Exploring Curved Space and Gravity: Understanding the Geodesic Line in Spacetime

[pervect]

The simple answer is the one Jesse gave. Various non-idealities such as "mascons" make the actual orbital path of a body non-ellilptical as well as non-parabolic, even ignoring air resistance. (Mascons, aka mass concentrations, are due to the fact that the Earth is not an ideal homogeneous sphere, but contains areas of higher and lower density, which slightly but measurably affect its gravitational field. They are present on both the Earth and moon, though mascons were first observed, IIRC, when trying to calculate landing orbits for the apollo program.)

The OP in this thread (Ratzinger) already knew that objects fall in parabolic paths in the low-velocity, low height approximation, the only one who appears to be really confused about this point is RandallB.

Unfortunately, I don't know of any way to un-confuse poor Randall (I and various other people have tried without success).

Jesse is definitely not "distorting the truth with handwaving" as RandallB claims, it is just RandallB "not getting it".

 

[RandallB]

The simple answer is the one Jesse gave. Various non-idealities such as "mascons" make the actual orbital path of a body non-ellilptical as well as non-parabolic, even ignoring air resistance.
(Ratzinger) already knew that objects fall in parabolic paths in the low-velocity, low height approximation, the only one who appears to be really confused about this point is RandallB.
it is just RandallB "not getting it".

NOT you too.
Your solution is to add yet another epicycle ("mascons") onto the epicycles already loaded up on this issue.
Of course the origin question understood that parabolic are observed! The question was how do you explain the parabolic curves – I assume Ratzinger was smart enough to know they should only occur with an escape velocity – hence the confusion as to how they come to be parabolic again at these low speeds.
With enough epicycles piled on here I assume you’ll eventually be able to convince yourself to go back to Ptolemy’s solution!

Why duck the real solution when is hiding right out in plain sight. Go back to basic physics 101 with fiction and air resistance.

Or do some reading on the Newton vs. Hook debates; Newton was long embarrassed by Hooke besting him in applying air resistance to the idea of imagining things falling to the center of the earth.
In this case you don’t even need to throw the ball just drop it leave gravity in place and assume you have a clear path to the center of the Earth solve for without air resistance and then with stationary air (not rotating with the point dropping the ball) resistance. If you cannot draw a picture for yourself find the one by Hooke, he solved this one over three hundred years ago!

This is to simple and obvious to be impressed by someone willing to shoot the messenger as “not getting it”.

Newton and GR work just fine at these levels to predict NON-parabolic elliptical paths. If you really don’t get that I’ll let keep it.

 

[JesseM]

Well at least you admit the real path is an ellipse

I think the real path is only an ellipse if you use Newtonian gravity, which is just as much of an approximation as the constant G-field is. We know that GR predicts Mercury's orbit is not a perfect elipse--the perihelion changes slightly with each orbit--so I would assume that this effect does not just vanish abruptly at some point, presumably if you calculated the exact path of a tossed ball which was able to travel straight through the earth, its perihelion would precess slightly too. Of course the difference from the Newtonian prediction and the GR prediction might be only a nanometer with each orbit or something, but the difference between the constant-G prediction of a ball's path between the time it is tossed and the time it hits the ground again would probably also differ from the decreasing-G prediction by only a microscopic amount (do you admit, by the way, that your earlier statement 'If you take air resistance out of it you will completely change the shape of the curve' was incorrect?)

The original question in post one was for clarification on if and how Newton’s or GR etc caused parabolic curves at low speeds tossing a ball on earth.

The original poster affirmed my guess that his question was about the conceptual question of how GR would explain curved paths in Earth gravity, therefore any answer that does not involve GR would be unhelpful.

The simple answer is - they don’t – air resistance does change the ellipse into a parabolic (or if we go back to the sphere a spiral to the center).

Would air resistance make it parabolic, even in theory? Not if the ball was tossed high enough that it would reach terminal velocity...what equation for air resistance as a function of velocity are you assuming here? It's possible that the idealized equation for air resistance at low velocities would transform it into an exact parabola, but I haven't seen this calculation before; in any case, any simple equation for air resistance as a function of velocity is probably just going to be an approximation as well.

We don’t get to experience them being tossed about in a vacuum where they would not move in a parabolic.

Again, if you toss a ball at ordinary velocities air resistance has virtually no effect on the path. That classroom demonstration where you toss a ball and it moves along a parabola projected on the wall would work just fine in a vacuum--do you disagree?

It does little good to cut out the top of an ellipse and call it a parabola. Cut out a smaller section in a demonstration and it looks “almost exactly like” a section of a circle – so let's call it that? Just because it’s approximately right doesn’t make it right.

Except there would be no mathematical justification for saying it looked like a circle, that would just be a matter of innacurate eyeballing. On the other hand, you can show show that in the limit as ratio between the height the ball is tossed above the surface and the radius of the planet approaches zero (ie the ball is tossed only a small height compared to the planet's radius), the difference between the parabolic path predicted by the constant-G assumption and the small section of the ellipse predicted by the decreasing-G assumption should approach zero. This is exactly analogous to how the difference between GR's prediction and Newtonian gravity's prediction approaches zero in the limit as mass and velocities approach zero; all good approximations should be justifiable in terms of such limits.

 

[RandallB]

I think the real path is only an ellipse if you use Newtonian gravity ... (do you admit, by the way, that your earlier statement 'If you take air resistance out of it you will completely change the shape of the curve' was incorrect?) The original poster affirmed my guess that his question was about the conceptual question of how GR would explain curved paths in Earth gravity, therefore any answer that does not involve GR would be unhelpful.

Wrong he confirmed the question was about how GR explained “the curve” that curve in question being parabolic! And as I said before IT DOES NOT.

Would air resistance make it parabolic, even in theory? Not if the ball was tossed high enough that it would reach terminal velocity...what equation for air resistance as a function of velocity are you assuming here? Again, if you toss a ball at ordinary velocities air resistance has virtually no effect on the path. That classroom demonstration where you toss a ball and it moves along a parabola projected on the wall would work just fine in a vacuum--do you disagree?

