Does gravity field really exist?

[ABW]

In GR gravity is associated with the curvature of space caused by the presence of massive bodies.
Along with GR, there is an approach in which the gravitational field
- is a real physical field, whose properties can be described by equations,
similar to the Maxwell equations for electromagnetic field.

Differential equations for the gravitational field
Part 1. Maxwell's equations for the electromagnetic field can be supplemented by the differential equations for the gravitational field :

The second equation describes the sources of the gravitational field: It is a common substance with density "Rho" , the electromagnetic field with intensity E and H , and the gravitational field strength g , ( g is the dimension of acceleration ), p is the pressure . The fact, that the gravitational field as any kind of matter, can be a source of gravity , while nobody was paying attention . G is Newton's constant . c - is the speed of light .
From equations we find formula for the energy density of the gravitational field :

Unexpected in this expression is that the gravitational constant G is in the denominator .
Compare this expression with the energy density of electric and magnetic fields in vacuum:

So for the gravitational field at the surface of the Earth g = 9.81 m/sec2 , we find :
W ~ - 10^11 Jowl/m3 . The sign “-” in the formula for the energy density of the gravitational field indicates that the energy density, and the corresponding energy of gravitational interaction is negative.

It is easy to give a formula for refined Newton's law, taking into account, that the gravitational field is also a source of gravitation :

M is the mass of the body , Ro is the radius of the body.

The gravitational field at large distances is higher compared to the classical value.
This formula can in our opinion to explain why the space station “pioneer-10 ”, NASA , currently outside the Solar system slows down faster than it should of calculations . Here is what says about this journal Relcom.ru : “Movement "Pioneer 10" in the space of interested scientists since it was discovered that the observed deceleration cannot be explained alone by the gravitational pull of the Solar system. It can serve as evidence for the existence of an as yet unknown to science forces, or associated with any properties of the spacecraft .”

 

[PeterDonis]

Along with GR, there is an approach in which the gravitational field
- is a real physical field, whose properties can be described by equations,
similar to the Maxwell equations for electromagnetic field.

This is not a different theory from GR; it is just a different way of expressing the theory of GR--more precisely, of GR in the weak field approximation. At least, that's what it looks like given the equations you've written down.

If you're referring to something else, something which is actually a different theory from GR (i.e., which makes different experimental predictions), then please give a reference.

 

[ShayanJ]

Take a look at here! Pioneer's anomaly is solved now.

 

[Jonathan Scott]

From equations we find formula for the energy density of the gravitational field :

I haven't checked the rest, but the Newtonian expression for the energy of the gravitational field is missing a factor of $4 \pi$. The standard result is as follows:

$$-\frac{g^2}{8 \pi G}$$

 

[Jonathan Scott]

It's unfortunately nowhere near that simple.

The parameterised post-Newtonian (PPN) approximation used to analyse the effects of relativistic gravity in the solar system already uses a correction to the potential for non-linearity of the form $(1 - \beta \, Gm/2rc^2)$ where GR predicts $\beta = 1$ and that result has been confirmed to good accuracy by observations of the perihelion precession of Mercury.

The gradient of the dimensionless scalar potential (the time dilation factor) is therefore approximately:
$$- \left ( 1 - \frac{Gm}{rc^2} \right ) \frac{Gm}{r^2 c^2}$$
This is approximately $\mathbf{g}/c^2$. In coordinate acceleration units for isotropic coordinates (where the coordinate speed of light is the same in all directions), velocities scale as the square of the time dilation and the approximate result is as follows:
$$\mathbf{g} = - \left (1 - \frac{Gm}{rc^2} \right )^5 \frac{Gm}{r^2} = - \left (1 - 5 \frac{Gm}{rc^2} \right ) \frac{Gm}{r^2}$$
In addition, it is not thought that any modification to the law of gravity based only on radial distance could explain the Pioneer anomaly, because there is no evidence of any similar effect on planets at the same distance. Any modification would need to rely on something unusual about Pioneer, such as the fact that its motion is in a more radial direction than that of planets.

Anyway, the Pioneer anomaly has now been satisfactorily explained as an effect of thermal recoil, due to the reflection of heat radiation from the spacecraft structure.

