Review
J Phys Astr, Volume: 7( 3)

Disruptive Gravity: An Alternative to General Relativity

*Correspondence:
RB Safo
CentraleSupelec, France, E-Mail: [email protected]

Received: June 6, 2019; Accepted: June 11, 2019; Published: June 18, 2019

Citation: RB Safo. Disruptive Gravity: An Alternative to General Relativity. J Phys Astron. 2019;7(2):183.

Abstract

Viewing gravity as a space-time bending force instead of just a space-time curvature, we come to the conclusion of rest mass relativity. A close analysis of Schwarzschild's metric lead us naturally to the Vacuum Apparent Energy Invariance principle from which we derive the metric equation. Using quantum physics in a curved space-time study, we derive a coherent quantum equation that includes gravity. Applying this theory to cosmology, we can explain galaxies redshift as a delayed gravitational redshift which fully explains Hubble diagrams, including Dark Energy.

Keywords

Quantum gravity, General relativity, Dark energy, MOND theories

Introduction

Gravity is currently understood as a space-time curvature. We will first see how gravity can be seen as a force able to bend space-time instead and then derive a new space-time bending equation thanks to a new principle equivalent to Einstein’s Equivalence Principle in low-intensity fields.

In this paper, Greek letters range from 0 to 3 (representing space-time) while roman letters range from 1 to 3 (representing space), contravariant coordinates have low indices except for four-potentials and the metric signature is equation We use Einstein’s summation convention.

A space-time bending force

Einstein’s General Relativity states that a body moving through gravity is just following the shortest path in curved space-time. This is summarized by the geodesic equations where the metric equation is derived from Einstein’s equation. Those equations are derived from the least action principle, with the following Lagrangian:

equation

If gravity were a force, the Lagrangian would be of the form:

equation

where equation is Minkowsky's metric and equation is the gravitational potential.

We know this Lagrangian is not correct since it would lead to incorrect geodesic equations.

How could we get to the same geodesic equations as General Relativity taking into account space-time curvature and a potential term? Let the Lagrangian be of the form:

equation

As such, we still wouldn't get the same geodesic equations as General Relativity. Is it possible to slightly change it in a physically acceptable way so it becomes equivalent to General Relativity's Lagrangian?

Speed of light cannot be modified since Special Relativity laws wouldn’t apply anymore. The only thing that could be changed is the mass of the body.

Let’s then write:

equation

where m0 is the rest mass in case of zero potential. So we have: equation

For more clarity, let's also write: equation

We then have: equation

We want to find the same geodesic equations with equation

For General Relativity, we have:

equation

For our new Lagrangian, we have:

equation

Since equation doesn't depend explicitly on equation we have:

equation

Leading to:

equation

It comes:

equation

We see the Lagrangian equation of L0 in the first and last terms of the equation:

equation

So we have the same geodesic equations as in General Relativity if and only if:

equation

Parametrizing with the body's proper time, we have: equation

It comes:

equation

Thus:

equation

Eventually:

equation

Gravity is then a space-time bending force if and only if the mass is relative such that:

equation

This is exactly what we wanted. Interpreting gravity as a force able to bend space-time instead of just a space-time curvature.

Mass changes seem to falsify the conservation of energy. For it to remain true, the energy has to be written as:

equation

We generalize this formula as such for a relativistic body:

equation

With: equation

We are now left with finding how gravity could bend space-time. What physical principle could explain that? Can we derive the metric through a new principle other than General Relativity strong equivalence principle?

Metric Field

Let’s postulate that the metric is of the form:

equation

Where time dilation and spatial curvature are disjoint. Then equation would be a equation dependent function and gswould be the spatial part of the matrix.

In this view, space-time is not bent by matter, but rather by gravitational potential.

Let’s derive equation and equation first.

Time Dilation

Let’s derive equation with two different methods.

We introduce the Apparent Energy as such:

equation

Thus:

equation

The apparent energy is the particle's energy under equation potential gravity as seen by an observer.

Applied to photons, we have: equation

Thus: equation

which is similar to a gravitational redshift.

From Einstein's gravitational redshift analysis, we have: equation

It comes: equation

Let’s have a quick look at Schwarzschild’s metric:

equation

We have: equation

The difference is really small: equation but still noticeable. Precise measurement of gravitational redshift would decide which time dilation factor is the most accurate. This theory could be falsified this way.

Let's now see what g00 comparing General Relativity's Lagrangian to a classic lagrangian. That would help us chose between Schwarzschild's result and ours.

The classic lagrangian is:

equation

And let's rewrite General Relativity's lagrangian this way:

equation

Then, for non-relativistic speeds, we have:

equation

with equation

Equating both lagrangians, it comes:

equation

And eventually: equation

Schwarzschild's solution would be a first-order approximation while ours seems more precise. But since Schwarzschild's solution is an exact solution of Einstein's equation, if the real physically observed time dilation factor was mathematically different from Schwarzschild's, it would mean that General Relativity's equations are false. There should be a more accurate theory. Since General Relativity derives from the strong equivalence principle only, we should conclude that the strong equivalence principle is false. Therefore, one would rather find a new physical principle to build a new theory upon. Which is the aim of the next section.