What on Earth are you talking about (small pun) how high you toss or drop a ball for a non rotating Earth has nothing to do with any kind of curve! It’s the horizontal component of initial speed that will give a curve and the horizontal air resistance that gives the parabolic. Staying in just the vertical is only going to give a straight line. And sure you need me to say it twice? IN A VACUUM IT’S ELLIPTIC not parabolic!

Except there would be no mathematical justification for saying it looked like a circle, that would just be a matter of innacurate eyeballing.

Exactly, just as there is no justification for inaccurately eyeballing a ellipse and calling it a parabolic curve!
I don’t know why you and pervect have not figured out you’ve conceptually blown this one badly – question is when you do figure out the concept correctly, are you willing to admit it and explain it here.
I can recommend as I did before find some old book that includes the diagram by Robert Hooke, who helped Newton get it right before he wrote the Principia. Not that he was any nicer about it than you guys, he was really rather abusive in embarrassing Newton in front of the rest of the Royal Society for some time over it.
It shouldn’t be hard to find, it’s famous. He shows three curves in it:
-One marked A,B,C,D,A for the circle an object follows that is supported by the structure of earth.
-A second marked A,F,G,H,A for the elliptical path that would be followed when all of Earth's structure is removed but gravity remained.
-And a third marked A,I,K,L,M,N,O,P,C for when only air and it’s resistance was replaced for the entire structure of earth.
It’s this third one that can be reduced to parabola if you redo the coordinates for a flat Earth - that will not happen in the non-air case.

So (even with constant g) don’t expect me to buy your argument that GR or Newton will produce a parabolic except in the case that they both agree on, escape v.

Just like I don’t believe pervect’s claim he can apply a nonlinear spring to mercury so that the Newtonian will duplicate the precession of GR – when he hasn’t even tried to produce the function of such a non-liner spring in the other thread.

So if you can match the diagram by Hooke and explain him wrong in his explanation of air resistance I’d like to see the detail - you’d be the first since Dec of 1679! – you’ll be famous.

 

[JesseM]

Wrong he confirmed the question was about how GR explained “the curve” that curve in question being parabolic! And as I said before IT DOES NOT.

Not exactly, but approximately. Just like GR doesn't say curves are exactly elliptical, although they are approximately so. If he had asked how GR explains elliptical paths, would you just thunder that GR "DOES NOT" predict such paths, or would you understand that he was speaking approximately?

What on Earth are you talking about (small pun) how high you toss or drop a ball for a non rotating Earth has nothing to do with any kind of curve!

Do you understand the concept of approximations based on limits in physics? If so, do you disagree that in the limit as the height the ball is tossed becomes very small compared to the radius of the planet, the path of the ball (in the absence of air resistance and other complicating factors) will approach a perfect match to the parabolic curve predicted by the constant-G assumption? Please answer this question yes or no.

It’s the horizontal component of initial speed that will give a curve and the horizontal air resistance that gives the parabolic.

Again, what mathematical calculation are you using to justify this? What specific equation are you assuming for air resistance, and can you show rigourously that the combined effects of gravity and air resistance produce a parabola? I have never seen such a claim before, it's possible it's correct but if you don't have some math to back it up it seems unfounded.

Secondly, you didn't answer my earlier question--do you agree that even in the absence of air resistance, the path of a tossed ball (with both horizontal and vertical velocity) will be very very close to the perfect parabola predicted by the constant-G assumption? Do you agree, for example, that the classroom demonstrations I linked to earlier would work just fine in a vacuum, and that the precise elliptical path predicted by Newtonian gravity would not depart from the parabolic path predicted by the constant-G assumption by more than a tiny amount? Again, please answer this question yes or no, it's not meant to be rhetorical.

Staying in just the vertical is only going to give a straight line.

The path will just be a straight line through space, but if you graph position vs. time it will be very close to a parabola again. And if you're just talking about the path through space alone, then the path won't look like an ellipse either when the ball's velocity has no horizontal component.

And sure you need me to say it twice? IN A VACUUM IT’S ELLIPTIC not parabolic!

Not exactly, according to GR--again, think of the precession of the perihelion of Mercury's orbit, showing that the orbit is not a perfect ellipse. But the ellipse is a valid approximation because in the Newtonian limit the orbit predicted by GR becomes arbitrarily close to an ellipse. Similarly, the parabola is a valid approximation because in the limit as the height of the ball gets arbitrarily small compared to the radius of the planet, the elliptical path predicted by Newtonian gravity becomes arbitrarily close to the parabolic path predicted by the constant-G assumption. Do you disagree with either of these statements about limits? Please answer yes or no.

Exactly, just as there is no justification for inaccurately eyeballing a ellipse and calling it a parabolic curve!

If you can show that in a certain limit the difference between the parabolic curve and the elliptical curve approaches zero, then it is reasonable to use the parabolic curve in situations where you are close to that limit. Do you think there is ever a rigorous justification for using approximations in physics? For example, do you think it is ever justified to use Newtonian mechanics when we know that these predictions differ slightly from those of GR (for example, GR will say that a planet's orbit is not precisely elliptical), and that GR's predictions are the more accurate ones? If you do think that the Newtonian approximation is all right, then you're being hypocritical in your attack on the constant-G approximation; if you don't agree that limits can be used to rigorously justify the use of approximations in any case, then you are disagreeing with pretty much the whole physics community on this one.

I don’t know why you and pervect have not figured out you’ve conceptually blown this one badly

..and pmb_phy, and every textbook on classical mechanics have "blown it" apparently, according to you. The use of approximations justified by mathematical limits is routine in physics, and the parabolic path approximation is a perfectly typical example of this.

