Edward ≪My question is, Dear colleagues, Is the assertion above correct?≫

No, it is mostly wrong.

A straightforward way to obtain correct results is to start with the Schwarzschild line element, and transform it to a coordinate system which moves close to the speed of light. It is a nice little exercise, which I sometimes have used for homework. It is slightly simpler if one starts with the line element in isotropic coordinates.

It is also interesting to compare the results with the similar ones for a fast moving electric charge, which f.i. can be found in the famous text by J.D. Jackson, Classical Electrodynamics. In both cases you find that the force points towards the instantaneous position of the source, not a previous position, as one would expect if the signal had to travel with the speed of light. These fundamental forces make extrapolations, to fool you to believe that they act instantaneously (they may f.i. have managed to fool Tom van Flandern).

You may have paid dearly to get your paper published in Journal of Modern Physics (not to be confused with the International Journal of Modern Physics). I am afraid you did not get your moneys worth from their refereeing system or editorial staff.

Edward ≪My question is, Dear colleagues, Is the assertion above correct?≫

No, it is mostly wrong.

A straightforward way to obtain correct results is to start with the Schwarzschild line element, and transform it to a coordinate system which moves close to the speed of light. It is a nice little exercise, which I sometimes have used for homework. It is slightly simpler if one starts with the line element in isotropic coordinates.

It is also interesting to compare the results with the similar ones for a fast moving electric charge, which f.i. can be found in the famous text by J.D. Jackson, Classical Electrodynamics. In both cases you find that the force points towards the instantaneous position of the source, not a previous position, as one would expect if the signal had to travel with the speed of light. These fundamental forces make extrapolations, to fool you to believe that they act instantaneously (they may f.i. have managed to fool Tom van Flandern).

You may have paid dearly to get your paper published in Journal of Modern Physics (not to be confused with the International Journal of Modern Physics). I am afraid you did not get your moneys worth from their refereeing system or editorial staff.

A very nice presentation of these issues can be found here: https://www.rand.org/content/dam/rand/pubs/research_memoranda/2006/RM2820.pdf

Regarding the question itself: the statistics of the particles, that make up matter isn't relevant in order for them to provide a source that can affect spacetime curvature. It's the energy-momentum tensor that matters and that's a classical quantity.

The electric and magnetic fields, that produce the Reissner-Nordstrom black hole solution to Einstein's equations, don't contain any particular information about the statistics of the charges that produced them-beyond that they follow Boltzmann statistics, since they're classical, too.

To professor Kare Olaussen,

I am honored to receive your feed back!.....To reiterate your advice, the best approach to conveying the theoretical notion (the generation of gravity via accelerated fermions) is to begin with the Schwarzchild line element describing a space-time curve in terms of (and thus a transformation to) isotropic coordinates that lie within the same inertial frame (i.e. residing at the precipice of the velocity of light). I am not disputing that this is the best course of action, however, as expressed in my papers, I thought it prudent to derive the Schwarzchild line element in the following step by step process:

1.) Establish the equation of Newtonian gravitational force between the dilated mass (m_p) of the accelerated fermions (asymptotically approaching the velocity of light) and an test mass m_0 separated by a distance R.

F = G(m_p)(m_0)/R^2

Where the dilated mass m_p of the accelerated particles can be presented such that:

m_p = m [1- (v^2/c^2)]^(-1/2)

and velocity "v" approaches the velocity of light "c" as the acceleration exerted on the particles increase.

2.) Integrate the resulting equation of gravitational force between the accelerated particles and a test mass m_0 with respect to distance R giving the corresponding value of gravitational potential energy.

