Why two mass attracts each other?

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This is my understanding as well, but if there is no force particle involved....what is a graviton and why are physicist looking for it?

The graviton is not part of GR; it is neither predicted nor required by the theory since, as you rightly say, no forces are involved. Gravitons arise only if we try to formulate a quantum field theory of gravity, which is a step beyond "classic" GR. Such a field theory would require a particle as mediator of the force, which is precisely the ( hypothetical ) graviton. Since the source term of gravity is a rank-2 tensor, the graviton would need to be a massless spin-2 boson.

If spacetime(all 4 dimensions) is a line, and that line bends....by definition that line must exist on a 2 dimensional plane.

No, you need to be careful here - GR deals with intrinsic geometry, not extrinsic one. The latter would require an embedding of the manifold in a higher dimensional space, but the former doesn't. Of course there exist models where our universe is indeed embedded in a higher dimensional manifold ( with very interesting results ! ), but "classic" GR does not require this to be the case at all.

Yaa, it is quite difficult to explain attractive force between two mass in terms of spacetime curvature.

There are no mechanical forces involved in this, just geometry.

This tells you that Einstein was thinking in terms of an elastic solid. He thought of space as something that could be curved and stretched and put under pressure, et cetera.

That is certainly not what Einstein was thinking.

$$T_{\mu \nu}$$ is merely the source term in the field equations, and describes the various forms of energy which form sources of gravity, and that includes things like density, flux, stress and momentum. This applies to any energy configuration, i.e. solids, fluids, fields etc; the total SEM tensor is simply the sum of all the various contributions. If $$T_{\mu \nu}=0$$ we are dealing with a vacuum ( exterior field equations ), which, by the way, does not imply that there is no curvature.
Space-time itself, in classical GR, possesses only one degree of freedom, and that is curvature. Space-time cannot be put "under pressure" or "under stress" or anything of that nature, it can only be curved - not surprising, since GR is explicitly restricted to using the Levi-Civita connection, which permits only curvature on the manifold and nothing else. That curvature is the result of the source terms contained in the stress-energy-momentum tensor as well as the non-linear self-interaction of the field itself; saying that Einstein thought of space as something that could be put "under pressure" or "stretched" is simply wrong, unless you can show that these are expressible in terms of curvature only. The geometry of space-time under GR is expressed as curvature only.

If you wish to see explicitly how a solution to these equations is obtained, and what the terms mean, then I shall be happy to provide that material. I have done the calculation myself on another forum ( exterior Schwarzschild metric ), so I can provide a link to that thread if you need it. It shows nicely the calculation of the curvature tensor and its contractions in terms of the Levi-Civita connection. I have also done the interior Schwarzschild metric where the SEM tensor does not vanish, but only on paper, not on the Internet. Maybe one day, when I have the muse, I'll type it up as well.

Just please don't try to sell statements as fact if really you aren't sure what you are talking about; you know very little about differential geometry, so better stay away from the maths of GR.

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gravity and photon

This is from Einstein's Paper and Einstein's Equation E=mc^2. Einstein in his paper proved that "inertia of a mass is dependent on its energy content". So, we can write m=E/c^2.

Gravitational Force F=G*m1*m2/r^2 can be re-written as F=G(E1/c^2)(E2/c^2)/r^2 or F=(G/c^4)(E1*E2)/r^2.

"Strong Nuclear Force" keep proton and neutron together. This nucleus keep the whole atom together. So, "strong nuclear force" keep the "energy content(E)" of a mass(m) together. So, inertia of a mass(m) is dependent on "Strong Nuclear Force".

As "Gravitational Force" is dependent on mass(m), so it is also dependent on "Strong Nuclear Force".

Newton's Equation of "Gravitational Force" also can be written in terms of photon particles.

Consider energy of photon Ep=h*f; where Ep is energy of photon, h is Planck's Constant, f is frequency of photon.

If mass m is converted into light energy E(where E=mc^2), it will generate E/Ep number of photons. Consider E/Ep=n, where n is a number. So, E1=n1*Ep or E1=n1*h*f. E2=n2*h*f.

We can write, "Gravitational Force" F=(G/c^4)(E1*E2)/r^2 or F=(G/c^4)(n1*h*f)(n2*h*f)/r^2 or F=((G*h^2)/c^4)(n1*n2)(f^2/r^2).

The graviton is not part of GR; it is neither predicted nor required by the theory since, as you rightly say, no forces are involved. Gravitons arise only if we try to formulate a quantum field theory of gravity, which is a step beyond "classic" GR. Such a field theory would require a particle as mediator of the force, which is precisely the ( hypothetical ) graviton. Since the source term of gravity is a rank-2 tensor, the graviton would need to be a massless spin-2 boson.

