Gedanken Experiment (Thought Experiment) about Gravo-Electric and Gravo-Magnetic Fields, and the Link to Gravitons and Gravitational Waves in the Early Universe

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, Physics Department, Chongqing University Huxi Campus, College of Physics, Chongqing University, Chongqing, China.

Journal of High Energy Physics, Gravitation and Cosmology, 2016, 2, 280-285. Received 4 November 2015; accepted 24 April 2016; published 27 April 2016.

Abstract

Our Gedanken experiment is a thought experiment as to what is called gravo-electric and gravo-magnetic potentials linked to gravo-electric and gravo-magnetic fields. We examine what Padmanabhan presented in an exercise as of a linkage of electromagnetic fields with Gravitation. The modifications we bring up take the nonrelativistic approximation as the beginning of an order of magnitude estimate as to gravitons, generated electromagnetic fields, and are by definition linked to the total angular momentum of an initial configuration of "particles" of space-time import. The innovation put into Padmanabhan's calculation is to, for total mass M used, substitute in M approximately equal to the number of gravitons times the graviton mass, where the graviton mass is about 10⁻⁶² grams, as well as specify distances, for the object spinning as being about Planck length in size, give or take a few orders of magnitude. The results are by definition very crude, and do not take into account relativistic effects, but are probably within an order of magnitude important comparison. We conclude with a comment as to the possibility of an additional polarization as due to a response function of an interferometer to "scalar" polarization as maybe indicate a scalar-tensor gravitational theory as a replacement for General Relativity.

Keywords: Biometric Gravity, Gravo-Electric, Gravo-Magnetic, Tensor-Scalar Gravity Theories, Inflaton.

1. Introduction

Reviewing what was done by Padmanabhan as far as gravo-electromagnetic waves being set up so as to have up for calculation of using the results of initial energy, and comparing it to a more general energy expression given below.

Our tack is to take the energy expression with a minimum energy given as follows, if the inflaton is the scalar field and the square of the scale factor and a non-dimensional perturbation of the "time" factor in a geodesic are as usual. Equation (1) states that the product of the time interval and the energy uncertainty is of the order of the reduced Planck constant, so that the energy uncertainty equals the reduced Planck constant divided by the product of the time interval and the time-time metric perturbation, which is itself the product of the squared scale factor and the inflaton.

And then what we do is to take the work of Padmanabhan to come up with a gravo-electromagnetic frequency which is then set as equal to give an initial import of energy according to a frequency. Equation (2) states that the energy uncertainty equals the reduced Planck constant times the gravitational angular frequency, so that this angular frequency goes as one over the product of the time interval and the time-time metric perturbation, that is one over the product of the time interval, the squared scale factor and the inflaton.

This frequency, as isolated in Equation (2), will be compared to the frequency generated by the gravo-electric and gravo-magnetic fields given in the following argument below. It will suggest something about the inflaton, as suggested in the last part of this document.

We will take the square of the frequency given in the second line of Equation (2) and compare it to the gravo-electric and gravo-magnetic generated frequency value, as our first principle linkage of electromagnetic waves with gravitons. We should keep in mind that the volume, which is for a complete cosmology, is small.

So, let us now come up with a gravo-electric and gravo-magnetic counterpart to Equation (2) above. To do that, we take an argument given in Padmanabhan's Gravitation, pages 278 to 279, exercise 6.15, of a magnetic and electric field being generated by a source with arbitrary density, as to have the following two lowest order perturbations. Here M is the total mass, the angular momentum tensor is built from the momentum density, itself the density times the four-velocity, and these yield the gravo-electric tensor and the gravo-magnetic potential. Note that we are referring to a volume V which will be for the entire spatial domain of the universe, but that our volume V is in early universe conditions of Planckian size dimensions. The author thanks the referee for the necessity of making this point obvious as to how to properly interpret Equation (1) given above. Here G is the usual gravitational "constant", c the speed of light, and what is called x is the spatial dimensions. The angular momentum tensor should be thought of, in this case, as having a relationship to electromagnetics.

Equation (3) states the results: the leading metric perturbation is given by twice G divided by the cube of the speed of light and the distance, multiplied by the position contracted with the angular momentum tensor, plus corrections of order one over the cube of the distance; the total mass is the volume integral of the density; the angular momentum tensor is the volume integral of position times momentum density, antisymmetrised; the gravo-electric potential is minus G times the mass divided by the distance; and the gravo-magnetic potential is minus G times the mass divided by the speed of light squared and the cube of the distance, multiplied by the spin crossed into position.

