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## Newton vs Einstein

Pervect: Thanks or the PPN link posted above.

I never saw all that nor discussions in a link within that article:

 A table comparing PPN parameters for 23 theories of gravity can be found in Alternatives to general relativity#PPN parameters for a range of theories.
http://en.wikipedia.org/wiki/Alterna...ral_relativity

 What is it about undergoing this labor intensive process of tensor calculus that allows is to calibrate the GPS and precession of Mercury that good old classical physics-special relativity can't handle?
So am I correct in assuming Diracpool's question now becomes 'why one approach works when 23 others aren't quite so good???
If so, we are back to the questions that have been discussed, but not answered, in these forums previously: Why does ANY of our man made math seem to describe the world around us? Why does some math work but not others? The only possible answer I have seen so far that registered with me is the possibility that
if there is a multiverse and maybe our 'other' math would apply there.

What I still find incredible is that 'Einstein intuition' seems to have allowed him to pick an 'off the shelf' math that is SO close overall and perhaps virtually perfect on large scales. And Diracpool may be asking in the future about a more accurate theory of 'quantum gravity'.....

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In post #2, Dalespam posted:

by DiracPool
 Mathematically the curving of this spacetime and the geodesics that arise from it are found through the continuous redefinition of the local coordinate axis due to the local mass energy density of the system in question. ...... Do I have this right?
Dalespam:
 Yes, I would say that is right.

Ok, so now I am sorry I did not ask what that 'redfinition' meant; I attributed it to my lack of understanding of mathematical details and interpretations.....

Has this do do with the characteristics of a smooth Riemannian manifold or metric space??...a metric space with geometric interpretations?

edit: Diracpool: "..the neccessity of a Geometrical approach over a Mechanics approach to address the motion of bodies in a gravitational field."

If I understand what you are positing, I'd reply "I don't think GR is the final answer, it's the best one we have. It does not take us back to the big bang, nor to the 'singularity' within a black hole....so we need a
better theory...like quantum gravity."

 What I still find incredible is that 'Einstein intuition' seems to have allowed him to pick an 'off the shelf' math that is SO close overall and perhaps virtually perfect on large scales.
Yes, it is amazing, and so iconoclastic for the time. Really, who would think not only to adopt a putatively unrelated "fringe" geometry maths, but to be so confident in its utility as to have the faith to stick with its development for upwards of a decade. And apparently get it so right that we are still "agog" over it, as evidenced right here. I still think that there's a more parsimonious model for gravity that will usurp Einsteins one day, and that will probably be the one that unifies the maths with QM. However, GR will always be notable if for no other reason than how the complexity of the model gave such accurate results. One could argue that there's beauty in complexity as well as parsimony.

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 Quote by DiracPool I still think that there's a more parsimonious model for gravity that will usurp Einsteins one day
I doubt that very much. GR has only two free parameters, the cosmological constant and the Newtonian gravitational constant. It is hard to see how any other model will get more accurate results with fewer. At least, all of the current competitive theories that have not already been falsified have more.

 Quote by DiracPool However, GR will always be notable if for no other reason than how the complexity of the model gave such accurate results. One could argue that there's beauty in complexity as well as parsimony.
I think you misunderstand what makes a model simple scientifically. It isn't the mathematical notation, but the number of free parameters in the theory. GR has only two, so it is a very simple and parsimonious model. That is precisely why it dominates over other current competitive theories.

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 Quote by DiracPool I still think that there's a more parsimonious model for gravity that will usurp Einsteins one day, and that will probably be the one that unifies the maths with QM. However, GR will always be notable if for no other reason than how the complexity of the model gave such accurate results. One could argue that there's beauty in complexity as well as parsimony.
Pretty much any quantum replacement theory that replaces GR is going to have to wind up looking very similar in the classical limit.

With the possible exception of redefining our notion of distance (and time) there isn't any way to escape the fact that the geometry of space-time is curved. Consider gravitational time dilation, for instance. There isn't any sort of "force" that's going to make a clock at a higher altitude run faster than one at a lower altitude. Something more fundamental is at work here, something that affects many different sort of clocks all in the same way.

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 Quote by Mentz114 The Good Old Newtonian Equations do not predict precessing ellipsoidal orbits, and that's what we got. So something else is needed to account for observations in our back yard. With the good algebraic software and powerful PCs available today, it is not so labour intensive any more.
"People today need "good algebraic software and powerful PCs" to do that? I remember doing the "precession of Mercury" example in a first-year maths degree course on dynamics and SR, with nothing more than pencil and paper.

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 Quote by AlephZero "People today need "good algebraic software and powerful PCs" to do that? I remember doing the "precession of Mercury" example in a first-year maths degree course on dynamics and SR, with nothing more than pencil and paper.
If you make the appropriate approximations, it's not too bad. Calculating the Chrsitoffel symbols would be more work than I cared to do by hand (you could always look them up I suppose). The real need for symbolic algebra shows up when you try to calculating the Riemann / Ricci tensors et al from the metric.

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Here is another distinction between Newton and Einstein I came across reading my notes on a related subject:

 Analogously to Newtonian mechanics, the four-momentum of a system is the sum of the four-momenta of its constituent particles, and the four-momentum of the system is conserved across any interaction, including particle annihilation and creation interactions. This means that a system's energy (timelike component of four-momentum), momentum (spacelike component of four-momentum), and mass ("length" of four-momentum) are also conserved and you get one conservation law which unifies three separate conservation laws from classical mechanics. To me it is one of the most elegant and compelling facets of relativity.
from pervect.... [Be careful what you say, I may quote you!!] [LOL]

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And another I just stumbled upon:

 ....the Austrian physicists Josef Lense and Hans Thirring,... predicted that the rotation of a massive object would distort the spacetime metric, making the orbit of a nearby test particle precess. This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation. The Lense–Thirring effect is very small—about one part in a few trillion.

http://en.wikipedia.org/wiki/Frame-dragging

Also the rotating massive object in free fall does not following the same geodesic as when non spinning.