[Edited to correct r to r^2 in gradient of potential]

 

[ABW]

I haven't checked the rest, but the Newtonian expression for the energy of the gravitational field is missing a factor of $4 \pi$. The standard result is as follows:

$$-\frac{g^2}{8 \pi G}$$

In the system of Gauss W is:
$$-\frac{g^2}{8 \pi G}$$
In the SI system W is:
$$-\frac{g^2}{2 G}$$

 

[harrylin]

In GR gravity is associated with the curvature of space caused by the presence of massive bodies.
Along with GR, there is an approach in which the gravitational field
- is a real physical field, whose properties can be described by equations,
similar to the Maxwell equations for electromagnetic field.

Just a little more about the question you asked in the title. According to Einstein's GR, the gravitation field of heavy bodies really exists: it's a real physical field, or in other words, a zone of influence around such bodies. The way the gravitational field affects their motion is described by so-called field equations.
Compare the introduction here and you will notice some similarities with your first post: https://en.wikipedia.org/wiki/Einstein_field_equations

 

[Jonathan Scott]

In the system of Gauss W is:
$$-\frac{g^2}{8 \pi G}$$
In the SI system W is:
$$-\frac{g^2}{2 G}$$

No, I'm sure that's wrong. Have you tried the integration yourself?

(And I'm not aware of any gravitational equivalent of Gaussian units).

 

[Jonathan Scott]

The electrostatic force constant (using SI conventions) is $1/4\pi\epsilon_0$ and the gravitational force constant is $G$. From straight substitution this makes $4\pi\epsilon_0$ equivalent to $1/G$ so $\epsilon_0$ is equivalent to $1/4\pi G$.

If you integrate the interaction term in $\mathbf{g}^2$ between the fields due to two masses, that is $2 \mathbf{g}_1.\mathbf{g}_2$, over all space, the result is $- 8 \pi G$ times the (negative) potential energy between those two masses, so this is what you have to divide by in order for the integral to be equal to the potential energy.

A better semi-Newtonian model for gravitational energy distribution is to assume that the energy of each source mass is reduced by the time dilation effect of the potential of the other source, which reduces its energy by the total potential energy (so the total energy for the two masses together is reduced by twice the potential energy), and then the energy of the field in space is added back in with a positive sign. This then gives a continuous positive energy distribution which is overall equal to the energy of the original source masses minus the potential energy of the system.

 

[ABW]

No, I'm sure that's wrong. Have you tried the integration yourself?

(And I'm not aware of any gravitational equivalent of Gaussian units).

Well, may be I was wrong with a factor of $4 \pi$, it is not difficult to check and correct.
More interesting to discuss the above equations in the absence of electric and magnetic fields.
$$\mathbf{rotg} = 0$$ $$\mathbf{divg} = - 4πGρ + \frac{g^2}{2c^2}$$
Or in other words, whether the assumption that the gravitational field itself can be a source of gravity, is correct .

 

[PeterDonis]

whether the assumption that the gravitational field itself can be a source of gravity is correct .

The answer to this is "it depends"--it depends on how you interpret the term "source of gravity".

In the Einstein Field Equation, which is the central field equation of GR, the source is the stress-energy tensor, which does not include any contribution from the "gravitational field". (The equations you are writing down are just restatements of the EFE in a particular coordinate system for a particular kind of stress-energy tensor, in the weak-field approximation.) So on this interpretation of "source", the answer is no, the gravitational field is not a source of gravity.

However, the EFE is nonlinear, which means you can't superpose two solutions to get a third solution. (Maxwell's Equations for electromagnetism, by contrast, are linear.) So if we think of the gravitational field when two gravitating bodies are present, it isn't just the sum of the fields from each body alone; there is an additional contribution which can be interpreted as due to the "gravitational field" acting as a source. (Another manifestation of this nonlinearity is that gravitational waves in GR carry energy.) So on this interpretation of "source", the answer is yes, the gravitational field is a source of gravity.

The real answer is that whether or not the gravitational field is a source of gravity is not an assumption of GR at all. GR is based on different assumptions; the term "gravitational field" is not among them, and you don't need to find an interpretation for the term "gravitational field", or decide whether or not it's a source of gravity, to solve problems in GR.