Then again by Padmanabhan, exercise 6.15, page 279, the following gravo-electric and gravo-magnetic fields appear. Equation (4) states that the gravo-electric field is minus G times the mass over the square of the distance, along the radial unit vector, and that the gravo-magnetic field is G divided by the speed of light squared and the cube of the distance, multiplied by the spin minus three times the component of the spin along the radial direction taken along that direction — the same dipole shape as an ordinary magnetic field.

Here the radial unit vector points outward and the spin is an angular velocity, which will then lead to the following angular frequency, again by exercise 6.15, page 279. Equation (5) states that the squared angular frequency goes as G times the mass divided by the cube of the radius, with a correction of order twice G times the spin divided by the speed of light squared and the fourth power of the radius.

From here, we will proceed to modify the mass and the spin by gravitational physics.

2. Modify M and S by Gravitational Physics

What we are going to do is to restrict the mass to the case of heavy gravity in the Planckian regime, call G the usual gravitational physics variable, and define the spin as following for the mass and the spin, with N being the number of initial gravitons and a radius of Planck length times ten to a positive power, so up to a good approximation.

Equation (6) states that the total mass is the number of gravitons times the graviton mass, and that the spin is the sum over gravitons of radius crossed into momentum, which is approximately the number of gravitons times the Planck length times ten to that positive power, times the graviton mass.

Then the maximum initial value of the angular frequency of Equation (5) is obtained by substituting these two expressions, which is Equation (7): the squared angular frequency becomes G times the number of gravitons times the graviton mass divided by the cube of the radius, with the spin correction now carrying the same graviton count and the Planck-length radius.

3. Modify M and S by Gravitational Physics with Numerical Inputs into Equation (7) for Frequency

The number of gravitons is approximately 10³⁷, due to a rest massive graviton mass of about 10⁻⁶² grams, plus a radial distance r from the source of the graviton production would lead to relic gravitational waves reduced dramatically from the beginning radii, presumably about 1 metre, to the present radii of the universe, presumably of the value of about 4.4 × 10²⁶ metres.

If so, then the energy would be represented, if the graviton wavelength is two pi times the graviton velocity divided by the graviton angular frequency. Equation (8) states the standard relativistic relation: the squared graviton mass times the fourth power of the speed of light, divided by the squared graviton energy, equals one minus the squared ratio of graviton velocity to the speed of light.

Then we have for gravitons an energy value of about, if m is the mass of a "massive" graviton, using in this case the relativistic formula to approximate to first order, Equation (9): the energy goes as two pi times the reduced Planck constant times the speed of light divided by the wavelength, times a square-root correction built from the graviton mass, the wavelength and the speed of light.

Compare this value of energy by making the following scaling, namely equate Equation (5) and Equation (9), which is Equation (10): the squared angular frequency written with the graviton count and graviton mass, as in Equation (7).

This is then compared with, and is implying, a frequency squared value given by Equation (11): the square of two pi times the speed of light divided by the wavelength, plus a term carrying the graviton mass, the wavelength and the fourth power of the speed of light, divided by the square of two pi times the speed of light.

To get to the present value of what the relic wavelength for produced gravitons initially would be, take the upper value of the wavelength given in Equation (11), with a graviton mass of about 10⁻⁶² grams, as would be the case if the radius in Equation (10) is 4.4 × 10²⁶ metres. Equation (12) gives a wavelength today of about 10⁹ to 10¹⁰ metres.

Whereas if the radius in Equation (10) is significantly less than 1 metre, emergent radiation would be, by Equation (13), a wavelength of about 10⁻²⁹ to 10⁻³⁰ metres initially.

Based upon Equations (12) and (13), Equation (14) gives an initial angular frequency of about 10³⁰ hertz or more, and an angular frequency today of about 10⁻¹⁰ hertz or less.

This is using extremely rough estimates.

