# Why two mass attracts each other?

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This makes no sense, since GR uses the stress-energy-momentum tensor as the source term of gravity; that tensor is a representation of mass and energy.

So is energy. Or momentum. Or potential. Or any number of other concepts in physics.

Actually, no. The scientific method only requires us to extract numerical predictions from the model as a whole, and then compare it to experiment and observation. The scientific method does not actually concern itself with the specific mechanisms of the model; you could, if you wanted to, write a model which explains gravity in terms of microscopic pink unicorns - so long as all predictions of that pink-unicorn explanation line up with experiment and observation, you'd have a valid model of gravity, until such time as a contradiction or a wrong prediction surfaces. The scientific method simply goes : Ask a question - formulate a hypothesis - extract predictions - test those against observation - analyse the outcome, and amend the hypothesis if necessary.
So far as predictions and observations go, GR does a pretty good job in explaining the effects of gravity.

As you know the Gravity Probe B measured the geodetic effect. Apparently Wellwisher doesn't know about this test for GR and spacetime geometry?

Can you explain the difference between 'effect of force on a mass' and 'effect of curvature(or geometry) of space-time on a mass'?

Well, I suppose the main difference would be that the geometric model gives the correct amount for the perihelion precession, whereas the force-based method gives wrong predictions
The crucial difference is that the Newtonian force field is linear, whereas the Einstein field equations are not; in other words, the gravitational field in GR is self-interacting. Hence the difference in predictions. GR does not use any forces - see also my GR primer thread for details.

Well, I suppose the main difference would be that the geometric model gives the correct amount for the perihelion precession, whereas the force-based method gives wrong predictions

I asked a general question. Instead of giving general answer, you are explaining a very particular case of perihelion precession.

The crucial difference is that the Newtonian force field is linear, whereas the Einstein field equations are not; in other words, the gravitational field in GR is self-interacting. Hence the difference in predictions.

What do you mean by "self-interacting gravitational field of GR"?

Can you explain a "contact force(F=ma)" in terms of GR(geometry of spacetime)?

I asked a general question. Instead of giving general answer, you are explaining a very particular case of perihelion precession.

I gave the general answer underneath. Newtonian force fields are linear, whereas the geometry of GR space-time is highly non-linear. That is the reason they don't give the same results, and that is where they differ.

What do you mean by "self-interacting gravitational field of GR"?

In GR, all forms of energy are a source of gravity. This obviously holds for the gravitational field itself as well, since it contains energy. Thus, gravity is self-interacting - the gravitational field is in itself a source of gravity. Mathematically, this is why the Einstein field equations are highly non-linear, in contrast to Newtonian gravity. This is also ultimately why the two models can never be equivalent for strong fields - the stronger the field, the more deviation from Newtonian results you will get.

Can you explain a "contact force(F=ma)" in terms of GR(geometry of spacetime)?

No, because GR is a model for gravity, not contact forces.
The concept of "forces" is not inherent in GR, but only a secondary derivation from it. Gravity can be modelled perfectly in terms of geometry alone, without any mention of forces. In fact that is ultimately the whole point of GR.

If you are asking whether F=ma can be generalised to curved space-times, then the answer is of course yes - the acceleration then becomes a 4-vector which is the covariant derivative ( defined in terms of space-time geometry ) of 4-velocity with respect to proper time, yielding a 4-force as result. But as I said already, this is "artificial" in that GR needs no forces to work.

I really do urge you to take a look at my GR primer thread, it is all explained there.

I gave the general answer underneath. Newtonian force fields are linear, whereas the geometry of GR space-time is highly non-linear. That is the reason they don't give the same results, and that is where they differ.

So in Newtonian case a mass will move linearly whereas in GR the mass will move non-linearly?

In GR, all forms of energy are a source of gravity. This obviously holds for the gravitational field itself as well, since it contains energy. Thus, gravity is self-interacting - the gravitational field is in itself a source of gravity. Mathematically, this is why the Einstein field equations are highly non-linear, in contrast to Newtonian gravity. This is also ultimately why the two models can never be equivalent for strong fields - the stronger the field, the more deviation from Newtonian results you will get.