Here is a paper on quantum gravity.

I'm almost positive this describes pudding. Very delicious.

Pretty close. If you placed 'your pudding and it's wristwatch' in a natural orbital path at r this derivation includes both the SR and GR component of time dilation.

The ratio

dTau_orbiting pudding/dt_Schwarzschild remote bookkeeper compares the tick rate between the local proper frame measurement dTau and the remote coordinate 'Schwarzschild boundary' measurement dt.

As Markus explained there's no forces associated with 'the natural path'. The 'puddings' natural path through the gravitational field is a path where no forces are present. Freefall.

The graviton is not part of GR; it is neither predicted nor required by the theory since, as you rightly say, no forces are involved. Gravitons arise only if we try to formulate a quantum field theory of gravity, which is a step beyond "classic" GR. Such a field theory would require a particle as mediator of the force, which is precisely the ( hypothetical ) graviton. Since the source term of gravity is a rank-2 tensor, the graviton would need to be a massless spin-2 boson.

Thank you! This explains a lot.

So...if quantum field theory is correct and the graviton is the force particle for gravity...would that also make it the inertial force particle during acceleration?

No, you need to be careful here - GR deals with intrinsic geometry, not extrinsic one. The latter would require an embedding of the manifold in a higher dimensional space, but the former doesn't. Of course there exist models where our universe is indeed embedded in a higher dimensional manifold ( with very interesting results ! ), but "classic" GR does not require this to be the case at all.

Ok...so space-time curvature is to space-time, as a bubble/ripple is inside a block of jello? The manifold is 4 dimensional space-time....or a block of jello. Gravity is inhomogeneous/curved space-time...or ripples/bubbles inside the jello block. If that's right, my confusion was largely due to semantics. Bending space-time isn't talking about bending the jello block...it's about imperfections within the otherwise smooth jello.

I'm not versed in the calculus like most of you appear to be, so I can't converse with the same fidelity. In general terms I understand how GR works pretty well, but none of the layman books go deep enough to fill in all the blanks. The only way to get answers for the questions I have are to learn the math or ask someone. I'm really hoping there are some patient people on here that'll help me understand questions I've had for years. They are like a splinter in my mind.

**Note: The reason I haven't learned the math isn't because I'm lazy. I'm a small business owner and ALSO a programmer for ISS. Two high stress jobs don't leave a lot of time for hobbies like this.

Pretty close. If you placed 'your pudding and it's wristwatch' in a natural orbital path at r this derivation includes both the SR and GR component of time dilation.

Time dilation is also delicious.

Newton's Equation of "Gravitational Force" also can be written in terms of photon particles.

Consider energy of photon Ep=h*f; where Ep is energy of photon, h is Planck's Constant, f is frequency of photon.

If mass m is converted into light energy E(where E=mc^2), it will generate E/Ep number of photons. Consider E/Ep=n, where n is a number. So, E1=n1*Ep or E1=n1*h*f. E2=n2*h*f.

We can write, "Gravitational Force" F=(G/c^4)(E1*E2)/r^2 or F=(G/c^4)(n1*h*f)(n2*h*f)/r^2 or F=((G*h^2)/c^4)(n1*n2)(f^2/r^2).
You can write that down. Ever hear of dimensional analysis?

Thank you! This explains a lot.

So...if quantum field theory is correct and the graviton is the force particle for gravity...would that also make it the inertial force particle during acceleration?

Ok...so space-time curvature is to space-time, as a bubble/ripple is inside a block of jello? The manifold is 4 dimensional space-time....or a block of jello. Gravity is inhomogeneous/curved space-time...or ripples/bubbles inside the jello block. If that's right, my confusion was largely due to semantics. Bending space-time isn't talking about bending the jello block...it's about imperfections within the otherwise smooth jello.

I'm not versed in the calculus like most of you appear to be, so I can't converse with the same fidelity. In general terms I understand how GR works pretty well, but none of the layman books go deep enough to fill in all the blanks. The only way to get answers for the questions I have are to learn the math or ask someone. I'm really hoping there are some patient people on here that'll help me understand questions I've had for years. They are like a splinter in my mind.

**Note: The reason I haven't learned the math isn't because I'm lazy. I'm a small business owner and ALSO a programmer for ISS. Two high stress jobs don't leave a lot of time for hobbies like this.