I can recommend as I did before find some old book that includes the diagram by Robert Hooke, who helped Newton get it right before he wrote the Principia. Not that he was any nicer about it than you guys, he was really rather abusive in embarrassing Newton in front of the rest of the Royal Society for some time over it.
It shouldn’t be hard to find, it’s famous. He shows three curves in it:
-One marked A,B,C,D,A for the circle an object follows that is supported by the structure of earth.
-A second marked A,F,G,H,A for the elliptical path that would be followed when all of Earth's structure is removed but gravity remained.
-And a third marked A,I,K,L,M,N,O,P,C for when only air and it’s resistance was replaced for the entire structure of earth.

OK, so presumably this is the basis for your claim that air resistance produces a parabola. If you're not sure what assumption Hooke made about the equation for air resistance as a function of velocity (presumably he used an equation which doesn't take into account the phenomenon of terminal velocity, for example), can you at least tell me the name of the book so I can look this up myself?

It’s this third one that can be reduced to parabola if you redo the coordinates for a flat Earth - that will not happen in the non-air case.
So (even with constant g) don’t expect me to buy your argument that GR or Newton will produce a parabolic except in the case that they both agree on, escape v.

I don't say that Newton or GR produce a parabolic exactly, just like GR doesn't produce elliptical orbits exactly. Once again, are you completely unfamiliar with the concept of approximations based on limits?

So if you can match the diagram by Hooke and explain him wrong in his explanation of air resistance I’d like to see the detail - you’d be the first since Dec of 1679! – you’ll be famous.

Sigh. Where have I ever said anything about Hooke being wrong? I simply said that I had never seen anyone claim that air resistance would produce an exact parabola, and asked you for more detail. I did say that this claim would obviously be wrong if you used a totally accurate equation for air resistance that takes into account terminal velocity, but presumably Hooke was using an approximation of some sort, if indeed your memory of what he said is accurate.

In any case, my claim was never about what the exact Newtonian prediction would be, it was always about an approximation based on what Newtonian mechanics predicts in the limit as the height that the ball is tossed becomes very small compared to the radius of the planet. It is you who is disagreeing with the entire physics community in claiming that the whole concept of limits based on approximations is unjustified, or that it is no more rigorous than just eyeballing a path and saying it looks like a certain curve.

 

[masudr]

For God's sake. Everyone knows that under decent, reasonable approximations the trajectory for an ordinary sized ball tossed with an ordinary human-sized toss in the Earth's gravitational field will result in a parabolic path.

Now, Newton's theory explains this via a constant force, and force being the first time-derivative of momentum. The OP's questions was how does GR explain this, with all it's machinery of spacetime and geodesics.

The OP did not ask, "what are the differences between the two approaches," since he/she (as well as everyone else) knows these differences will not be appreciable, and since we have already made very reasonable assumptions in claiming that Newton's theory predicts parabolic trajectories, we can assume these differences are negligible.

The OP wanted to know how we could arrive at the answer of parabola straight from GR machinery.

EDIT: so my point was, stop confusing the issue! Incidentally, I think someone answered the OP question way back in page 1...

 

[Ratzinger]

The OP did not ask, "what are the differences between the two approaches," since he/she...

I'm male, even though I think posting under a female name would give more replies. So if you like, keep thinking I'm a twenty years old brazilian girl gotting interested in GR at the Newtonian limit and struggling with the conceptual understanding.

I think someone answered the OP question way back in page 1...

I think post no. 2 cleared me up already. Also found this link http://astro.isi.edu/notes/gr.pdf . It's called GR for the faint of heart, seven pages long, page five handles my original question.

 

[masudr]

I'm male...

I just always thought it could (not necessarily that it would) be quite offensive if I assumed someone was a certain sex. It's certainly true that those with names that imply they are female get more attention on this forum.

 

[RandallB]

Please answer this question yes or no.

How many times do I need to say NO, are you not reading?
Based on you’re quick reply you’re certainly not thinking.

GR reduces to the Newtonian in our real world yet you the micro almost impossible to detect the flaw in Mercury’s orbit as justification for making irrational approximations to convert ecliptics into parabolas - I don’t believe I’m seeing from someone of science.
I was giving you Hooke as a means to help you see your problem with just a little common sense. But you don’t seem to have the time to look or think.

You need help finding a book ask a librarian,
As to common sense get face to face with a good sensible instructor maybe you can work it out.

This is too simple and basic, to fuss over here any more.

 

[JesseM]

How many times do I need to say NO, are you not reading?

OK, you were responding to this:

Do you understand the concept of approximations based on limits in physics? If so, do you disagree that in the limit as the height the ball is tossed becomes very small compared to the radius of the planet, the path of the ball (in the absence of air resistance and other complicating factors) will approach a perfect match to the parabolic curve predicted by the constant-G assumption?

So are you saying "no" you don't understand the concept of approximations based on limits, or don't think such approximations have any place in physics? Or are you saying "no" you don't agree that in the limit as the height of the ball becomes very small compared to the radius of the planet, its path approaches that of a parabola? If the latter, would you disagree that in the limit as the height of the ball becomes small compared to the planet's radius, the difference in the gravitational force between the point where the ball is tossed and the maximum height it reaches will approach zero? Would you disagree that in the idealized case where the difference in the gravitational force between different heights is zero (ie constant-G), the ball's path will be a perfect parabola in a vacuum? If you disagree with either of these statements, I'd be quite happy to prove them; if you agree with both but don't see how they imply that the path of a tossed ball will approach a parabola in the limit as the difference between the ground and the maximum height becomes arbitrarily small compared to the radius of the planet, then you really need to think about it a little more, it's pretty trivial.

If you want to see some outside confirmation, look at the section titled "low energy trajectories" of the wikipedia article on Orbital Equations:

If the central body is the Earth, and the energy is only slightly larger than the potential energy at the surface of the Earth, than the orbit is elliptic with eccentricity close to 1 and one end of the ellipse just beyond the center of the Earth, and the other end just above the surface. Only a small part of the ellipse is applicable.

...