4. Considerations as to BICEP2, the Matter of Scalar-Tensor Polarizations as an Alternative to General Relativity and Alternate Gravitational Theories and Experimental Tests of General Relativity via Interferometric Methods

We have the following to consider, namely trying to determine restraints upon the nature of gravity, that is, whether it is consistent with general relativity or whether we have an alternative situation as given in the following quote. We hope that getting a consistent model of inflaton physics will help clarify the following alternatives:

Quote: "This fact rules out the possibility of treating gravitation like other quantum theories, and precludes the unification of gravity with other interactions. At the present time, it is not possible to realize a consistent Quantum Gravity Theory which leads to the unification of gravitation with the other forces. On the other hand, one can define Extended Theories of Gravity those semiclassical theories where the Lagrangian is modified, in respect to the standard Einstein-Hilbert gravitational Lagrangian, adding high-order terms in the curvature invariants (terms like R squared, and so on) or terms with scalar fields non minimally coupled to geometry (terms like the squared scalar field times the Ricci scalar)." End of quote.

We claim that the strength of the inflaton term, as we will give in Equation (15), may allow us to determine if we have to use a semi-classical set of terms which add more terms to the space curvature of early universe Planckian physics space-time geometry. We also though have to temper this quest in requiring that the following holds, namely:

Quote: "Recent data from Planck matches well with the minimal Lambda-CDM model. A likelihood analysis using Planck, WMAP and a selection of high resolution experiments (highL), tensor to scalar ratio r at 0.002 is found to be less than 0.11 when the running of the spectral index vanishes." End of quote.

Our inflaton, which is given in Equation (15), must be made consistent with the requirements of a low scalar-to-tensor ratio, and this requires exquisite fine tuning of inputs into the inflaton.

We find that the resulting inflaton measurement, which is the conclusion of our document, follows from assuming that the initial start value of the scale factor is about 10⁻⁵⁵. That is, is our inflaton consistent with just two standard polarizations, or is there a third polarization necessary so that the following inflaton forms? If there is not a response function of an interferometer to an additional "scalar" polarization we define, we stick to general relativity, whereas though if Equation (15) necessitates an additional polarization, we are looking at a scalar-tensor gravitational theory. Needless to say we will require careful analysis of Equation (15), which states that the initial inflaton is approximately one over the Planck time, divided by the square of the initial start value of the scale factor of about 10⁻⁵⁵, and divided by the initial angular frequency of about 10³⁰ hertz or more.

This enormous value for the inflaton, initially, needs to be examined further. It further should be linked to Corda's pioneering work with "gravity's breath", that is, traces of the inflaton, and that is the justification of Equation (15) above. We can use this to determine what to make of the stochastic background of pre space-time physics.

5. Avoiding the BICEP2 Mistake: What We Can Do with Equation (15)

What we are doing is examining the stochastic regime of space-time where the following holds:

Quote: "Omni-directional gravitational wave background radiation can arise from fundamental processes in the early Universe, or from the superposition of a large number of signals with a point-like origin. Examples of the former include parametric amplification of gravitational vacuum fluctuations during the inflationary era, termination of inflation through axion decay or resonant preheating, Pre-Big Bang models inspired by string theory, and phase transitions in the early Universe; the observation of a primordial background will give access to energy scales of 10 to the 9 power, up to 10 to the 10 power GeV, well beyond the reach of particle accelerators on Earth." End of quote.

Needless to say though, we need above all to avoid getting many multiple stochastic signals in what we process for primordial gravitational waves, and to use instead tests to avoid getting dust signals, which are what doomed BICEP2. In all, what we are doing is consistent with the requirements given in the author's other article, as is given in the following quote:

Quote: "The main agenda will be in utilization of Equation (27) to help nail down a range of admissible frequencies which will be to avoid conflating the frequencies of collected gravitational wave signals from relic cosmological conditions (or would be signals) with those connected with dust generated gravitational wave signals, especially from dust conflated with Galaxy formation in the early universe. More than anything else, we need to find likely narrow frequency ranges, which will be commensurate with Equation (27), and to use advanced detector technology. Of course such a search will be hard. But it also will be a way, with due diligence, as to answer questions raised by the Author. In doing so, the relative flatness of the early universe and its departure from curved space conditions will be a great way to answer the suppositions raised." End of quote.

Understanding inflaton physics properly will also give credence to considerations as to the degree of flatness, or lack of it, in the early Universe.

Acknowledgements

This work is supported in part by National Nature Science Foundation of China grant No. 11375279.

(The numbered reference list is omitted here; the complete text, with its equations set as mathematics, is at the source.)