The math Markus is referring to is generally introduced in graduate school. There's another way to study this. Start with the metric solutions to the EFE and the principle of least action [Noether's Theorem]. You can do 'most all' the derivations describing the natural phenomena predicted by relativity theory with algebra, calculus, quadratic formula for finding limits, and understanding how to do a weak field approximation to simplify weak field analysis [very simple]. The only formal math classes I've had was algebra and geometry in high school. I learned the rest on my own with some help from others when needed. During the learning process it becomes apparent why Einstein modeled the theory with tensor equations and differential geometry.

Ok...so space-time curvature is to space-time, as a bubble/ripple is inside a block of jello?

Well, so long as you realize that this is just an analogy, then yes, you can think of it in that way. Just be careful to realize that space-time can only be curved, it can't be stretched or compressed or put under pressure or anything like that. Just curvature.

In general terms I understand how GR works pretty well, but none of the layman books go deep enough to fill in all the blanks.

That's true, and it is due to the fact that analogys only go so far. There comes a point at which, in order to develop a deeper understanding, you have no choice but to learn at least the basics of differential geometry.

I'm really hoping there are some patient people on here that'll help me understand questions I've had for years. They are like a splinter in my mind.

But again, there are some questions that are very hard to answer without going into the maths. That is just in the nature of GR.

The reason I haven't learned the math isn't because I'm lazy. I'm a small business owner and ALSO a programmer for ISS. Two high stress jobs don't leave a lot of time for hobbies like this.

I fully understand - I am in a similar situation, having to work two full time jobs to make ends meet for my family. I try to put in about an hour per day studying from textbooks, but it is hard going and often I am just too tired.

So...if quantum field theory is correct and the graviton is the force particle for gravity...would that also make it the inertial force particle during acceleration?

I am not going to pretend that I can answer that question for you. QFT in flat space-time is hard enough, but on a curved space-time background it becomes a veritable beast. I know very little about it, so not wanting to give you any wrong info I'd rather refrain from replying altogether. Sorry

Space-time is a mental abstraction, which lacks substance. This abstraction is used as a mental grid to show how substance interacts. Picture wearing a movie projector hat, which projects a coordinate system onto reality (green light grid (x,y,z,t)), like on the windshield display of a jet fighter. At first, one would be constantly aware the coordinate system projection is separate from tangible reality. But as time goes on, people forget this and begin to think the grid is part of reality. The brain will merge these.

When space-time bends, the movie projector is bending the abstraction coordinate system to simulate the relationship between mass particles due to the force of gravity. Answers to why mass attracts, which include the projection as a tangible part of the solution, have lost touch with the overlay of abstract and reality. It does not address the real in terms of real. It will satisfy the math but not the validity of a solid conceptual model; math cart leads the horse.

The trick for answering the question, is you first need to shut off the projector, and use only substance. My approach is to look at where mass substance is heading in terms of final affects within substantial reality. The projector grid is useful for modeling the approach and how the variables relate in terms of the grid.

An analogy is buying a new sports car. At first, you will notice all the new gadgets relative to the old car. But as time goes on, and you adapt to handling and gadgets, the car becomes an extension of you, so the lines between you and car, blur. This is how the mind works but the car is not organic but is abstractly connected in the imagination.

Or say you break your leg and have to wear a cast and use crutches. At first the cast and crutches are awkward, because they feel separate from you, But as time goes on, you get used to the cast and become one with the crutches until you can balance on a basketball. The mind forgets the distinction, until you plus cast and crutches become a unit.

The mind is the most important tool of science, but there is no calibration requirement in science.

Two Masses Attract O><O

If the two masses did not attract-- mass-attraction ---our finite Universe would not exist.

Mesons( medial/medium ) = OO OO = two quarks = 1440 degrees = surface area of two tetrahedrons.

4 equaltorial/bisecting/great circles compose and define the spherical/spheroidal Vector Equilirbium/cubo-octahedron aka the operating system of Universe.

The strong nuclear force( mesons 2 quarks ) between hadrons-- proton neutron etc ---is often times confused with the strong sub-nuclear force( gluons ) of the nucleus that holds 3 quarks together as a hadron( proton or neutron ).

Hadron( heavy ) = OO OO OO = 6-GrCPP's of the spherical/spheroidal cube are same as the 6 GrCPP's of tetrahedron.

r6

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... Just be careful to realize that space-time can only be curved, it can't be stretched or compressed or put under pressure or anything like that. Just curvature. ...

Can you curve a string without stretching or compressing it? Curvature implies there is stretching or compression.

What about "Lorentz length transformation"? Isnt this stretching or compression of length/space?