 

[Jonathan Scott]

... whether the assumption that the gravitational field itself can be a source of gravity, is correct .

In Newtonian terms, potential energy doesn't just appear or disappear when a system of masses is changed to move masses closer together or further apart. It has to be converted to or from some other form internally, such as kinetic energy, or added or removed externally. This suggests that "gravitational energy" isn't really a separate stuff, but rather an accounting term for energy transferred by the mechanism of gravity.

In a semi-Newtonian approach, where energy is conserved in the conventional way, it is clear that the effective source mass of a system must effectively take into account potential energy, as an adjustment relative to the energy that the same source masses would have separately, and the obvious adjustment is that the time dilation factor due to the potential of other masses. However, as I've previously mentioned, if the potential energy correction is applied to each mass separately, this reduces the energy of the system by twice the potential energy, but the energy of the field can be added back into get the correct total energy. This is very similar to the model of electromagnetism where the Poynting Theorem shows how the flow of energy and momentum is conserved.

However, when one starts looking the sort of tiny corrections to gravity which this scheme might produce, it is immediately clear that they are so small that one would also need to take into account other GR aspects such as the curvature of space, and we already know that GR predictions match experiment very precisely at least in the solar system. This means that there isn't any obvious reason to try to stretch Newtonian theory into this area.

Personally, I like to look into the relationship between Newtonian and GR viewpoints of gravity, as I feel it helps me to gain a more intuitive understanding.

 

[Dale]

whether or not the gravitational field is a source of gravity is not an assumption of GR at all. GR is based on different assumptions; the term "gravitational field" is not among them,

Well said.

 

[Jonathan Scott]

I've just spotted a MNRAS paper "Gravitational field energy density for spheres and black holes" from 1985 by D Lynden-Bell and J Katz which takes the GR Schwarzschild solution (for the static spherical case) and applies the model where the effective energy density of the field is $\mathbf{g}^2/8\pi G$ (positive) and the effective energy of the central mass is reduced by the time dilation (which as they point out reduces it by twice the potential energy). In this model the entire energy of a black hole resides in the field outside the event horizon!

This is exactly the same model that up to now I've only ever seen discussed in Newtonian terms. They even use isotropic coordinates (which I've always preferred to Schwarzschild).

It is however rather weird that in this model the location of the "effective energy" is clearly not the same thing as the source of the field!

 

[bhobba]

Interesting question.

Without delving into it you will find the following book by Ohanian explores it very thoroughly:
https://www.amazon.com/dp/1107012945/?tag=pfamazon01-20

I do not 100% endorse his views - I do not agree that tidal forces are always the give away of gravitational forces vs acceleration. It was the first serious book I learned GR from before moving onto my bible - Wald.

But on these type of issues it really has no equal.

Thanks
Bill

 

[bhobba]

whether or not the gravitational field is a source of gravity is not an assumption of GR at all. GR is based on different assumptions; the term "gravitational field" is not among them,

Not so fast Kemosabe :p:p:p:p:p:p:p:p

I didn't want yo get into it because the book I pointed to by Ohanian examines the issue in detail - but it's not quite that simple.

You can model gravity in exactly the same way as EM and you get linearised gravity which actually is perfectly OK in explaining many phenomena.

The issue though is it contains its own destruction and inevitably leads to GR. First, it can be shown, particles do not move in flat space-time - but rather as if space-time had an infinitesimal curvature. Most importantly however it ignores the issue of gravity gravitating so it must be non-linear.

The interesting thing however is GR is rather strange, because, as Ohanion shows, the linear equations imply the non linear ones and you get full GR.

This is also related to the issue - is space-time curved or does the gravitational field simply make clocks and rulers behave as if it was curved. Experimentally there is no way to tell the difference.

BTW my bible on GR is Wald - do you can guess which side on that debate I come down on - but it's debatable 100% for sure.

Thanks
Bill

 

[bhobba]

The fact, that the gravitational field as any kind of matter, can be a source of gravity , while nobody was paying attention .

