Why two mass attracts each other?

Serious question Farsight: are there physicists you believe understand general relativity better than you do? If so, could I ask for some examples?

I ask because I recall you saying you understood electromagnetism better than Dirac, and you seem equally confident in your understanding of the (considerably more complex) theory of general relativity. Thanks.

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I don't know, Guest. One of the relativity "heavyweights" is George Ellis, see his papers on arXiv. They include a paper on entropic gravity, a paper on the flow of time, and Note on Varying Speed of Light Cosmologies which I took to be a slap down for Magueijo. It doesn't square with Einstein or the hard scientific evidence wherein optical clocks run slower when they're lower. This kind of thing:

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The light goes slower when it's lower, so the speed of light isn't constant, just like Einstein said.

My pleasure Guest. Like everything else, when you "get" it, it isn't all that complicated.

 

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So would it be fair to say that if George Ellis told you that you actually misunderstood some part of general relativity, you would accept his correction? What about other heavyweights, e.g. Hawking, Geroch and so on. Would you listen to them, or would you assume to know better?

I guess I don't "get it" like you do - I think general relativity is extremely complicated!

 

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If he explained it, and backed that up with evidence and references to what Einstein said, I'd accept it. But if he just "told" me and said I had to accept it because he was the expert, I wouldn't. Especially if what he told me contradicted Einstein and the evidence.

I'll listen to anybody, Guest. I don't care who they are.

I'd say the maths is complicated, but the "big picture" isn't. Have a look at my post #158 to get a handle on that. The important thing to appreciate is that the bowling-ball pictures aren't showing you curved space, they're showing a curvature in your plot of your measurements in space over time. Like those you do with the light-clocks in an equatorial slice through the Earth. The light-clocks have light moving in them, and the clocks nearer the Earth run slower because the light goes slower there. When you plot all your measurements you end up with a curved "metric", just like the bowling-ball pictures of Riemann curvature. But it isn't showing you curved space, it's just depicting a graph of the speed of light around the Earth.

You have to read what Einstein said to "get" this. In 1916 he wrote Relativity: The Special and General Theory which was translated into English in 1920. The English translation says this:

"In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light)".

There's a bit of a problem with it: the word "velocity". People immediately think of the vector quantity, but look closely. If this was really the vector quantity instead of the "high velocity bullet" common usage, why is Einstein referring to the SR postulate? That was the constant speed of light. And if it really was the vector quantity wherein the change of velocity was a change in direction, he'd be saying "light curves because it curves". That's obviously not right. To get to the bottom of it you look at the original 1916 version in German, and then you read "die Ausbreitungs-geschwindigkeit des Lichtes mit dem Orte variiert". That's "the propagation speed of light varies with position", totally in line with those parallel-mirror light clocks. And it varies because space is inhomogeneous, just like Einstein said, and so light curves like the car veers. IMHO it's fairly simple to explain and understand. What I can't explain is why it isn't common knowledge or why people can't accept it. Look at the image of the light clocks above. It's a bit exaggerated, but it isn't misleading. And yet to hell with Einstein and what's in front of their noses, there are people out there who will insist that the two light pulses are moving at the same speed. Crazy.

 

Farsight, slow down a bit. I am pretty sure Guest254 was not born yesturday.

You know I disagree with your interpretation of much of what Einstein said... That aside your insistence on Einstein as THE authority on Special or General Relativity, as it is known today is laughable. Einstein began something which has surpassed him. I am pretty sure he would have no other way. There are many folks out there today that are greater authorities on GR than Einstein was. Even during his lifetime his theory and his understanding of it, was modified and evolved, through the influence of others.

 

There is no such thing as "space" or "time" in GR, there is only "space-time". You cannot separate the two, which is why most of your post is meaningless.

Wrong. Space-time has curvature, and curvature is the only degree of freedom which the Levi-Civita connection permits. See my previous post.

Yes. They are geodesic equations which determine geodesics in space-time.

No, it is the intrinsic geometry of space-time.

I presume you are referring to the metric tensor of a Riemann manifold. Again, this refers to space-time, not space.

It would appear that you are the one here who is confused about space, time and space-time. What is it with your laughable attempts at separating space and time ? You can't. They are aspects of the same manifold - there is only "space-time", anything else is meaningless. When we are talking about motion in the context of GR, we are dealing with a foliation of parametrized hypersurfaces; we are considering a world line in space-time. All of your misconceptions stem from your failure to realize that space and time cannot be separated.

Go and learn some differential geometry instead of throwing around textual quotes !

 

It is the stress energy tensor on the RHS that contains these terms; that is the source term in the field equations. The LHS contains curvature only.