Will also see a third way to derive g00 applying a new principle which doesn’t depend on quantum physics in the next section

Vacuum Apparent Energy Invariance

We naturally want to change Schwarzschild’s metric into the following metric:

equation

This way, in Cartesian coordinates, we would have:

equation

What could be the physical meaning of this? Let equation be a mass density:

equation

It’s just as if it was the ratio of the apparent energy of virtual vacuum masses under zero gravity potential over the apparent energy of those same masses under equation potential gravity.

For an infinitely small space volume =dxdydz, we can write:

equation

It comes:

equation

So we naturally introduce the Vacuum Apparent Energy Invariance principle (VAEI) as follows:

"The apparent energy of the vacuum is invariant."

Let's apply this principle to derive equation and equation

At a given point in time t, in an infinitely small volume equation under zero gravity (flat space) with vacuum energy density equation we have:

equation

and under equation gravity potential, we have:

equation

Applying VAEI, we have: equation

It comes:

equation

Let’s apply VAEI in the time domain to have a more rigorous way to find equation

The reasoning is a bit similar to the one for the derivation of the gravitational redshift. We reason in terms of observational events.

Let E0be the total vacuum energy and N be the number of observational events.

The apparent total vacuum energy by time unit for an observer under a global 0-potential is:

equation

The apparent total vacuum energy by time unit for the same observer under a global Φ-potential is:

equation

Applying VAEI, we have: equation

It comes:

equation

With equation it eventually comes:

equation

We don’t need a quantum argument anymore which is really important for this theory not to be dependent on quantum physics and the energy conservation argument wasn’t strong enough since gravitational redshift of photons implying a violation of energy conservation is currently interpreted otherwise.

Space Metric

We still don’t fully know gs. We only know its determinant.

Spherical potential from a point like mass is a special case in which space is only dilated radially. Which gives locally in equation basis:

equation

Applying the VAEI principle, it comes:

equation

In this coordinate system, the spatial part of the metric is local:

equation

with equation

Changing coordinates, we get:

equation

where MT is the change of basis matrix from equation So with equation we have:

equation

then:

equation

It comes:

equation

And using orthogonal matrices properties:

equation

Eventually:

equation

That doesn't depend on the choice of equation and equation It can be rewritten this way, with equation for brevity:

equation

Which is only true for a spherical potential from a point like mass. How can we generalize it to any kind of potential?

The above formula is also locally true for a linear mass distribution along equation passing through the observer's location in space.

Let:

equation

Let's rewrite the formula this way:

equation

This way, λ is a renormalization parameter ensuring that equation while conserving the basis change invariance. In this special case, we obviously have λ = 1.

If we are in presence of mass distribution, we want to add up the potential influence from every direction to derive gs. We then introduce the angular potential distribution equation where equation is the observed direction. We have:

equation

where equation is a solid angle element.

For an infinitely small solid angle in equation direction, applying the previously derived formula, with equation we have:

equation

Integrating over the whole observation sphere we have:

equation

It eventually comes:

equation

This metric solution is easily verified in the case of a point-like mass distribution. In the case of a homogeneous and isotropic mass distribution cross-terms are null so we have:

equation

In case of a Newtonian potential in a static mass distribution no need for solid angles formalism. Adding up the potential influence from every mass, with equation for brevity, we have:

equation

Gravitational Field Tensor

Gravitation seen as a force is very similar to electromagnetism. A direct analogy gives us the gravitational potential as a Lorentzian vector:

equation

And the gravitational tensor as:

equation

The lagrangian becomes:

equation

Where we now separate the vacuum potential equation from the particle’s gravitational potential. Einstein’s statement of equality of inertial and gravitational mass is not necessary anymore. And the potential is not necessarily a Newtonian one analogous to electromagnetism, leaving open doors to modified Newtonian gravity laws. If it was, we would have the equivalent of Maxwell equations for gravity. The last thing, gravitational waves are not dependent on a gauge choice contrary to General Relativity since the potential is Lorentzian by definition.

Electromagnetism

Any potential could be added to the Lagrangian, so including electromagnetism is pretty straightforward.

Let equation be the electromagnetic four-potential. Including electromagnetism contribution to the Lagrangian, we have:

equation

We see that the stronger the gravity field, the lesser the influence of other forces. Other forces can be neglected if gravity is strong enough.

Quantum Gravity

Gravity being a force again, we now have a coherent way to blend gravity into the quantum realm. What follows is based on Fock's equation (V. Fock, Z. Phys. 57, 261 (1929)) as a curved space-time version of Dirac equation:

equation

Were equation are the generalized gamma matrices defining the covariant Clifford algebra (H. Tetrode, Z. Phys. 50, 336 (1928))

equation

were equation is the space-time metric, whose signature is equation is the spinorial affine connection and equation is the electromagnetic four-vector potential.