Including gravity itself - which is why the full equations of GR are non-linear.

Thanks
Bill

 

[ShayanJ]

This is also related to the issue - is space-time curved or does the gravitational field simply make clocks and rulers behave as if it was curved.

Sorry, but that doesn't make sense to me!
Considering that possibility means accepting that gravitational fields know how to make our instruments to go wrong. But because we have different methods for time and distance measurements, it means gravity knows how to deal with different types of measurement devices (which means "he" knows the mechanisms we use!) and, because we can still make new devices, it means gravity can learn how to fool our new instruments too. But...come on...who are we talking about here? Frank Abagnale?

 

[bhobba]

Sorry, but that doesn't make sense to me!
Considering that possibility means accepting that gravitational fields know how to make our instruments to go wrong. But because we have different methods for time and distance measurements, it means gravity knows how to deal with different types of measurement devices and, because we can still make new devices, it means space-time can learn how to fool our new instruments too. But...come one...who are we talking about here? Frank Abagnale?

Weird hey.

It's more reasonable to assume its curved.

The guy who first clued me into this stuff was Steve Carlip when he posted a lot on sci.physics. relativity. He is no crank by a long shot.

Thanks
Bill

 

[ShayanJ]

Weird hey.

It's more reasonable to assume its curved.

The guy who first clued me into this stuff was Steve Carlip when he posted a lot on sci.physics. relativity. He is no crank by a long shot.

Thanks
Bill

Yeah...that's also much more beautiful!

 

[bhobba]

How can it come up with methods to fool our mechanism?

Who knows - but that's not the point is it?

Also such a theory would be too complicated, and by Occam's razor, too ugly because it should accommodate a mechanism for gravity to realize how we're measuring space and time and mechanisms for fooling our different devices and a mechanism so that gravity can learn to fool our new devices. Also any time a new method of space and time measurement is proposed, the theory should change to accommodate that too! That doesn't seem right!

Its like LET (Lorentz Ether Theory). It has exactly the same issues - how does it shorten rods and change clock rates. But take my word for it - you can't prove its believers wrong.

And Occams Razor - well simplicity is in the eye of the beholder - in LET for example those that believe in it believe the loss of absolute time is simply far too big a price to pay.

Thanks
Bill

 

[ShayanJ]

Its like LET (Lorentz Ether Theory). It has exactly the same issues - how does it shorten rods and change clock rates. But take my word for it - you can't prove its believers wrong.

And Occams Razor - well simplicity is in the eye of the beholder - in LET for example those that believe in it believe the loss of absolute time is simply far too big a price to pay.

Yeah...I know what you mean. But after experiencing the revolutionary change of minds by SR and QM, it seems very natural to expect people to understand that we can't understand nature as we could in classical physics.
But well, it seems we always have people who don't accept what they don't understand. Who knows?! Maybe we always should have them!

 

[bhobba]

Yeah...I know what you mean. But after experiencing the revolutionary change of minds by SR and QM, it seems very natural to expect people to understand that we can't understand nature as we could in classical physics. But well, it seems we always have people who don't accept what they don't understand. Who knows?! Maybe we always should have them!

Don't get me wrong - I believe they - colloquially - have rocks in their head.

But the issue is in principle - not what's reasonable.

Thanks
Bill

 

[harrylin]

Sorry, but that doesn't make sense to me!
Considering that possibility means accepting that gravitational fields know how to make our instruments to go wrong. But because we have different methods for time and distance measurements, it means gravity knows how to deal with different types of measurement devices (which means "he" knows the mechanisms we use!) and, because we can still make new devices, it means gravity can learn how to fool our new instruments too. But...come on...who are we talking about here? Frank Abagnale?

Not Frank Abagnale but Albert Einstein.

Rephrasing Bill's sentence: he mentioned the issue if gravitational fields "curve" an invisible and never observed physical thing called "space-time", or if they simply have a common effect on all matter, so that it is most conveniently described by means of geometric equations. Physics can certainly not decide which view is correct. But as you can get from my phrasing, on this metaphysical question I side with Einstein. ;)

PS. In a way such explanations can be regarded as two ways to express more or less the same thing, as anyway a "field" implies an invisible and never directly observed physical something. We can only observe the effects of the presence of mass, not the cause.