No it doesn't. The metric tensor ( I can only assume that is what you mean ) is defined as the inner product of basis vectors on our manifold :

\(\displaystyle{g_{\mu \nu }=\left \langle \partial _{\mu }|\partial _{\nu } \right \rangle}\)

Since the set of basis vectors in our case has four elements, the indices in the above definition run from 0...3, and we have a metric signature of (+,-,-,-), we are dealing with space-time. And it does not describe any "states" either, it merely defines the notions of angles as well as line/surface/volume elements on our manifold via the above inner product. This is all elementary differential geometry.

You see, this is what happens if you rely on textual quotes without any understanding of the maths behind it. The above is wrong.
The geometric degrees of freedom of our space-time manifold aren't the result of the metric tensor, but of the connection used. In GR this is the Levi-Civita connection, which is an affine connection with the property that torsion vanishes everywhere on it; the only degree of freedom it permits is curvature. This connection is unique, and the direct result of the fundamental theorem of Riemann geometry.

Now, I am telling you that what you write above is wrong; if you are unwilling to accept that fact then I will insist on a formal mathematical derivation to show that the metric tensor on a pseudo-Riemannian (3+1) manifold "features" pressure and shear stress. In fact you will need to define those terms first in the context of a metric tensor, because so far as differential geometry is concerned these concepts are meaningless. There are no such things as "pressure" or "stress" on a connection, there is curvature and torsion only. I will show you now that the Levi-Civita connection, which is the one used in GR, admits only curvature.

Start with the metric tensor on a general Riemann manifold

\(\displaystyle{g_{\mu \nu }=\left \langle \partial _{\mu }|\partial _{\nu } \right \rangle}\)

A connection on such a manifold is defined via the relation

\(\displaystyle{\nabla _{\partial _{\mu }}\partial _{\nu }=\sum_{k}\Gamma _{\mu \nu }^{k}\partial _{k}}\)

In order for torsion to vanish we must have

\(\displaystyle{\nabla _{\partial _{\mu }}\partial _{\nu }=\nabla _{\partial _{\nu }}\partial _{\mu }}\)

or in other words

\(\displaystyle{\Gamma _{\mu \nu }^{k}=\Gamma _{\nu \mu }^{k}}\)

Now insert back into the definition of the metric tensor

\(\displaystyle{\left \langle \nabla _{\partial _{\mu }}\partial _{\nu }|\partial _{\lambda } \right \rangle=\Gamma _{\mu \nu }^{k}g_{k\lambda }=\Gamma {_{\nu \mu }^{k}}g_{k\lambda }}\)

via permutation of indices. This means that our connection leaves the metric tensor unchanged, thereby fulfulling the vanishing torsion condition above. Since the Riemann curvature tensor is a function of this connection and its derivatives, and therefore inherits its symmetries, we get a manifold that has non-zero curvature, but everywhere vanishing torsion.
I invite you to verify this yourself for any given, specific metric.

Your assertion that a pseudo-Riemannian manifold is only "space" and has no curvature is just completely ludicrous, and even more ludicrous is that you are trying to sell your own laughable ignorance as facts.

My primary reference for all of the above is "Differential Forms and Connections" by R.W.R. Darling, where this is all made mathematically precise and rigorous.
A scondary source then is Fecko's "Differential Geometry and Lie Groups for Physicists", which is an excellent book and a standard text for any undergrad physics student. So don't you say I have nothing to back it up with - I can provide references for every single bit of maths and physics I presented above, if that is necessary.

I urge you once again not to comment on things which you do not understand and know nothing about, like differential geometry, making your claims look like facts. You are misleading other members or casual readers who might come here looking for real information; this behaviour of yours is what got you confined to the "Pseudo" section on other forums. What you are asserting about stress and pressure in metric tensors, and about "space" not having curvature, is plain and simply meaningless and wrong. The entire model of GR is based on mathematics which are well defined and well understood; in fact these maths were in existence long before Einstein came along, he simply made use of them. You cannot just go and reinterpret things as you please, inserting your own opinions on how things should be according to yourself. GR models only space-time, not space and not time, and that space-time has curvature as the non-vanishing curvature tensor clearly shows us. Claiming anything else about GR is wrong.

 

Try \nabla.

 

Works like a charm, thank you

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No problem. .

I don't insist on that actually. I insist on the hard scientific evidence as THE authority.

If those greater authorities contradict the hard scientific evidence AND Einstein, then I'm afraid they aren't greater authorities. It's that simple. Look at the parallel mirror light clocks above and the Einstein quotes. Look at the Shapiro delay too. Don't just believe what some "greater authority" tells you. Physics is a science, not a religion. Do your own research and think for yourself. This is a good little article to look at. See where it says this:

"Since Einstein talks of velocity (a vector quantity: speed with direction) rather than speed alone, it is not clear that he meant the speed will change, but the reference to special relativity suggests that he did mean so. This interpretation is perfectly valid and makes good physical sense..."

