Practical calculations for gravity

[michelcolman]

Hi,

Could somebody please explain to me how you can calculate the effects of gravity in general relativity in a practical way? (Or point me to a site where these things are explained in an understandable way)

I keep trying to read articles and courses about it, but whenever it gets interesting, all the words disappear from the page and are replaced by curly d's and upside down triangles with subscripts and superscripts all around them. I could swear they are even moving about on the page ;-)

OK, I'm exaggerating a bit, I do have a fairly good background in mathematics, I even managed to get a master's degree somehow, but GR just seems to be too much about symbols and too little about genuine understanding (unlike SR, which I've got a pretty good grasp of). So my mind basically goes tilt whenever it sees a page full of GR incantations.

- How would you calculate, for example, the curvature of a ray of light when it passes the sun in Einstein's famous solar eclipse experiment?
- Or how do you explain the precession of Mercury's perihelion?
- Or for something simpler, how do you calculate the gravitational effects between two moving objects (with distances, masses, etc... depending on the reference frame)?

What exactly makes the relativistic result differ from Newtonian calculations? For example, for the curvature of light, Is it the fact that light slows down near a heavy mass (as observed from far away) and therefore curves a bit more when it passes the sun? Or is there more going on?

Can you just use Newtonian attraction and throw in some corrections for slower clocks, shorter rods, slower light and a finite speed of gravity? Or do you have to abandon Newtonian attraction completely and make a full switch to tensor calculus to get any accurate results?

I imagine adjusting Newton's laws isn't easy, since different observers don't agree on distances, speeds, how long it took gravity to go from source to destination, or the mass of the objects involved. But maybe they would still apply in the frame of an observer using his measurements and a few corrections?

If not, how exactly do you calculate the acceleration of an object or the curvature of a ray of light in a field of gravity? Any tricks or shortcuts to make the math bearable and actually understand what you're doing?

Thanks!

Michel

 

[Naty1]

The symbols in GR have so far thrown me for a loop as well...after four of five substiutitions I have no idea what I am looking at either! So I know better than to try math here!

With your math background you might find this paper helpful and not too stressful...on tensors... I have started it but not finished yet...its seems like a way to ease into GR.

http://www.lerc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

How would you calculate, for example, the curvature of a ray of light when it passes the sun in Einstein's famous solar eclipse experiment?...What exactly makes the relativistic result differ from Newtonian calculations?

I'm afraid the answers are not as simple as you and I would like: After Einstein developed special relativity and calculated the curvature of light he got the wrong answer! He did not realize at that point that spacetime curvature would have an additional curving effect...At that point no one even knew if light was curved gravity. He got lucky because GR progressed far enough that he revised his theoretical prediction in late 1915 before confirming experiments were made by Eddington...these made made Einstein famous because a competing theory predicted no light curvature... (Subsequently, the accuracy of Eddingtion's result were questioned but relativity was not.) So light curves and undergoes frequency shifting as it moves in a varying gravitational gradient.

In Wikipedia there is a decent description
http://en.wikipedia.org/wiki/General_relativity#Light_deflection_and_gravitational_time_delay
of light bending under LIGHT DEFLECTION AND GRAVITATIONAL TIME DELAY.

There are some descriptions, discussed in other threads in this forum, which suggest 1/2 the bending of light is classical and 1/2 relativistic...since the final theory predicts twice the curvature of special relativity...and that's a simple introductory way to think about it...but a more accurate way is to view the bending as incompletely described without full GR...the full effects are only properly described via GR.

 

[Vanadium 50]

It's true that GR's math is not simple - tensors, differential geometry, Christoffel symbols. However, except in a few simple cases, it's not avoidable. It's not like the people who developed GR said to themselves "well, even though we can do this with high school algebra, let's toss in some Christoffel symbols to spice things up!"

The complicated math is in there because it has to be.