 This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation.
Ah, thar ya go Naty, 25 posts and finally what I was looking for. A good quote is worth a thousand pictures. And you got quite a few quotes in your quiver I see.

I wonder if Einstein knew this when he set about the problem of calculating the precession of Mercury with his new field equations. Or if he just sort of blindly tried his new model with the unexplained anamolies of the day to see if it worked. Anyone know?

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 Quote by DiracPool Yes, it is amazing, and so iconoclastic for the time. Really, who would think not only to adopt a putatively unrelated "fringe" geometry maths, but to be so confident in its utility as to have the faith to stick with its development for upwards of a decade. And apparently get it so right that we are still "agog" over it, as evidenced right here. I still think that there's a more parsimonious model for gravity that will usurp Einsteins one day, and that will probably be the one that unifies the maths with QM. However, GR will always be notable if for no other reason than how the complexity of the model gave such accurate results. One could argue that there's beauty in complexity as well as parsimony.
This is possible, but I think that it is likely to be wrong for at least two reasons.

First, since they move us further from everyday experience, new theories tend have greater technical complexity and/or a greater level of abstractness than the theories that they "replace".

Secondly, current non-relativistic quantum mechanics is already more complex than general relativity.
 Quote by George Jones In my opinion, students could find physics courses in general relativity easier than courses in quantum mechanics. I think that students become more familiar with quantum mechanics because they spend more time studying it. For example, when I was a student, I: saw bits of special relativity stuck here and there into a few courses; did not have the opportunity to take any lecture courses in general relativity; was required to take three semesters of quantum mechanics as an undergrad and two semesters of advanced quantum mechanics as a grad student; was required to take two semesters of linear algebra, which gives the flavour of much of the mathematics of quantum mechanics; was not required to take any maths courses that give the flavour of the mathematics used in general relativity. Because of the importance and widespread applicability of quantum mechanics, my programme offered much more opportunity to learn quantum mechanics than to learn relativity. If physics students spent as much time studying general relativity and its mathematical background (say 4 or 5 semesters) as they spend studying quantum mechanics and its mathematical background, then general relativity would be understood by possibly millions of people. I understand why students spend much less time studying relativity than they spend studying quantum theory, and I am not necessarily saying that students should spend more time studying relativity (see the post above by Haelfix), but I do think that this time difference is a big part of the reason that general relativity still has a bit of a reputation. Fortunately, there are many more good technical books on general relativity (pedagogical, advanced, physical, mathematical, etc.) available now than were available 25 years ago.

Quote by George Jones
Time to put a myth to bed.
 Quote by f95toli *GR is very complicated if you do it "properly" and it is very unlikely that you will get the mathematical background as part of you undergrad math courses.
 Quote by D H *General relativity. As others have noted, the math is a bit on the advanced side even for the typical senior physics major.
The mathematics of non-relativistic quantum mechanics is, in my opinion, more difficult than the mathematics of general relativity. Students acquire more facility with the mathematics of quantum mechanics because they spend more time studying it.

At the level of mathematics taught by physicists, the mathematics of non-relativisitic quantum mechanics is somewhat more difficult than the mathematics of general relativity. Typically, an undergrad physics major is introduced to quantum mechanics in a Modern Physics course, and then takes two more semesters of quantum mechanics. In a one=semester general relativity course, the techniques and mathematics of general relativity are presented at light speed, and this perpetuates the myth that the mathematics is difficult. If the techniques and mathematics of general relativity were spread out over 2+ semesters, I don't think that things would seem nearly so difficult.

At the level of honest mathematics, functional analysis, the mathematics of non-relativisitic quantum mechanics is substantially more difficult than the differential geometry used in general relativity. For example, if operators $A$ and $B$ satisfy the canonical commutation relation $\left[ A , B \right] = i \hbar$, then at least one of $A$ and $B$ must be unbounded. Say it is $A$. Then, by the Hellinger-Toeplitz theorem, if $A$ is self-adjoint, the domain of physical observable $A$ cannot be all of Hilbert space! Also, the spectral decomposition for $A$ will be given by the the spectral theorem for unbounded self-adjoint operators. It would be crazy, if not impossible, to teach quantum mechanics this way!

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 Ah, thar ya go Naty, 25 posts and finally what I was looking for. A good quote is worth a thousand pictures. And you got quite a few quotes in your quiver I see.
At my age I can't remember all the stuff I'd like! When I find an explanation that creates an 'aha moment', into my notes she goes!!

 Fortunately, there are many more good technical books on general relativity (pedagogical, advanced, physical, mathematical, etc.) available now than were available 25 years ago.

There may be a few good books out there, but try to find some video tutor-age on it. I've done quite a bit of searching on the web, and have found only two sources that go into any detail, and one of those two sources isn't even a focused GR treatment, its a treatment on tensor analysis only. For those who are interested, the GR treatment is Lenny Susskinds course on the Stanford channel, and the other is the guy from digital-university.org. If anybody knows of any other, please let me know.

So we have two sources to choose from, and even these aren't very accessible to someone with a fairly decent math background watching them cold.

Now take QM, as George points out rightly. Here we have literally hundreds of video tutor treatments by many dozens of amateur and not so amateur presenters. So I think the proof is in the pudding here, and it's not plum pudding, I mind you.