Energy content of a mass is same as energy content of its gravitational field?

No, because GR is a model for gravity, not contact forces.
The concept of "forces" is not inherent in GR, but only a secondary derivation from it. Gravity can be modelled perfectly in terms of geometry alone, without any mention of forces. In fact that is ultimately the whole point of GR.

If you are asking whether F=ma can be generalised to curved space-times, then the answer is of course yes - the acceleration then becomes a 4-vector which is the covariant derivative ( defined in terms of space-time geometry ) of 4-velocity with respect to proper time, yielding a 4-force as result. But as I said already, this is "artificial" in that GR needs no forces to work.

I really do urge you to take a look at my GR primer thread, it is all explained there.

Let us consider a practical example. Consider "precession of a gyro". If a 'contact force' is applied to a "gyro", it will precess. Can this "precession of a gyro" be explained by GR?

So in Newtonian case a mass will move linearly whereas in GR the mass will move non-linearly?

The correct statement is that:

1. In Newtonian physics, the equation of motion is:

$$\frac{d^2u}{d \phi^2}+u=\frac{m}{h^2}$$

where

$$\frac{1}{r}=u=\frac{m}{h^2}(1+e cos (\phi))$$ is the solution of the above ODE

whereas

2. In GR, the equation of motion is:

$$\frac{d^2u}{d \phi^2}+u=\frac{m}{h^2}+3mu^2$$

The term $$3mu^2$$ is responsible for the existence of the non-Newtonian orbits.

So in Newtonian case a mass will move linearly whereas in GR the mass will move non-linearly?

No, this is not what I said. I stated that the field equations in GR are non-linear, leading to a geometry of the gravitational field which is different from the Newtonian ( linear ) description. Hence the differences between Newton and GR. In other words - if formulated in terms of forces, the gravitational field in GR is not a simple inverse square law, but something much more complicated.

Energy content of a mass is same as energy content of its gravitational field?

I didn't say that ( though it's an interesting point ). All I was saying is that the gravitational field itself is also a form of energy, and thus a source of gravity. This is the physical result of the non-linearity of the field equations.

Let us consider a practical example. Consider "precession of a gyro". If a 'contact force' is applied to a "gyro", it will precess. Can this "precession of a gyro" be explained by GR?

Of course. You will indeed find that, just like in the case of Mercury's orbit, the Newtonian prediction for gyroscopic precession becomes increasingly inaccurate the stronger the gravitational influence in question is. Once again Einstein gives the correct numbers here - in GR this effect is called Lense-Thirring precession :

http://en.wikipedia.org/wiki/Lense–Thirring_precession

It is one of the classic tests of General Relativity, and ultimately is the result of frame dragging which is a purely geometric effect.

Quasi-physical "force"---"purely geometri"c vs spin

MH}...It is one of the classic tests of General Relativity, and ultimately is the result of frame dragging which is a purely geometric effect

"Purely geometric", here again is reminiscent of shape/form/pattern of vehicle affecting air-flow around the vehicle i.e. the position a collective set of atoms/molecules that create a 2ndary ~~wave/shape/pattern~~~ ergo there can be no puerly geometric affect, without the existence of a medium that occupies a space and a set of positions in space.

prevous quote below..MH}.."I didn't say that ( though it's an interesting point ). All I was saying is that the gravitational field itself is also a form of energy, and thus a source of gravity. This is the physical result of the non-linearity of the field equations".

Ultra-micro ergo quasi-physical yet affects the more macro-physical, tho we do not see the connection, except indirectly between loss of energy via two observations far away of very massive set of two orbiing each other celestial objects-- binary pulsars or something like the -- and one lab experiement that shows neutrons falling but hesitating at discrete levels--- apparrently similar or reminiscent of to early discrete levels of electron shell identificaltion via photons coming in or going out.