Yaa, it is quite difficult to explain attractive force between two mass in terms of spacetime curvature.

There are no mechanical forces involved in this, just geometry.

There is energy involved in this, where potential energy is converted into kinetic energy. So, there has to be some force involved in this.

Can you curve a string without stretching or compressing it?

You are misunderstanding this - we are talking about space-time itself, not some object in space-time. So yes, intrinsic curvature is a well defined and physically meaningful concept - no stretching and compression is required.
Unfortunately there isn't any way to visualise this, because not matter what example I choose it would of necessity always be something which is embedded in space-time. This is one of those things where you really need the maths, because we cannot easily step out of our own frame of reference.

What about "Lorentz length transformation"? Isnt this stretching or compression of length/space?

Lorentz transformations are mathematical relations between frames of reference on flat Minkowski space-time; they have no connection to space-time curvature. A single observer in Minkowski space-time will never observe any length contraction since this is a relation between at least two frames, but he can detect gravitational fields. These two things aren't comparable.

There is energy involved in this, where potential energy is converted into kinetic energy. So, there has to be some force involved in this.

No, there are no forces involved. You are still thinking in terms of classical mechanics. You can define tidal forces as secondary phenomena through gradients in gravitational potential, but the curvature itself has nothing to do with forces.
Just to demonstrate the principle, consider two observers standing at different points along the equator. They now start walking north, and the further they walk, the closer they get. At the north pole, they meet. Why is that ? Is there a force between them, pulling them together ? No, they approach merely because of the geometry of the surface they are on ( a curved sphere ), even though they will each say that they have been walking "straight" all along. If you do the same thought experiment on a flat sheet, the observers just keep walking straight into infinity, without ever meeting. Likewise in space-time; two massive objects will, as they age into the future, approach each other due to the underlying curved space-time. The "walking north" roughly corresponds to "aging into the future".
I reiterate that this is just a very simple analogy to demonstrate the principle of how things can approach without forces acting between them, nothing more. Yes, there is of course energy involved as well, but in GR energy is equivalent to geometric properties of space-time, not mechanical forces.

...this sounds a lot like space-time displacement. Like adding a ball to a sealed tank full of air. The ball displaces the air, but since the air can't escape it becomes compressed...the PSI rises and it adds stress to the whole system. Kind of like that?
Yes. A better analogy is a gin-clear tank of jelly, and you inject more jelly in the middle. A field is where the jelly is stressed. When the stress propagates, it's a wave. An important thing to note is the wave nature of matter. Matter is like standing waves in the jelly. The jelly is all you've got to play with.

MeNotYou said:
Wait...would that make the Higgs-Boson the ball/tensor/thing that displaces(stresses) space-time??
No. But see this for mention of ex-CERN physicist John Ellis referring to the Higgs field as a kind of relativistic aether. That's like saying the Higgs field is the jelly. I think it's better to say just space is the jelly myself.

MeNotYou said:
This is my understanding as well, but if there is no force particle involved....what is a graviton and why are physicist looking for it?
A graviton is a hypothetical "virtual" particle, a field quantum. Think of a field quantum as small portion of the stressed jelly rather than some little billiard-ball thing that flies around. Physicists aren't so much looking for it as trying to develop a theory that employs a calculation method wherein the jelly is mentally divided up into little cubes. It's important to understand the general idea of this. Virtual particles are not actual particles. One can treat the electromagnetic field as being made up of virtual photons, but these are not actual photons flying back and forth. Hydrogen atoms don't twinkle, and magnets don't shine.

That is certainly not what Einstein was thinking.

$$T_{\mu \nu}$$ is merely the source term in the field equations, and describes the various forms of energy which form sources of gravity, and that includes things like density, flux, stress and momentum. This applies to any energy configuration, i.e. solids, fluids, fields etc; the total SEM tensor is simply the sum of all the various contributions. If $$T_{\mu \nu}=0$$ we are dealing with a vacuum ( exterior field equations ), which, by the way, does not imply that there is no curvature.
Space-time itself, in classical GR, possesses only one degree of freedom, and that is curvature. Space-time cannot be put "under pressure" or "under stress" or anything of that nature...
I'm afraid it is, Markus. You're confusing space and spacetime. See my post 120 where I explained the distinction. Let's run through it again.

Spacetime is an abstract mathematical space in which motion does not occur because it models space at all times. You can draw world-lines in it, and you can draw them curved, but that worldline represents the motion of a body through space over time. The body doesn't actually move through spacetime. People tend to talk of "the spacetime around the Earth" and suggest that light moves through it, but that's wrong. Pay careful attention to Einstein's 1920 Leyden address where he said this:

“According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν), has, I think, finally disposed of the view that space is physically empty”.