Integ.(F)dr = G(m_p)(m_0)/R

3.)Set the obtained value of gravitational energy equal to the relativistic energy of test mass m_0 (i.e.(m_0)c^2 ) at the velocity of light "c" such that:

G(m_p)(m_0)/R = ((m_0)c^2)

4.) We eliminate test mass m_0, giving the Schwarzchild radius of:

R= G(m_p)/c^2

5.) The Value of the Schwarzchild radius above, is now expressed in terms of the dilated mass m_p of the accelerated fermions. Therefore, the compression factor can be expressed such that:

a(r) = (1 - G(m_p)/rc^2 )

6.) Thus, the corresponding Non- isotropic version of the Schwarzchild metric is expressed in terms of the dilated mass of the accelerated fermions m_p such that:

c^2dt^2 = a(r)c^2 - (a(r)^-1)dr^2 - r^2( d^2(theta) + sin^2(theta)d(phi))

Where the compression factors a(r) and b(r) in terms of the dilated mass m_p of the accelerated particles correlating to Isotropic Schwarzchild metric would assume values of:

a(r) = (1 - G(m_p)/4rc^2 )

b(r)=(1 + G(m_p)/4rc^2 )

Thus giving the isotropic Schwarzchild metric in terms of the dilated mass m_p of the accelerated particles of:

((a(r))^2)/(b(r))^2)dt^2 - (b(r)^4)(dx^2 +dy^2 +dz^2)

Acting on isotropic spatial coordinates x, y, and z.

With the exception of the expression of the isotropic Schwarzchild metric above, this is the process that I used to describe "Schwarzchild" space-time in the original paper and its corrected version which is given by the first link. One error that I will point outs within the paper is that I set gravitational energy equal to kinetic energy at the velocity of light (k=(mc^2)/2) as opposed to setting it equal to relativistic rest mass energy (E=mc^2).

> Respectfully, would this constitute a correct description (within the six steps )according to what you have stated?

>Additionally, despite of the electromagnetic forces accelerating the particles, is it in your opinion possible that the dilated mass of the fermions generate an increasing gravitational curve and thus force as the particles infinitesimally approach the velocity of light?

But Edward, your expressions are all wrong. I don't believe you tried to do the coordinate transformation. You did not even bother to look up the isotropic form of the line element on the web, and understand how it occurs. Without such understanding you cannot know how to transform to cartesian coordinates, and appriciate why the isotropic form is simpler to work with. And, without such understanding, and transformation, you are not in position to make the transformation to the rapidly moving frame.

You are just making arbitrary and unfounded replacements in some expressions you have found somewhere, based on some other expressions you found somewhere else. This is not how physics works.

Added: I could not at first read the tail of your post, which indeed provides the isotropic line element. Congratulations, and apologies for my misjudgments! Now you should do the transformation to the rapidly moving frame, taking into account that the original expression involves the rest mass.

No-the ``dilated mass'' doesn't have anything to do with generating a source for curving spacetime and, once more, the fact that the particles are fermions doesn't enter anywhere, because spacetime is a classical quantity, whose properties are described by Einstein's equations.

The closest topic would be neutron stars and magnetars, cf. http://www.tat.physik.uni-tuebingen.de/~kokkotas/My_Papers_files/KokkotasJPSL.pdf

Dear Dr. Nicolis,

I thank you for your response. In reference to my specification of fermions, as you know, I simply mean massive particles. I have to take some time to read your presentation. But I will offer that one of my descriptions incorporates the energy momentum tensor of general relativity. I have to ascertain if there is any semblance.

Edward

Dear Dr. Olaussen,

Thank you for your attention to my question, apology accepted! And please accept my apology, with constantly switching devices throughout the day, I did not see your add on prior to writing my last post addressed to you lol! ...........However, I will consider and perform the transformation to the rapidly moving frame and may adjustments accordingly.

Sincerely,

Edward

Once more: whether the particles are massive or massless isn't relevant for gravity, because only the energy-momentum tensor couples to the metric and, thus, affects spacetime geometry.

Dear Dr.Nicolis,

I am taking heed to what you are asserting. You are in the correct, the metric tensor does couple with the metric tensor. I've done research with the interaction lagrangian of the theoretical graviton. You have shared a link that I haven't had an opportunity to read in-depth yet. Otherwise I would like to at some point having more mathematically explicit conversation with you pertaining to what you are trying to convey.

Regards,

Edward

Interesting.

@Edward

c2dτ2 = ((a(r))^2)/(b(r))^2)c2dt^2 - (b(r)^4)(dx^2 +dy^2 +dz^2)

Related to the coordinates of a rapidly moving frame (T, X, Y, Z), you have

ct = cosh η × cT - sinh η × X, x = -sinh η × cT + cosh η × X, y = Y, z = Z

with cosh η = γ, sinh η = γβ. From this you can express r, dt, dx, dy, dz in terms of the new coordinates, and find the line element. The expressions are a bit messy, so you may prefer to work only to first order in 1/r to start with. That makes it simpler to interpret the results.