 

[michelcolman]

The symbols in GR have so far thrown me for a loop as well...after four of five substiutitions I have no idea what I am looking at either! So I know better than to try math here!

I'm glad I'm not the only one :-)

With your math background you might find this paper helpful and not too stressful...on tensors... I have started it but not finished yet...its seems like a way to ease into GR.

I've just read the introduction, and it looks like this is exactly the kind of thing I was looking for! Thanks!

I'm afraid the answers are not as simple as you and I would like: After Einstein developed special relativity and calculated the curvature of light he got the wrong answer!

Great, Einstein and I have something in common then :-)

In Wikipedia there is a decent description

I used to be a huge Wikipedia fan (the articles on special relativity paradoxes like the ladder paradox, train paradox, etc... have really given me a feel for SR) but I've gotten a bit more careful after reading the article about gravitational time dilation, which has a number of serious errors in it, especially the "inside a non-rotating spherer" bit.

But I'll have a look on the page you suggested anyway.

There are some descriptions, discussed in other threads in this forum, which suggest 1/2 the bending of light is classical and 1/2 relativistic
(...) but a more accurate way is to view the bending as incompletely described without full GR...the full effects are only properly described via GR.

No shortcuts then... bummer.

Thanks for the help!

It's not like the people who developed GR said to themselves "well, even though we can do this with high school algebra, let's toss in some Christoffel symbols to spice things up!"

It does seem logical if you put it that way :-)

I was just hoping that GR was so difficult because it was trying to explain everything in one big formula, while parts of it might be explained a bit more simply. But I guess not, then...

Thanks!

Michel

 

[A.T.]

I was just hoping that GR was so difficult because it was trying to explain everything in one big formula, while parts of it might be explained a bit more simply. But I guess not, then...

Many of the simple looking formulas with nice symbols won't help you to calculate particular things. They describe differential equations which in general cannot be solved analytically. So you have use numerical integration and available solutions for some simple cases.

 

[Jonathan Scott]

- How would you calculate, for example, the curvature of a ray of light when it passes the sun in Einstein's famous solar eclipse experiment?
- Or how do you explain the precession of Mercury's perihelion?
- Or for something simpler, how do you calculate the gravitational effects between two moving objects (with distances, masses, etc... depending on the reference frame)?

You can do most things in GR using Newtonian theory with corrections, but the difficult bit is knowing what you can and cannot ignore in a given case.

The curvature of a ray of light is the easiest one. You can assume that time rate and ruler size both vary to first order by a factor $(1-Gm/rc^2)$ and the result pops out. Basically, something moving at speed $v$ has its momentum deflected by $(1+v^2/c^2)$ times the Newtonian gravitational acceleration, which for light is simply twice the Newtonian deflection.

Calculating the perihelion of Mercury is a lot trickier, in that it depends on second order accuracy in the approximation. In isotropic coordinates, the time rate can be approximated more accurately by an expression of the form $(1 - Gm/rc^2 + k (Gm/rc^2)^2)$, where the value of $k$ varies between different relativistic gravity theories, but if you use the GR theoretical value then you get the correct perihelion precession for Mercury.

Calculating the gravitational effect between moving objects is a combination of impossible and trivial. GR is so complicated that there are no known exact analytic solutions involving more than one gravitational source, so it can only be used exactly when one of the sources is a test mass. However, once you know the effect in any frame, you can trivially transform it to any other by a Lorentz transformation on the appropriate tensor.

 

[michelcolman]

You can do most things in GR using Newtonian theory with corrections, but the difficult bit is knowing what you can and cannot ignore in a given case. The curvature of a ray of light is the easiest one. (...) Calculating the perihelion of Mercury is a lot trickier (...) Calculating the gravitational effect between moving objects is a combination of impossible and trivial. GR is so complicated that there are no known exact analytic solutions involving more than one gravitational source, so it can only be used exactly when one of the sources is a test mass. However, once you know the effect in any frame, you can trivially transform it to any other by a Lorentz transformation on the appropriate tensor.