Then look at the end of the article where it says this:

"Finally, we come to the conclusion that the speed of light is not only observed to be constant; in the light of well tested theories of physics, it does not even make any sense to say that it varies".

Did you catch that? The article says that what Einstein said makes good physical sense, and then says it does not even make any sense. That's the sort of nonsense you end up with when you let some "greater authority" do your thinking for you. Do not fall for it.

 

No, that is the kind of nonsense you end up with if you rely on textual quotes instead of a good understanding of the mathematical principles behind GR, which, by the way, predate Einstein. Bernhard Riemann is the real authority here, so is Ellie Cartan and to some extent Hermann Minkowski. All Einstein did was write down a tensor equation for a simple physical principle ( which he can take credit for coming up with ); however, the language he used - differential geometry - was already in existence. What his equation really means is made mathematically precise and rigorous by Riemannian differential geometry.

Hard scientific evidence is that GR makes the correct predictions, as a tensor relation between the presence of energy and curved space-time, as correctly understood by established mainstream science and every single textbook on geometrodynamics which is out there. Actually, to make it really precise we would need to also demand that the Lie derivative of the metric tensor with respect to any arbitrary smooth differentiable vector field must vanish - do you know why ? You won't find the answer to that in Einstein's words, only in the understanding of the geometry behind GR...

 

No Markus, the sort of nonsense you end up with this:

Finally, we come to the conclusion that the speed of light is not only observed to be constant; in the light of well tested theories of physics, it does not even make any sense to say that it varies.

Because we know that optical clocks run slower when they're lower. NIST have demonstrated this with a vertical separation of only a foot. And we know that the same applies to the parallel-mirror light clock. So here you have it:

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Yes there is. Einstein referred to space repeatedly, as I've shown. Did you miss 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, and as a result the metrical qualities of the continuum of space-time exhibit curvature. It's a curvature in our plot of our measurements. Now read the paragraph again and tell me again that there's no such thing as space in GR.

It's not me who's confused. it's you. And you're totally disregarding my clear simple explanation along with what Einstein said, and then you're getting emotional about it. I'm sorry Markus, but Einstein said inhomogeneous space. That's what he said, along with "die Ausbreitungs-geschwindigkeit des Lichtes mit dem Orte variiert". You know what it means.

And I urge you to read what Einstein said along with my clear simple explanation. Until you do this and understand it, along with the distinction between space and spacetime, you do not understand GR at all. Putting up mathematics in an attempt to persuade other readers of your "greater authority" is no substitute for this. You do not really understand something unless you can explain it to your grandmother. I can, you can't. And moreover you cannot address my clear simple explanation to point out where it is wrong. Because it isn't.

 

What I miss is the second part of the sentence, the bit where I said "there is only space-time". Why did you cut that out ?

There is no such thing as space in GR, there is only space-time.

The inhomogeneity refers to curvature of space-time since, as I have shown you, that is the only geometric degree of freedom in GR. Einstein wasn't precise in his words. Of course the geometry on an Einstein manifold ( space-time ) is not homogenous, since there is no single coordinate chart with which you can cover the entire manifold. Again, that is basic DG. The entire geometry is captured within the Riemann curvature tensor.

As for the second part, it is imprecise also. Precisely because you can't cover the entire manifold with a single coordinate chart it makes no sense to say something like "the propagation velocity of light varies", because the speed of light is always a local measurement. At any given point of space-time the speed of light is always exactly c; the reason why light deflection exist is because light traces out null geodesics in space-time, and such geodesics have the same intrinsic curvature as space-time itself, which is exactly your "inhomogeneity".

If you follow his maths ( which are precise and correct ! ) instead of his words, none of these misconceptions would arise !

 

An excellent demonstration of how curvature varies throughout space-time. There's your inhomogeneity - an inhomogeneity of the geometry of space-time in the vicinity of massive objects.

 

i think considering like this is better:
if velocity of light varies with position without any curvature in space-time,we can say that its velocity varies.if light's velocity varies with position because space-time itself is curved,then light is not changing its velocity at all.

 

Two Masses Attract O><O

I'm new here a so really I have no bias of the individuals involved.

I do not understand the maths-- tensors and differential geometries etc -- tho I have owned for several years and read and Micho Kaku's "Hyperspace" book and referenced it over the years.

Based on my my little knowledge I lean toward Markus's attitude and presentation of GR and spacetime as being the most correct.

So is spacetime a something-- fermionics or bosonic --- that occupies a space?

I understand the graviton is the needed boson to quantify gravitational spacetime i.e. make the geometry a something--- bosonic force --- that occupies space.

To say mass curves space or spacetime means space or spacetime is a somethingness--- ergo a boson or fermion --to be curved.