 

[harrylin]

[..] after experiencing the revolutionary change of minds by SR and QM, it seems very natural to expect people to understand that we can't understand nature as we could in classical physics.[..]

SR and GR are relatively easy to understand (pun intended!); however you do have a point with QM - that's a very hard nut to crack!

 

[ShayanJ]

SR and GR are relatively easy to understand (pun intended!); however you do have a point with QM - that's a very hard nut to crack!

What I said doesn't refer to how hard the theory is to understand. I was talking about the philosophical deviations from the thoughts that were the essentials of classical physics which of course are present in SR and GR no less than QM.

 

[PeterDonis]

is space-time curved or does the gravitational field simply make clocks and rulers behave as if it was curved

I agree that, as long as we consider only experiments that have already been done (or even ones we can contemplate doing in the near future), there is no way to tell the difference.

However, if we go beyond that to consider theoretical models, there is a possible difference: modeling gravity as a field on flat spacetime limits you to solutions with the same topology as flat spacetime. You can't, for example, model the full Schwarzschild geometry as a field on flat spacetime, because its topology is not R^4. Modeling gravity as spacetime curvature has no such restriction.

(I have not read the Ohanian book, btw, but I am putting on my list of things to read.)

 

[bhobba]

However, if we go beyond that to consider theoretical models, there is a possible difference: modeling gravity as a field on flat spacetime limits you to solutions with the same topology as flat spacetime. You can't, for example, model the full Schwarzschild geometry as a field on flat spacetime, because its topology is not R^4. Modeling gravity as spacetime curvature has no such restriction.

For sure.

I don't believe it, and the elegance of no-prior geometry compared to the other approach as detailed in Ohanion makes you wonder why anyone would bother. Ohanion makes a big deal out of you can tell the difference between acceleration and gravity by tidal forces - I don't 100% agree with that - some distributions of matter like two large slabs do not exhibit tidal forces - but of course they don't really exist - they are contrived purely for that purpose.

That said the advantage of the other approach is it works better as a limit of a QFT of spin 2 particles. Then again I believe that approach explains why space-time is curved rather that it being a field in flat space-time.

Thanks
Bill

 

[PAllen]

Not Frank Abagnale but Albert Einstein.

Rephrasing Bill's sentence: he mentioned the issue if gravitational fields "curve" an invisible and never observed physical thing called "space-time", or if they simply have a common effect on all matter, so that it is most conveniently described by means of geometric equations. Physics can certainly not decide which view is correct. But as you can get from my phrasing, on this metaphysical question I side with Einstein. ;)

PS. In a way such explanations can be regarded as two ways to express more or less the same thing, as anyway a "field" implies an invisible and never directly observed physical something. We can only observe the effects of the presence of mass, not the cause.

Do you think Einstein had only one view of this? I have seen (what I interpret as) multiple flip flops by Einstein on the 'nature of spacetime' in his post 1915 writings. Just as he flip flopped at least 3 times on the reality of gravitational waves (ending with the position they are real and carry energy).

 

[harrylin]

Do you think Einstein had only one view of this? I have seen (what I interpret as) multiple flip flops by Einstein on the 'nature of spacetime' in his post 1915 writings. Just as he flip flopped at least 3 times on the reality of gravitational waves (ending with the position they are real and carry energy).

No, why? I have no idea if Frank Abagnale had only one view of things, but I think that Einstein flip-flopped his thinking much of the time - that was perhaps part of the reason why he was so creative.

 

[harrylin]

What I said doesn't refer to how hard the theory is to understand. I was talking about the philosophical deviations from the thoughts that were the essentials of classical physics which of course are present in SR and GR no less than QM.

Thanks for the precision. :)

 

[stevendaryl]

Not so fast Kemosabe :p:p:p:p:p:p:p:p

I didn't want yo get into it because the book I pointed to by Ohanian examines the issue in detail - but it's not quite that simple.

You can model gravity in exactly the same way as EM and you get linearised gravity which actually is perfectly OK in explaining many phenomena.