The part of the ellipse above the surface can be approximated by a part of a parabola, which is obtained in a model where gravity is assumed constant. This should be distinguished from the parabolic orbit in the sense of astrodynamics, where the velocity is the escape velocity. See also trajectory.

GR reduces to the Newtonian in our real world

Not exactly, no--only in the limit. In any real example, there will be a slight difference between GR's prediction and the Newtonian prediction. Do you disagree?

yet you the micro almost impossible to detect the flaw in Mercury’s orbit as justification for making irrational approximations to convert ecliptics into parabolas - I don’t believe I’m seeing from someone of science.

The parabolic path predicted by the constant-G assumption will also differ from the elliptical path predicted by Newtonian gravity by only a "micro almost impossible to detect" amount for a situation like a ball tossed a few feet in the air in a vacuum, and in the limit as the height the ball is tossed becomes arbitrarily small compared to the radius of the planet, the difference between the between the two predictions also becomes arbitrarily small (just for the section of the path between the ball leaving the ground and hitting it again, not if you extrapolate the path through the planet). Again, this is a very obvious fact that any physicist could confirm for you, and everyone else who has chimed in on this thread has agreed it is trivially true as well; if you disagree with this, you need to do some reviewing of basic Newtonian mechanics, and possibly of how limits work in calculus.

If you like, we could also work out some numerical examples involving a ball tossed a few meters up, to show how the difference between the parabolic-path prediction and the elliptical-path prediction would be miniscule. If I calculated the equations for both paths in the case of a ball tossed a meter up, and looked at 10 different times while the ball was in the air and showed that the height at each time predicted by both equations differed by only a microscopic amount, would this lead you to reconsider your position?

I was giving you Hooke as a means to help you see your problem with just a little common sense. But you don’t seem to have the time to look or think.
You need help finding a book ask a librarian,

Hard to do if you don't tell me what the title of the book is. In any case, you obviously don't remember the actual derivation since you never respond to my request to provide Hooke's equation for air resistance, and if you "think" a bit you will see that if Hooke took into account the phenomenon of terminal velocity there's no way a falling object's path could stay parabolic for long, since an object moving at terminal velocity will have a constant speed. So if Hooke did provide a proof of what you say he did, he must have used an equation for air resistance that becomes totally inaccurate at high velocities. Anyway, I have my doubts that your memory of what Hooke proved is accurate in the first place, since you have been very vague on the details, including where you read it.

In any case, the question of whether an object's path could be made precisely parabolic by assuming some approximate equation for air resistance is irrelevant to what I was arguing, namely that the path of an object thrown at a small height is approximately parabolic, because the actual Newtonian path gets arbitrarily close to a parabola in the limit as the height becomes small.

This is too simple and basic, to fuss over here any more.

It should be, but your refusal to listen to anyone else or answer questions in detail has made this discussion go on forever. If you continue to deny the obvious while refusing to answer any of my questions or take me up on the offer to look at an actual numerical example, I think I'll just report your post under the "wrong claims" rule, since any admin with an understanding of classical mechanics will surely agree it's wrong to deny that the path of a tossed ball becomes arbitrarily close to a parabola in the limit as the height the ball is tossed becomes very small compared to the radius of the planet.

 

[RandallB]

. .

O good grief you can’t find your way in a library? You need to use more than wikipedia. Try biographies like “Isaac Newton and His Times” by Gail Christianson or others that include details on the Newton - Hooke letters of 1679. The diagram there comes from others copies of his letters, just as many other complete biographies use it as well, just find a good one. “provide Hooke's equation for air resistance” why would you think Hooke would use formulas to define his “Elleptueid” curves for these cases, this was years before Principia made them ellipses and decades before calculus was published.

As to you PROOF - I’m waiting

Pick and altitude for your peek vertical height and define the trajectory by the speed of the object perpendicular to the radius to that point.

Just to be sure we are on the same page – free falling objects like this are described by Newtonian and GR alike as conic sections (circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory).
With the key speed being Vc that of a circular orbit, at this speed there is no change in altitude (circular orbit)
Above Vc our described point is the perigee of an elliptic orbit.
Except where the speed is greater than (√2)Vc where it’s not an orbit at all but an escape hyperbolic trajectory.
And the special case of the speed being exactly (√2)Vc for an escape on a parabolic trajectory.
Describing this parabolic path in some detail; if you send objects in opposite directions from this point the limit of both these trajectories are parallel lines with the mid point line between them being the extended line of the radius to our starting point and perigee.

Now for speed less than Vc we have elliptic orbits again this time with our point being the apogee or peak altitude.
If you disagree with any of the above and cannot clear up you confusion in wikipedia let us know.


NOW for your proof; somehow at lower speeds the hyperbolic returns?
Please be complete in describing this parabolic – do use two objects going in opposite directions.
1) define the limit lines of each
2)Are these limits parallel to each other?
3)What reference line(s) can we compare them to as perpendicular or parallel too?
4)And at what speed below which do these slower speeds change from the expected elliptic orbits into your parabolic trajectory, defining some a factor of Vc will be fine.

This I got to see.

 

[JesseM]

O good grief you can’t find your way in a library?

Of course I can, but again, a library is not much use to me until you tell me what book I should be looking for, which you haven't up till now.

You need to use more than wikipedia. Try biographies like “Isaac Newton and His Times” by Gail Christianson or others that include details on the Newton - Hooke letters of 1679. The diagram there comes from others copies of his letters, just as many other complete biographies use it as well, just find a good one. “provide Hooke's equation for air resistance” why would you think Hooke would use formulas to define his “Elleptueid” curves for these cases, this was years before Principia made them ellipses and decades before calculus was published.

I'll look up the book--but if Hooke didn't calculate how air resistance would affect the path of a moving object by using a formula giving air resistance as a function of velocity, then would the assumptions he made about how air resistance affects the paths of moving objects be considered remotely accurate today?