Here above, is it even fair for me to state it as a 'gravitational spacetime', or should it just be spacetime?

When I read Lee Smolins 3 Roads to Quantum Gravity, he said people didn't understand that Eienstein was referring only to geometry not a somethingness force, yet Smolins was working on just that with LQG, showing that space or spacetime is this grainy/foamy somethingness, that occupies space, ergo will affect EMRadiational photons( occupied space ) as the move through the occupied space.

r6

 

Only because I wanted to draw your attention to Einstein's reference to space. After you said:

There's space all right. Inhomogeneous space. And because of that, the speed of light is not constant, and because of that, when light moves through space, it curves. Just like Einstein said. Light doesn't move through spacetime because spacetime is static. It models space at all times, like a block of movie frames.

He was precise enough when he said empty space in its physical relation is neither homogeneous nor isotropic.

Yes. But look again at the plot of gravitational potential. It's a fair depiction of Riemann curvature, and you can understand that we can make a plot like this with those equatorial light clocks, which are like that parallel mirror gif. OK? OK, zoom in on a sloping portion. Zoom right in and there's no detectable curvature. Curved spacetime is synonymous with Riemann curvature which equates to the tidal force. You can't detect this in the room you're in, but you can detect a falling brick. That brick doesn't fall down because the spacetime in the room you're in is curved. It falls down because the space in the room is inhomogeneous. Your brick falls down because of the wave nature of matter and because the speed of light near the floor is lower than near the ceiling. Yes you need the Riemann curvature to get the plot off the flat and level, but it's just a plot of your measurements. The brick doesn't fall down because your measurements exhibit a curvature. Your measurements exhibit a curvature because space is inhomogeneous. The brick falls down because space is inhomogeneous. The map is not the territory.

Einstein didn't talk nonsense. Go and look at the parallel-mirror gif and try asserting that those two light pulses are moving at the same speed. Then go and take a look at how we define the second. The NIST caesium clock uses the hyperfine transition and microwaves - light in the wider sense. There’s a peak frequency which is found and measured by the detector. But note that frequency is measured in Hertz, which is defined as cycles per second, and the second isn't defined yet. So you can forget frequency. Then you appreciate that the detectors effectively count incoming microwave peaks. When they get to 9,192,631,770, that's a second. Then we use it to define the metre as "the distance travelled by light in vacuum in 1⁄299,792,458 of a second". Local measurement always yields 299,792,458 m/s because we use the motion of light to define the second and the metre, which we then use to measure the speed of light.

I'm sorry Markus, but that's confusing cause and effect. Your curved spacetime is a plot. Light does not move through it. It moves through space. Inhomogeneous space.

Au contraire Markus. If you look at the patent scientific and avoid dismissing what Einstein said, then none of those misconceptions would arise. Like I said to OnlyMe, think for yourself.

 

No you don't. Physics is a quantitative science, so in order to tell that a theory is consistent with hard evidence you need to be able to do calculations. You also need to be mathematically literate in order to be able to tell that certain mathematics actually has something to do with what the text surrounding it is saying. I've never seen you show any commitment to that whatsoever. Instead you seem to keep trying to change the rules. Whenever it becomes an issue whether an idea actually works, you always fall back on something along the lines of "read what Einstein said". Yeah. We did that, and we either disagreed with it or we thought you were misrepresenting it. Then what are you going to do?

General relativity has a mathematical foundation in terms of Riemannain geometry set in the context of 3+1 dimensional spacetime. That is where its quantitative, testable predictions are derived from. Because you don't know that mathematical foundation, and it is hardly a secret that you don't, you are not qualified to discuss what is and isn't a valid interpretation of it. You can certainly read something and think it sounds interesting, and ask if the idea is plausible or correct or not. But if an actual physicist says "no, that doesn't make sense" or "equation 3 doesn't follow from equation 2" or "no, the metric and the stress-energy tensor are not the same thing", then how in the world do you expect to be able to argue with that on rational or scientific grounds?

Seriously, in all the years you've been posting here, I've never seen you give a good response to this.

 

Wrong again. Riemann curvature does not directly equate to a tidal force. Take for example the Lagrange point between earth and moon - there are no net tidal forces acting on a body placed there, but space-time is still curved at that point, i.e. the Riemann curvature tensor does not vanish even though there are no tidal forces. The object which measures tidal forces for movement along geodesics is the Weyl tensor, which is not the same thing as Riemann curvature. It can be computed from the Riemann tensor and its contractions, though.

It should be noted that there are various different notions of curvature in GR, namely Riemann curvature, Ricci curvature, scalar curvature, sectional curvature and Weyl curvature. These are not all the same.

 

Hey Markus, earlier I mentioned an interesting result; that the force was perpendicular to the relativistic rod.

Any examples of such interesting results in GR?