The issue though is it contains its own destruction and inevitably leads to GR. First, it can be shown, particles do not move in flat space-time - but rather as if space-time had an infinitesimal curvature. Most importantly however it ignores the issue of gravity gravitating so it must be non-linear.

The interesting thing however is GR is rather strange, because, as Ohanion shows, the linear equations imply the non linear ones and you get full GR.

This is also related to the issue - is space-time curved or does the gravitational field simply make clocks and rulers behave as if it was curved. Experimentally there is no way to tell the difference.

BTW my bible on GR is Wald - do you can guess which side on that debate I come down on - but it's debatable 100% for sure.

Thanks
Bill

The route to GR via starting with a linear approximation and adding self-interaction consistently is interesting for the following reason:

Standard GR has the form:

$G = \kappa T$

where $G$ is a tensor formed from the metric and its first and second derivatives, and $T$ is the stress-energy tensor for nongravitational sources. So that's the sense in which gravity doesn't act as a source for gravity--there are no gravity terms on the right-hand-side.

If you start by treating the metric as a field whose source is stress-energy, and then you include self-interaction, you get something equivalent to standard GR, but not quite the same left-hand and right-hand sides. Instead, you get something like this: (This is intentionally hand-wavy, because I don't have a reference handy)

$G_0 = \kappa (T_{non-grav} + T_{grav})$
where $G_0$ is a different tensor (not $G$) involving the metric tensor and its derivatives, and $T_{grav}$ is the stress-energy due to the metric, treating it as a field.

This is equivalent to GR because the combination of $G_0 - \kappa T_{grav}$ happens to be equal to the tensor $G$.

So my very hand-wavy understanding is that there are terms that can be put on the left-hand side of the equation, and they become part of the Einstein curvature tensor. Or they can be put on the right-hand side, and they become the gravitational part of the stress-energy.

I was never able to follow the mathematics (described very briefly in Misner, Thorne and Wheeler) for how to bootstrap yourself from a linear spin-2 field theory to full GR, which is another reason this is hand-wavy. However, as I understand it, you can't directly figure out stress-energy from equations of motion. You have to concoct a Lagrangian that reproduces your equations of motion, then use that to compute the stress-energy. So there's conceptual an infinite iteration going on:

1. Start with a Lagrangian for non-gravitational fields and matter in flat spacetime.
2. Compute the stress-energy tensor from that.
3. Use that tensor as the source for the first approximation to an equation for the metric (as a spin-2 field).
4. Modify your lagrangian to give that field equation as the Euler-Lagrange equations of motion when you vary the metric.
5. Compute the stress-energy tensor for the modified Lagrangian.
6. Use that stress-energy as the second approximation to an equation for the metric.
7. Modify the Lagrangian again to give that field equation.
8. etc.

But it's possible to be clever and figure out (or guess) the "limit" theory.

 

[bhobba]

I was never able to follow the mathematics (described very briefly in Misner, Thorne and Wheeler) for how to bootstrap yourself from a linear spin-2 field theory to full GR, which is another reason this is hand-wavy.

I nutted it out from Feynmans Lectures many moons ago when I as really into GR:
https://www.amazon.com/dp/0813340381/?tag=pfamazon01-20

It took a bit of mucking around and tooing and frowing with Ohanions text.

Thanks
Bill

 

[PeterDonis]

there are terms that can be put on the left-hand side of the equation, and they become part of the Einstein curvature tensor. Or they can be put on the right-hand side, and they become the gravitational part of the stress-energy.

Yes, this is true. However, one key thing to note is that, with the terms on the LHS (i.e., part of the Einstein tensor), both sides of the equation have zero covariant divergence, which is an important conservation law. If you move the "gravitational field energy" terms to the RHS, the covariant divergence of the LHS and RHS is no longer zero.

 

[stevendaryl]

Yes, this is true. However, one key thing to note is that, with the terms on the LHS (i.e., part of the Einstein tensor), both sides of the equation have zero covariant divergence, which is an important conservation law. If you move the "gravitational field energy" terms to the RHS, the covariant divergence of the LHS and RHS is no longer zero.

That's a very good point, and a good argument for putting gravity on the LHS.