As to you PROOF - I’m waiting

Well, the thing I offered to prove was that as the ratio between the height of the tossed ball and the radius of the Earth approaches zero, the difference in gravitational force between the point the ball was tossed and its maximum height will approach zero. Do you disagree with this? If so I can provide a proof, it's pretty simple to demonstrate (provided that in your limit, the radius of the planet does not also approach zero).

I also offered to look at a numerical example where we compared the predicted path of Newtonian gravity with the predicted path of uniform gravity, so we could check how small the difference is. Do you want me to do this? If so, please first answer my earlier question:

If I calculated the equations for both paths in the case of a ball tossed a meter up, and looked at 10 different times while the ball was in the air and showed that the height at each time predicted by both equations differed by only a microscopic amount, would this lead you to reconsider your position?

Finally, for a more general proof about when the constant-G approximation works, I came across this paper (summarized http://www.lhup.edu/~dsimanek/scenario/secrets.htm):

http://arxiv.org/abs/physics/0310049

The paper does show that you need a few extra conditions than the one I mentioned to make the limit work, though. For one, not only does the ratio between the vertical height and the planet's radius have to approach zero, but so does the ratio between the horizontal distance the ball travels and the Earth's radius; this is a fairly obvious one, but I forgot about it. They also show that it is necessary to assume the maximum curvature of the ball's path is much greater than the curvature of the Earth's surface, a condition I wasn't aware of. But if all these assumptions hold, the paper shows that the difference between the parabolic path predicted by the constant-gravity assumption and the elliptical path predicted by Newtonian gravity will approach zero in the limit.

Pick and altitude for your peek vertical height and define the trajectory by the speed of the object perpendicular to the radius to that point.
Just to be sure we are on the same page – free falling objects like this are described by Newtonian and GR alike as conic sections (circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory).

I agree that this is true in Newtonian mechanics, I don't think it's always true in GR, because the perihelion of a non-circular orbit will precess--see the animated diagram at Perihelion Advance of Mercury.

With the key speed being Vc that of a circular orbit, at this speed there is no change in altitude (circular orbit)
Above Vc our described point is the perigee of an elliptic orbit.
Except where the speed is greater than (?2)Vc where it’s not an orbit at all but an escape hyperbolic trajectory.
And the special case of the speed being exactly (?2)Vc for an escape on a parabolic trajectory.
Describing this parabolic path in some detail; if you send objects in opposite directions from this point the limit of both these trajectories are parallel lines with the mid point line between them being the extended line of the radius to our starting point and perigee.
Now for speed less than Vc we have elliptic orbits again this time with our point being the apogee or peak altitude.
If you disagree with any of the above and cannot clear up you confusion in wikipedia let us know.

Your description of the exact paths predicted by Newtonian gravity is fine.

NOW for your proof; somehow at lower speeds the hyperbolic returns?

For the millionth time, do you have no understanding of the difference between exact results and approximations based on limits? At lower speeds, the exact trajectory predicted by Newtonian gravity will always be an ellipse. However, for a ball tossed a small height above the Earth's surface, the difference between the exact piece of an ellipse predicted by Newtonian gravity and the parabola predicted by the uniform-G assumption will approach zero in the limit described in the paper I linked to above (the limit where the the height the ball is tossed becomes arbitrarily small compared to the radius of the earth, as does the horizontal distance it travels, but the maximum curvature of the path does not become arbitrarily close to the curvature of the earth). Thus the parabolic path is a perfectly good approximation even though it is never exactly correct for any finite height, just like Newtonian orbits are perfectly good approximations for GR even though they are never exactly correct for any gravitational source with finite mass/energy.

Please be complete in describing this parabolic – do use two objects going in opposite directions.

How can one use two objects going in opposite directions for a tossed ball? Do you want one ball tossed upward and another falling downward through the earth? That won't work, because the limit only works for the small section of the path that's above the earth. Do you want two falling objects moving in opposite horizontal directions from the peak of the path of a single ball, or what?

1) define the limit lines of each

What exactly is a "limit line"? Googling this phrase, and searching for it on the mathworld site, it doesn't look like it's a standard mathematical term. Are you thinking of something like the asymptotes of a hyperbola, or the tangent line to a curve at a particular point?

4)And at what speed below which do these slower speeds change from the expected elliptic orbits into your parabolic trajectory, defining some a factor of Vc will be fine.

Try thinking about what I mean when I keep talking (over and over and over) about the idea of approximations based on limits, and the fact that the path will never be exactly parabolic for any finite V below the escape velocity, it is just that the difference between the exact elliptical path and the parabolic approximation approaches zero in the limit as the height and horizontal distance become small compared to the radius of the earth, with the extra caveat mentioned in the paper above that the maximum curvature of the path does not become arbitrarily close to the curvature of the Earth in the limit.

 

[RandallB]

for a more general proof about when the constant-G approximation works, I came across this paper

http://arxiv.org/abs/physics/0310049

The paper does show that you need a few extra conditions ...
...I don't think it's always true in GR,

Are we starting to waffle here ?
Are you about to say you didn’t intend assumptions to actually be accurate?

For the millionth time, do you have no understanding of the difference between exact results and approximations based on limits? the parabolic path is a perfectly good approximation even though it is never exactly correct for any finite height, just like Newtonian orbits are perfectly good approximations for GR

I question your counting as much as I disagree with the usefulness these approximations limits or otherwise.

Just look at the web DOC you suggested – see figure 1
The diagram on the left cuts out a section of an ellipse that looks much more like a hyperbolic than a parabolic. But then the less than faithful redrawing in a “flat” frame on the right might look parabolic, so what if you can distort the view to make look like anything someone wants. You aren’t going to let that pass as real science when they already plainly agree the true path is an ellipse are you?
The point of the paper is complaining that most texts do a bad job of defining hyperbolic curves at low speeds! Are they talking about CREN??
And then inexplicably to me they proceed to interpret a parabola from the apogee of an ellipse! Parabolas don’t have an apogee just a perigee! Why don’t they use the IDENTICAL in shape “parabola” cut from the other side of the ellipse they display at perigee! Then describe a “parabolic” path displaying some kind of negative gravity as the ball is rising in altitude! Their concluding proviso that it shouldn’t apply for speeds much less than escape? They really mean speeds much less than circular – this is just a pure diversion from the reality. I consider it no more than a magician’s need to justify a misdirection even if it has fooled themselves as well.