In order to take into account gravity, we just write equation and we take into account the gravitational four-vector potential equation We get:

equation

This way, we finally have a coherent quantum gravity equation! It only concerns 1/2-spin charged particles though. Same work should be done for quantum electrodynamics and quantum field theory in general.

Cosmology

Let's see how global vacuum gravitational potential evolves in a homogeneous and isotropic universe. The potential is induced by the mass in a ct radius sphere where is the age of the universe. Space dilation can be neglected in weak field approximation. We have:

equation

Where equation is the universe matter density at time t and equation is the potential of the gravitational field by the mass unit at a distance r. In the special case of Newton's law, it would beequation Time dilation is neglected in the integral in weak field approximation and equation is obviously not dependent on space dilation.

Let equation and equation be two galaxies at a distance D away from each other. An observer in equation at time t0 would see equation as it was in the past at the time equation The gravitational potential of equation and the gravitational potential of equation at the time it's being observed are then:

equation

With the time dilation factor the observed redshifted frequency is:

equation

It comes:

equation

Thus:

equation

And eventually:

equation

One could integrate and have the exact solution but it's easier to compare the distance derivative of the redshift to the observational data. We have:

equation

For very small distances, observational data show that equation is constant (Hubble law). Given that equationequation in that case,equation must be constant. If equation evolved in the early universe, it must have been in time no greater than the time light would take to reach the nearest galaxies.

Let's write equation

For small distances compared to the size of the universe, we have:

equation

It comes:

equation

Eventually:

equation

This is equivalent to Hubble's law with an acceleration term. From Hubble diagrams, we deduce equation which is verified for a Newtonian potential.

That gives a good explanation of Hubble diagrams with no need for any kind of Dark Energy.

Redshift is related to Hubble's constant as follows:

equation

Identifying it to our formula, we have:

equation

If we could measure redshifts in smaller distances, we would be able to have more accurate information about equation A way to do so would be sending a signal and making it bounce back to were it has previously been emitted. The global gravitational potential will have changed and redshifted the signal [1].

Let equation be the time for the signal to come back. Doing the same reasoning as previously done in the case of galaxies, we have:

equation

It comes:

equation

Since the local potential is not time-dependent, we have:

equation

So we just have to replace D by equation in the previous equations:

equation

It comes:

equation

In other words after a variable change:

equation

Plotting equation against tsignal we can derive equation as:

equation

It's quite wonderful that there is a theoretical way to ''see'' the early stages of the creation of the universe.

Using the average mass density of the early universe, we can rewrite the redshift this way:

equation

This could possibly be detected through laser interference modulation using a large interferometer. The path difference equation and equation are directly related: equation One can then choose equation to have constructive interences such as:

equation

This is to better see the modulation phenomenon. Let's assume that equation It's possible to have laser frequencies of about equation Hubble constant is approximately equation With a length of about 3 km for the interferometer, we can have a maximum difference path of 6 km thus equation That gives us:

equation

It's about equation oscillating period. It's roughly 8 days only! The longer the oscillating period, the lesser the average universe mass density before equation [2].

Conclusion

This theory can be proven less accurate than General Relativity by fine measurements of the gravitational redshift. It could either falsify this theory or falsify General Relativity.

We never mentioned the Quantum Vacuum. This theory could have been created without knowing the existence of Quantum Vacuum, thus predicting its very existence. Quantum Vacuum being neutral, an electric field wouldn't induce any potential energy since negative and positive charges would nullify their potential energy. Quantum Vacuum being isotropic, its potential vector is null, justifying the fact that we only took into account the scalar potential throughout the whole paper.

Magnetic fields could change Quantum Vacuum energy since particles spins would tend to line up with field lines and thus have negative potential energy.

This theory is compatible with any violation of the weak equivalence principle and any non-newtonian gravity potentials.
The last important thing to mention is the retraction of the metric on the potential. Space dilation implies a modification of the way the potential is derived which in turn implies a modification of the space dilation until a balance is found. The maths of this effect can be done in the case of a spherical potential leading to really interesting discussions which are off topic.

The last section about cosmology is quite disruptive. Contrary to Einstein's equations, our theory doesn't imply an expanding universe but can predict what we observe and interpret as being an expansion as it also gives a natural explanation to what we interpret as Dark Energy. This theory says nothing about the very early moments of the universe though. But as you saw at the end of the last section, we have a powerful way to investigate these moments through what one may call Signal Bouncing Gravitational Redshift.

As you can imagine, there is a lot more to say about this. It implies many things about the nature of Quantum Vacuum, of gravity, or even of time itself but this is not the topic of this paper. The topic was to show how to interpret gravity as a force turning its quantization into a trivial thing given all the previously done research about quantum physics in curved space-time.

References

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