Finally I'll be able to understand the curvature of light, thank you! It will take a bit of work for me to work out the details, but it looks like it should be within my reach. Not sure if I can say the same for the perihelion of Mercury...

But if there are no GR solutions for more than one mass, how do people make cosmological models of the universe, then? How do they come up with things like, say, the amount of dark energy needed to accelerate the expansion of the infinite universe, if they can't even calculate the gravitational effect between a few masses? Are they just using Newtonian gravity?

 

[Jonathan Scott]

But if there are no GR solutions for more than one mass, how do people make cosmological models of the universe, then? How do they come up with things like, say, the amount of dark energy needed to accelerate the expansion of the infinite universe, if they can't even calculate the gravitational effect between a few masses? Are they just using Newtonian gravity?

The usual approach in GR is to assume the universe is homogeneous, which is effectively equivalent to assuming it is a single large object (of extremely small density)!

 

[michelcolman]

The usual approach in GR is to assume the universe is homogeneous, which is effectively equivalent to assuming it is a single large object (of extremely small density)!

And how would such a calculation go, then? Is there any way I could understand without deeper knowledge of GR tensor calculus? I can deal with shorter rods, slower clocks, curved space time diagrams, etc..., but am hopelessly lost as soon as strange symbols with subscripts and superscripts start appearing, if you know what I mean.
(Although I'm trying to get up to speed on that, too).

Just guessing, I imagine you should use comoving coordinates to get an isotropic view of the universe (which would be the only kind of reference frame in which it can be considered homogenous, but which doesn't correspond to any real observer) and then transform them back into a "real" reference frame (our own)? Then somehow translate curvature from comoving coordinates to our own to figure out the accelerations?

I just want to know why the expansion of the universe should slow down if it weren't for dark energy. What exactly is making stuff want to come closer together if there's the same amount of matter everywhere?

 

[CompuChip]

I was just hoping that GR was so difficult because it was trying to explain everything in one big formula, while parts of it might be explained a bit more simply. But I guess not, then...

Actually, the idea of GR is very simple: things left free to move in a gravitational field will move along special paths which are called geodesics (a generalization of straight lines in a plane). Geodesics are determined by the matter that is around (which "curves" the spacetime) and conversely, by the famous rubber-sheet analogy.

This is, if you want, also the "problem" of GR: the above statement implies that the curvature of space both influences matter distributions, whereas the matter distributions themselves influence the curvature of space. Therefore, the equation is rather simple:
(something with curvature of space, i.e. Einstein tensor) = (something with matter distribution in space, i.e. stress-energy tensor).
The problem is solving it, because of this "circular" effect: they each influence each other.

The complicated mathematics that one uses to describe GR is in a sense the easiest form, or at least: the most natural one. Differential geometry (because that's what it is) is made for describing things concerning curvature of spaces, "distances" in them, relations between different points, and so on, in the most natural generalization of our ideas and intuitions about "flat" space(time).

I don't know if this is a good analogy, but remember when you had never heard of differential equations? You could do classical mechanics then, sure, but not really well. You had formulas like: "the average velocity over some time interval is the distance over the time" and you could solve F = m a for simple cases of F (F = m g, without friction and other external potential, for example). However, to really understand the foundations and to calculate useful things (friction: F = - k v or "real" gravitation: F = - G M m / r^2, for example), it would (almost) always come down to writing down some differential equation and solving it.

 

[HallsofIvy]

"The Mathematical Theory of Relativity", written by Sir Arthur Eddington, way back in 1928, is a good introduction to general relativity and does some of these calculations.

 

[michelcolman]

"The Mathematical Theory of Relativity", written by Sir Arthur Eddington, way back in 1928, is a good introduction to general relativity and does some of these calculations.

And it's even freely downloadable from a number of acedemic sites I found with Google, terriffic!