I’ve already covered the point that a segment of an ellipse can look like part of a circle now here we have what looks like a hyperbola till they distorted it to seem like a parabola. SO WHAT, who cares that you can contort the measuring frame to make something like else. Don’t you want to deal with reality?
Or is a flat map of the world a good enough approximation to convince you that Greenland is larger than the USA and Iceland is nearly as large as Texas!
The graininess of observation (approximation) here required to fail to see elliptical paths is huge by miles more than the graininess of the mercury observations – so stop making that comparison it’s just foolish.

How can one use two objects going in opposite directions for a tossed ball? Do you want one ball tossed upward and another falling downward through the earth? ...
Do you want two falling objects moving in opposite horizontal directions from the peak of the path of a single ball, or what?

Of course what’s hard about that and why do you continue to define a tossed ball by it’s vertical direction or speed?? That is totally meaningless here. When defining your toss, I don’t care what angle – the only thing we need is its horizontal speed at apogee or perigee when vertical speed reaches zero. So just pick a horizontal speed to start with and pitch it flat to the north from the equator. To see the whole curve just toss a ball north and a second ball south same speeds what’s the problem, you can’t be worried about air resistance.

What exactly is a "limit line"?

A parabolic trajectory is on a path to reach “apogee” 180 on the opposite side of the Earth (or main body being orbited) that apogee on a straight line from our stating perigee point though the center of orbit as a major axis to whatever distance needed to reach the apogee point. Just like the elliptic path that as that orbit moves away from the major axis it cannot start to move back towards it until the tangent of the path is parallel to that major axis line at some distance from it. The point at which it does become parallel defines the mid point of the ellipse and the crossing point for the minor-axis making a perpendicular line across the major-axis to reach the point where the ball tossed the other way has also reached a parallel tangent in its path. The issue with a parabolic is it can never move toward an apogee or even back towards what would be the major axis as its tangent never reaches this line except as a limit at infinity. What are you using to define the limit of this parabolic, if it’s an asymptote what do you use as references to it? Does it ever become perpendicular to the Earth's surface? – how and when?
.
Approximations are only useful if they accurately describe true results. And I see no useful propose here for these misrepresentations.
Using approximations to convince people that GR actually predicts parabolic paths at speeds much below escape velocity or circular velocity should be considered malpractice – I don’t care if it’s you or if CERN actually did it – it’s just wrong and grossly misleading of the truth in reality.

As I said before, but not a million times yet, GR only predicts a parabola at escape velocity not at lower speeds. Galileo’s parabolas require air resistance.

 

[JesseM]

for a more general proof about when the constant-G approximation works, I came across this paper

http://arxiv.org/abs/physics/0310049

The paper does show that you need a few extra conditions ...
...I don't think it's always true in GR,

Are we starting to waffle here ?
Are you about to say you didn’t intend assumptions to actually be accurate?

Not waffling, just admitting that I was wrong about the correct definition of the limit--it's not just the limit as the height becomes small compared to the radius of the planet, but also the limit as the horizontal distance becomes small compared to the radius, and the maximum curvature of the curve does not become small compared to the curvature of the planet. In my defense, I had mainly been thinking about the simplest case where the ball is tossed vertically and it's the graph of position vs. time that you want to approach a parabola--in this case, the only limit you need to worry about is the height being small compared to the radius of the planet.

As for my second statement that "I don't think it's always true in GR" (referring to planets traveling in ellipses), if you think this is "waffling" you haven't been paying attention to my earlier posts, I've said all along that elliptical orbits would only be valid in GR as an approximation in the Newtonian limit. For example, in post #38 I said:

I think the real path is only an ellipse if you use Newtonian gravity, which is just as much of an approximation as the constant G-field is. We know that GR predicts Mercury's orbit is not a perfect elipse--the perihelion changes slightly with each orbit--so I would assume that this effect does not just vanish abruptly at some point, presumably if you calculated the exact path of a tossed ball which was able to travel straight through the earth, its perihelion would precess slightly too. Of course the difference from the Newtonian prediction and the GR prediction might be only a nanometer with each orbit or something, but the difference between the constant-G prediction of a ball's path between the time it is tossed and the time it hits the ground again would probably also differ from the decreasing-G prediction by only a microscopic amount (do you admit, by the way, that your earlier statement 'If you take air resistance out of it you will completely change the shape of the curve' was incorrect?)

Then in post #40 I emphasized this point many times:

Just like GR doesn't say curves are exactly elliptical, although they are approximately so. If he had asked how GR explains elliptical paths, would you just thunder that GR "DOES NOT" predict such paths, or would you understand that he was speaking approximately?

...

Originally Posted by RandallB
And sure you need me to say it twice? IN A VACUUM IT’S ELLIPTIC not parabolic!

Not exactly, according to GR--again, think of the precession of the perihelion of Mercury's orbit, showing that the orbit is not a perfect ellipse. But the ellipse is a valid approximation because in the Newtonian limit the orbit predicted by GR becomes arbitrarily close to an ellipse.

...

For example, do you think it is ever justified to use Newtonian mechanics when we know that these predictions differ slightly from those of GR (for example, GR will say that a planet's orbit is not precisely elliptical), and that GR's predictions are the more accurate ones?

...

I don't say that Newton or GR produce a parabolic exactly, just like GR doesn't produce elliptical orbits exactly. Once again, are you completely unfamiliar with the concept of approximations based on limits?

And in post #45 I repeated this:

GR reduces to the Newtonian in our real world

Not exactly, no--only in the limit. In any real example, there will be a slight difference between GR's prediction and the Newtonian prediction. Do you disagree?