And similar to our even more indirect assessment of virtual particles popping into and out of spactial existence, to rapidly for us to directly observe. This reminds me of my belief that gravity may operate just a fraction greater than the speeds-of-radiation. There is a least one experiement to verify the speed-of-gravity using Jupitor--- as best as I recall ---and if we take the allowed tolerance of errors given, we see that on the far side of error, gravity could be operating at .2% faster than speed-of-radiation.

So, if a virtual particle pops in and out of existence to quickly to be observed, why not have a gravitational force( gravitons ) that is too fast to observe?

So, in summary;

Newton = linear static background space

Einstien = complex( 4D ) dynamic background graviational spacetime

Precession = affects of bodies upon other bodies--- Fuller approximation "usually at 90 degrees see moon moving nearly 90 degrees to radial line to Earth"

Frame dragging = spin( vector? ) affecting/interaction-with gravitational spacetime and that may or may not translate to "purely geometric".

Trying to bring GR down to the common human comprehension abilities. The sports fans stand and sit in sequence to create a geometric pattern/shape/form called a ~~~wave~~~ or more specific as a people wave.

Air/wind blows the water to create water ~~~waves~~~~. No wind leaves no water waves on semi-static pond surface.

r6

No, this is not what I said. I stated that the field equations in GR are non-linear, leading to a geometry of the gravitational field which is different from the Newtonian ( linear ) description. Hence the differences between Newton and GR. In other words - if formulated in terms of forces, the gravitational field in GR is not a simple inverse square law, but something much more complicated.

I didn't say that ( though it's an interesting point ). All I was saying is that the gravitational field itself is also a form of energy, and thus a source of gravity. This is the physical result of the non-linearity of the field equations.

Of course. You will indeed find that, just like in the case of Mercury's orbit, the Newtonian prediction for gyroscopic precession becomes increasingly inaccurate the stronger the gravitational influence in question is. Once again Einstein gives the correct numbers here - in GR this effect is called Lense-Thirring precession :

http://en.wikipedia.org/wiki/Lense–Thirring_precession

It is one of the classic tests of General Relativity, and ultimately is the result of frame dragging which is a purely geometric effect.

No, this is not what I said. I stated that the field equations in GR are non-linear, leading to a geometry of the gravitational field which is different from the Newtonian ( linear ) description. Hence the differences between Newton and GR. In other words - if formulated in terms of forces, the gravitational field in GR is not a simple inverse square law, but something much more complicated.

You are just explaining the difference between 'GR field equations' and 'Newtonian equations'. This is not the exact answer to my question. If you refer my original question in the post #759, it is basically to find the difference between 'effect of GR field equations on a mass' and 'effect of Newtonian equations on a mass'.

I didn't say that ( though it's an interesting point ). All I was saying is that the gravitational field itself is also a form of energy, and thus a source of gravity. This is the physical result of the non-linearity of the field equations.

EFEs may be non-linear but is there effect on a mass is also non-linear?

Of course. You will indeed find that, just like in the case of Mercury's orbit, the Newtonian prediction for gyroscopic precession becomes increasingly inaccurate the stronger the gravitational influence in question is. Once again Einstein gives the correct numbers here - in GR this effect is called Lense-Thirring precession :

http://en.wikipedia.org/wiki/Lense–Thirring_precession

It is one of the classic tests of General Relativity, and ultimately is the result of frame dragging which is a purely geometric effect.

Newtonian prediction of precession is inaccurate because at that time 'frame-dragging effect' was not known. Otherwise his calculation was correct.

So, from this a conclusion can be made that "frame dragging" has a effect of "anti-gravity" or "repulsive gravity" which causes 'anomaly in perihelion precession of Mercury'.

...In GR, all forms of energy are a source of gravity. This obviously holds for the gravitational field itself as well, since it contains energy. Thus, gravity is self-interacting - the gravitational field is in itself a source of gravity. Mathematically, this is why the Einstein field equations are highly non-linear, in contrast to Newtonian gravity...