Space is inhomogeneous, not curved. And because it's inhomogeneous, motion through this space over time is curved, so we say spacetime is curved. Do refer to the The Foundation of the General Theory of Relativity and note that Einstein refers to the equations of motion. Also note that metric is to do with measurement. The metrical qualities of the continuum of space-time differ because space is neither homogeneous nor isotropic. You can understand this by placing gedanken light-clocks in an equatorial plane through the Earth. The light-clocks run at different rates, and when you plot your measurements your plot is curved just like the wikipedia plot of gravitational potential, which is in turn like the bowling-ball-in-the-rubber-sheet pictures. But the light-clocks didn't run at different rates because your plot was curved, they ran at different rates because the space they're in is inhomogeneous. Google on inhomogeneous vacuum for more information. One of the papers you come across is Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime. Relate this back to Einstein's reference to inhomogeneous space.

...saying that Einstein thought of space as something that could be put "under pressure" or "stretched" is simply wrong, unless you can show that these are expressible in terms of curvature only. The geometry of space-time under GR is expressed as curvature only.
That geometry is the geometry of motion through space. It is not the geometry of space. The Einstein field equations do include pressure terms and shear-stress terms. And note Einstein's 1929 presentation on field theory:

"It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds, and it is natural to suspect that this only appears to be so because the structure of the physical continuum is not completely described by the Riemannian metric."

The Riemannian metric describes the state of space. And it features pressure and shear stress. These and other terms result in curved motion through that space, to which we apply the label curved spacetime. It's like a car encountering mud at the side of a road. The road isn't curved, nor is the mud. But the car veers left. It's path is curved.

Markus Hanke said:
Just please don't try to sell statements as fact if really you aren't sure what you are talking about; you know very little about differential geometry, so better stay away from the maths of GR.
With respect Markus, I understand this, and you are confusing space and spacetime. It's important that you understand the difference.

MeNotYou: this is interesting. I'm going to volunteer some responses.

So...if quantum field theory is correct and the graviton is the force particle for gravity...would that also make it the inertial force particle during acceleration?
You've touched on something interesting here. Inertial mass and gravitational mass are equivalent. See what I said above about field quanta. Also remember E=mc² and the mass of a body is a measure of its energy content, along with the photon in the box which increases the mass of the system. It's only a photon in that box, it's the only particle there. You can divide up its field into virtual photons. And you can divide that self-same field up into virtual gravitons. How come? I think it goes something like this: mentally divide your gin-clear jelly into little cubes. Then stress the jelly. If your little cubes are no longer cubes and instead are trapezoidal, you call them virtual photons. If they're cuboid (rectangular) you call them virtual gravitons. But they're the same cubes. You don't have two sets of cubes. You can stress the jelly in one point and get trapezoids. Then when you stress is at another point you get more trapezoids. But look closely, and you will see that between these two points some of the cubes are now cuboid. Add more stress points and get it right, and all your cubes are cuboid.

Ok...so space-time curvature is to space-time, as a bubble/ripple is inside a block of jello? The manifold is 4 dimensional space-time....or a block of jello. Gravity is inhomogeneous/curved space-time...or ripples/bubbles inside the jello block. If that's right, my confusion was largely due to semantics. Bending space-time isn't talking about bending the jello block...it's about imperfections within the otherwise smooth jello.
See what I said above to Markus. It's important to understand that the jello is an analogy for space rather than spacetime. In a gravitational field the jello isn't bent, instead motion through the jello is bent. If your jello was marked out with a cubic lattice, then if you stressed it to emulate a gravitational field and looked at it through a microscope, I think it would would look something like this: View attachment 6251. But note that this is only an analogy. You might want to ask around and see if people think it's a good analogy.

Space-time is a mental abstraction, which lacks substance. This abstraction is used as a mental grid to show how substance interacts. Picture wearing a movie projector hat, which projects a coordinate system onto reality (green light grid (x,y,z,t)), like on the windshield display of a jet fighter. At first, one would be constantly aware the coordinate system projection is separate from tangible reality. But as time goes on, people forget this and begin to think the grid is part of reality. The brain will merge these...
Well said, wellwisher. Space is real, it supports waves and fields, that's what's "out there". Spacetime however is a an abstract mathematical "space". It's used for modelling, and has provided just about the most robust and well-tested theory we've got. But spacetime just isn't the same thing as space, and yet people continue to confuse the two.

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