OK, back to your post:

For the millionth time, do you have no understanding of the difference between exact results and approximations based on limits? the parabolic path is a perfectly good approximation even though it is never exactly correct for any finite height, just like Newtonian orbits are perfectly good approximations for GR

I question your counting as much as I disagree with the usefulness these approximations limits or otherwise.

Whether it is "useful" or not is not the issue here, the issue is just whether the notion of an approximation becoming arbitrarily close to accurate in a certain limit is a mathematically rigorous one, and it is. You're free to accept this but still not use these approximations because "anything less than perfect accuracy is EEEVIL" or whatever, although I'll note again that if this is your attitude it's hypocritical of you to say Newtonian mechanics is appropriate to use in any situation (even, say, planetary orbits), because we know it always differs slightly from the predictions of GR, even the difference between Newtonian predictions and GR predictions becomes arbitrarily small in certain limits. Again, I'm pretty sure that GR would say the orbits of planets are never exactly elliptical, so that'd be an example of what I'm talking about.

Just look at the web DOC you suggested – see figure 1
The diagram on the left cuts out a section of an ellipse that looks much more like a hyperbolic than a parabolic. But then the less than faithful redrawing in a “flat” frame on the right might look parabolic, so what if you can distort the view to make look like anything someone wants.

In the limit as the width and height of the region you're considering becomes arbitrarily small compared to the radius of the planet, the polar coordinates become arbitrarily close to cartesian coordinates--for example, the angle between different radial lines in the polar coordinate drawing approaches zero in the region, and the curvature of the constant-radius lines approaches zero in the region. Do you disagree?

You aren’t going to let that pass as real science when they already plainly agree the true path is an ellipse are you?

For the last time Randall, I agree the true path in Newtonian mechanics is an ellipse too, my argument (and theirs) is just that in a certain well-defined limit, the true elliptical path becomes arbitrarily close to the approximate parabolic path. Similarly, in GR the true path of an orbit is not an ellipse, but in a certain well-defined limit, the true non-elliptical path predicted by GR becomes arbitrarily close to the elliptical path predicted by Newtonian mechanics.

The point of the paper is complaining that most texts do a bad job of defining hyperbolic curves at low speeds! Are they talking about CREN??

The CERN page did not even attempt to define what the appropriate limit is where the parabolic approximation becomes valid, it just took for granted that tossing a ball on Earth is a situation where it's fine to use this approximation, and the paper would validate this assumption, even if the authors may feel it's important to make the nature of the limit explicit.

And then inexplicably to me they proceed to interpret a parabola from the apogee of an ellipse! Parabolas don’t have an apogee just a perigee!

So what? The point is that the apogee of the ellipse becomes arbitrarily close to the perigee of the approximate parabola in this limit, and they demonstrate this mathematically. There's no rule that says it must be the perigee of an ellipse which approaches the perigee of a parabola in the limit. If you disagree with their proof, please point out which step you think is incorrect.

Why don’t they use the IDENTICAL in shape “parabola” cut from the other side of the ellipse they display at perigee!

Why should they? The perigee would be somewhere deep beneath the surface of the earth, and would have no relevance to the question of whether the portion of the path immediately above the surface becomes arbitrarily close to a parabola in some limit. In any case, if mathematics proves that the apogee of an ellipse becomes arbitrarily close to the perigee of a parabola in some well-defined limit, it doesn't make any sense to "dispute" this proof by spluttering "but one is an apogee and the other is a perigee!"

Then describe a “parabolic” path displaying some kind of negative gravity as the ball is rising in altitude!

Er, what? If the initial velocity of the ball at the surface is in the upward direction, then of course it will rise (this is just as true of the elliptical path as it is of the parabolic approximation), positive gravity just says that everything must accelerate towards the Earth (as the ball is doing, since its upward velocity is decreasing as it goes up), not that every object's velocity must point towards the Earth at all moments. "Negative gravity" would be if it was accelerating upwards, ie if its velocity away from the Earth was increasing as it travelled.

Their concluding proviso that it shouldn’t apply for speeds much less than escape?

That's not an additional proviso, they just say that this proviso is equivalent to the condition on the radius of curvature of the object's path vs. the Earth that they mentioned earlier, namely $$\alpha \epsilon << 1$$.

They really mean speeds much less than circular – this is just a pure diversion from the reality. I consider it no more than a magician’s need to justify a misdirection even if it has fooled themselves as well.

Why is this a "misdirection"? As they say earlier in the paper, "The condition $$\alpha \epsilon << 1$$, then, is simply the condition that the motion is very slow compared to typical “orbital” (as opposed to “trajectory”) motions." Would you disagree that a ball tossed by human hands near the surface of the Earth will have a velocity very small compared to that of a ball orbiting the earth?

I’ve already covered the point that a segment of an ellipse can look like part of a circle now here we have what looks like a hyperbola till they distorted it to seem like a parabola. SO WHAT, who cares that you can contort the measuring frame to make something like else. Don’t you want to deal with reality?

I know of no well-defined limit where the elliptical path predicted by Newtonian gravity can be shown to become arbitrarily close to a circular path--do you? This certainly would not be true in the case of the limit they discuss in their paper, where the height and horizontal path length are very small compared to the radius of the planet, and the maximum curvature of the path is much greater than the curvature of the planet.

The graininess of observation (approximation) here required to fail to see elliptical paths is huge by miles more than the graininess of the mercury observations – so stop making that comparison it’s just foolish.

No it isn't. Again, I'd be happy to look at an actual numerical example where we consider a ball tossed with a small initial velocity, and find the difference between the exact elliptical path and the approximate parabolic path at a bunch of different points along the trajectory. I am confident we'd find that the difference is microscopic--so again, if you take me up on this offer, then please tell me in advance whether you'd reconsider your position if it turns out I'm correct that the difference would be microscopic in such an example. I would certainly reconsider my position on the usefulness of this approximation if it turned out the difference was non-microscopic (I would not reconsider the position that as the height and horizontal length of the path becomes arbitrarily small compared to the radius of the planet, the parabolic approximation becomes arbitrarily close to perfect, but if it turned out the difference was non-microscopic in the case of a ball tossed a few meters than I'd conclude that this height was not close enough to 'arbitrarily small' compared to the radius of the Earth for the approximation to be useful for typical examples).