How do you reconcile your sofar unchallenged claim above, repeated also in #767, with the oft quoted passage from http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
Does "gravitational energy" itself act as a source of gravity? Now, the Einstein field equations are

Gmu,nu = 8pi Tmu,nu

Here Gmu,nu is the Einstein curvature tensor, which encodes information about the curvature of spacetime, and Tmu,nu is the so-called stress-energy tensor, which we will meet again below. Tmu,nu represents the energy due to matter and electromagnetic fields, but includes NO contribution from "gravitational energy". So one can argue that "gravitational energy" does NOT act as a source of gravity. On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other (unlike light waves in Maxwell's (linear) theory). So one can argue that "gravitational energy" IS a source of gravity.
That last bit is best understood as an "it-sure-looks-like-it-here-but-looks-are-deceiving" style of statement - i.e. the real position is that stress-energy tensor has no gravitational energy density term - period. I happen to believe gravitational field should be a source term (as is the case in e.g. much maligned Yilmaz Gravity), but in GR it is not. As another German GR buff put it elsewhere: "gravity does not somehow walk around from the LHS to the RHS of the EFE's and become it's own source." But you obviously disagree with that individual and above reference too. Why?

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Newtonian prediction of precession is inaccurate because at that time 'frame-dragging effect' was not known.

You are mixing up two very different phenomena: orbital precession and the gyroscopic precession.

Otherwise his calculation was correct.

No, it wasn't, the Newtonian equation I gave you above shows a (closed) ellipse with no precession while GR equation of motion shows an open trajectory with precession.

Newtonian prediction of precession is inaccurate because at that time 'frame-dragging effect' was not known.
You are mixing up two very different phenomena: orbital precession and the gyroscopic precession.

What is the difference in "basic principle(not effect)" for "orbital precession" and "gyroscopic precession"?

Otherwise his calculation was correct.
No, it wasn't, the Newtonian equation I gave you above shows a (closed) ellipse with no precession while GR equation of motion shows an open trajectory with precession.

If your above statement is right; how Newton could calculate "perihelion precession" for all the planets(except Mercury) in the Solar system almost correctly.

If your above statement is right; how Newton could calculate "perihelion precession" for all the planets(except Mercury) in the Solar system almost correctly.

He didn't. That is the whole point.

You have reference?

Any book on GR. have you ever opened one? Maybe now is the time for you to get started. I can recommend a few, are you interested?

Any book on GR. have you ever opened one? Maybe now is the time to get started.

YES. See chapter 12.12 and 12.13. Newtonian calculations for precession of all the planets in the solar system is mentioned there.

YES. See chapter 12.12 and 12.13. Newtonian calculations for precession of all the planets in the solar system is mentioned there.

Yes, did you understand this:

"If the calculation described in the previous section is carried out more accurately, taking into account the slight eccentricities of the planetary orbits, as well as their small mutual inclinations, and retaining many more terms in the expansions (1015) and (1017), then the perihelion precession rate of the planet Mercury is found to be $5.32$ arc seconds per year. However, the observed precession rate is $5.75$ arc seconds per year. It turns out that the cause of this discrepancy is the general relativistic correction to Newtonian gravity. "

This is precisely what I have already shown you a few posts ago.

Yes, did you understand this:

"If the calculation described in the previous section is carried out more accurately, taking into account the slight eccentricities of the planetary orbits, as well as their small mutual inclinations, and retaining many more terms in the expansions (1015) and (1017), then the perihelion precession rate of the planet Mercury is found to be $5.32$ arc seconds per year. However, the observed precession rate is $5.75$ arc seconds per year. It turns out that the cause of this discrepancy is the general relativistic correction to Newtonian gravity. "

This is precisely what I have already shown you a few posts ago.

Did you mean the above by your statement in post #773? This fact I already mentioned in my post #772.

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