Dear Dr.Olaussen,

Thank you! I greatly appreciate you taking the time to write out the correlating equations . I am going to workout the transformations that you have recommended. If you are not too busy( I am always cautious about impeding) at some point in the near future, I will send you my workings for verification. I look forward to making addendum to the preprint.

With kindest regards,

Edward

Dear Professor Korteweg,

Many thanks for your wonderful input to my query.I greatly appreciate it and I look forward to an opportunity to read your research. A portion of my research expounds the theoretical idea of gravitation by accelerated particles on the verge of the velocity of light. Perhaps your research will compliment my own.

With kindest regard,

Edward

No. The particles are accelerated to high energy, but gravitational mass does not increase.

Since 1960's variable mass has lost support. The majority now support a concept of non linear inertia at high speed relative to the laboratory.

Dear Dr. Decker,

Many thanks for your valuable input and your answer is Duly noted! However, considered the momentum as it relates to momentum below.

P = (integral (a)dt) (m_p)

Where (a) denotes acceleration and m_p denotes the mass of a particle, a relativistic expression of momentum of the same particle of mass m_p would be:

P= v * m_p * [1- (v^2/c^2)]^(-1/2)

Where v denotes relative velocity and c the velocity of light, now setting the two expressions of momentum equal gives:

(integral (a)dt) (m_p) = v * m_p *[1- (v^2/c^2)]^(-1/2)

Now if acceleration (a) is set to a value that is multiples greater than the velocity of light per unit of time such that:

a = Nc/delta (t) for N > 1

Then the relative velocity v would asymptotically approach the velocity of light c (of course never achieving the velocity of light), hence, as we know, the relative mass would increase according to the inequality:

m_p *[1- (v^2/c^2)]^(-1/2) > m_p

Where M is the source mass, the increase in gravitational force according to Newton’s gravitational force equation between the ass M and m_p separated by distance r would be expressed by the inequality :

(G * M * m_p)/r^2 < (G * M * (m_p *[1- (v^2/c^2)]^(-1/2)))/r^2

Where the corresponding Schwarzchild line element is given such that:

c^2dt^2 = ((a(r))^2)/(b(r))^2)dt^2 - (b(r)^4)(dx^2 +dy^2 +dz^2)

and the corresponding compression factors are given by (This was formulated in an early post to this question):

a(r) = (1 - G(m_p)/4rc^2 )

b(r)=(1 + G(m_p)/4rc^2 )

Therefore, the assertion is that there is not a literal increase in mass m_p, but space-time curves to compensate for the dilation and thus increase in (mathematically) mass m_p of the particle or particles. This can be interpreted as a "gravitational curvature" in space-time due to accelerated particles, in theory.

I believe that this assertion would have to be verified experimentally, however, all insights on this assertion are welcomed and highly considered.

Despite the support of non linear inertia, I believe that this assertion should be revisited. The force of gravity would indeed be minute in a particle accelerator, however, I believe that as more powerful particle accelerators are developed it is worth the effort in attempting to detect small distortions in space-time using laser interferometers or simply attempting to detect small amounts of gravitational force near accelerated particles using ultra sensitive sensors.

Best,

Edward

@ Edgar Korteweg

I apologize for the delay in my response. You are a professor in my book LOL! I thank you once more for your input to my question. Your statement on relativistic effects on massive and thus heavier particles that are at relativistic velocity regimes within the atomic structure are enriching and factual. Furthermore, within particle accelerators, a gravitational field produced by accelerated particles would be difficult to detect due to the inherent weakness of gravitational force. A substantial amount of power would have to be put into a particle accelerator in order to detect a resulting gravitational field thus proving or disproving the assertion. No such experiment has been performed, simply disregarded. If more "main stream" physicist would not simply dismiss the possibility, then great discoveries in gravitation could be on the horizon. And if not at least it would have been a direction that is has been FULLY considered.

God Bless!

Edward