Of course what’s hard about that and why do you continue to define a tossed ball by it’s vertical direction or speed?? That is totally meaningless here. When defining your toss, I don’t care what angle – the only thing we need is its horizontal speed at apogee or perigee when vertical speed reaches zero. So just pick a horizontal speed to start with and pitch it flat to the north from the equator. To see the whole curve just toss a ball north and a second ball south same speeds what’s the problem, you can’t be worried about air resistance.

OK, I see what you're saying--you want two balls tossed in opposite directions from their maximum height, with their vertical velocity set to be zero at the moment they're tossed. Yes, then the two balls together will give the same curve as a ball tossed upwards and horizontally from the surface with a certain velocity--but if the two situations give the same path, what difference does it make? Perhaps you're just suggesting that we start from the apogee because it's easier to calculate the equation of the elliptical path with this information?

What exactly is a "limit line"?

A parabolic trajectory is on a path to reach “apogee” 180 on the opposite side of the Earth (or main body being orbited) that apogee on a straight line from our stating perigee point though the center of orbit as a major axis to whatever distance needed to reach the apogee point. Just like the elliptic path that as that orbit moves away from the major axis it cannot start to move back towards it until the tangent of the path is parallel to that major axis line at some distance from it. The point at which it does become parallel defines the mid point of the ellipse and the crossing point for the minor-axis making a perpendicular line across the major-axis to reach the point where the ball tossed the other way has also reached a parallel tangent in its path. The issue with a parabolic is it can never move toward an apogee or even back towards what would be the major axis as its tangent never reaches this line except as a limit at infinity. What are you using to define the limit of this parabolic, if it’s an asymptote what do you use as references to it? Does it ever become perpendicular to the Earth's surface? – how and when?

You're talking about what would happen to the ball after it falls right through the surface of the earth, but in case I haven't made this clear, the parabolic approximation is only supposed to become arbitrarily close to the true elliptical path for the small subsection of the path above the surface of the earth, I completely agree that the parabolic path will begin to wildly diverge from the elliptical path if you extrapolate it through the crust and down past the center of the earth! In this case, as you say, the elliptical path will eventually turn around and start approaching the major axis again, finally hitting it at the perigee, while the parabolic path will never do so. The approximation was never meant to hold for such an extrapolated path, just for the section of the path that is actually physically meaningful for a ball tossed near the surface (since a real ball won't be able to travel right through the surface of the Earth like a neutrino).

As I said before, but not a million times yet, GR only predicts a parabola at escape velocity not at lower speeds. Galileo’s parabolas require air resistance.

And as I've said, the fact that the path of a tossed ball looks like a parabola has nothing to do with air resistance. Again, if we were to actually calculate the difference between an elliptical path and a parabolic path for a ball tossed above the surface (not continuing the paths past the surface) at ordinary velocities, I am confident that the difference will be microscopic. Are you willing to actually look at such an explicit numerical calculation and reconsider your position if it turns out I am right?

 

[RandallB]

And as I've said, the fact that the path of a tossed ball looks like a parabola has nothing to do with air resistance. Again, if we were to actually calculate the difference between an elliptical path and a parabolic path for a ball tossed above the surface (not continuing the paths past the surface) at ordinary velocities, I am confident that the difference will be microscopic.

And as I've said well within the microscopic range of a circle and a hyperbola as well

Are you willing to actually look at such an explicit numerical calculation and reconsider your position if it turns out I am right?

Sure if you can work out the numbers that translate these curve shapes to a flat surface I'll look at them. Why not include what you expect the curve shape with air resistance to be as well, if it’s not a parabola it be worth knowing what it matches up best with, ellipse at perigee, circle, ellipse at apogee, hyperbola.

 

[JesseM]

And as I've said, the fact that the path of a tossed ball looks like a parabola has nothing to do with air resistance. Again, if we were to actually calculate the difference between an elliptical path and a parabolic path for a ball tossed above the surface (not continuing the paths past the surface) at ordinary velocities, I am confident that the difference will be microscopic.

And as I've said well within the microscopic range of a circle and a hyperbola as well

Nonsense. Again, as far as I know there is no well-defined limit in which the curve of a ball tossed a small height would approach a circle or a hyperbola (again, only looking at the section of the curve above the surface, not extrapolating it further), but there is such a limit for a parabola. And if I come up with an actual numerical example, calculate the exact heights at different times using the exact elliptical solution, then calculate the heights using the parabolic approximation and find only a microscopic difference, you will not be able to give me the equation of a circle or hyperbola that fits the elliptical curve so well that there is only a microscopic difference between them.

Are you willing to actually look at such an explicit numerical calculation and reconsider your position if it turns out I am right?

Sure if you can work out the numbers that translate these curve shapes to a flat surface I'll look at them.

I didn't ask you whether you'd look at them, I asked you whether you would reconsider your position if it turns out I'm correct that in a typical numerical example there is only a microscopic difference between the exact Newtonian prediction and the parabolic approximation (and if it turns out I'm also correct that it's not possible to find a circle or a hyperbola that fits the exact solution with such accuracy). Will you or won't you reconsider your position if this is the case? And are you in fact predicting that I am wrong that this is what will happen when we look at a numerical example, or are you unwilling to commit to a definite prediction?

Why not include what you expect the curve shape with air resistance to be as well, if it’s not a parabola it be worth knowing what it matches up best with, ellipse at perigee, circle, ellipse at apogee, hyperbola.

I'm not sure I'll know how to solve the differential equations for the path when air resistance is included, but I can give it a try.