10²⁶ metres, corresponding to about 10³⁰ hertz or more initially and 10⁻¹⁰ hertz or less today — extremely rough estimates by the author’s own statement.Section 3, Equations (12) to (14)
If an interferometer shows a response function to an additional scalar polarisation, beyond the two polarisations general relativity allows, that points to a scalar–tensor gravitational theory as a replacement for general relativity; if it does not, general relativity stands.Section 4, and the paper’s Abstract
The initial value of the inflaton implied by this frequency, with an initial scale factor of about 10⁻⁵⁵, is enormous, and any such model must be reconciled with the low tensor-to-scalar ratio allowed by Planck and WMAP data, which requires fine tuning of the inputs.Section 4, Equation (15); Section 5
The way in
https://doi.org/10.4236/jhepgc.2016.22023How to cite it
Andrew Walcott Beckwith (2016) Gedanken Experiment (Thought Experiment) about Gravo-Electric and Gravo-Magnetic Fields, and the Link to Gravitons and Gravitational Waves in the Early Universe doi:10.4236/jhepgc.2016.22023