How do relativity explain elliptical orbits of planets?

[niin]

How do relativity explain elliptical orbits of planets?

 

[D H]

You don't need to learn about relativity to understand why planets have elliptical orbits. Any central force that follows an inverse square law will result in orbits that are conic sections (circles, ellipses, parabolas, and hyperbolas). This is Newtonian mechanics, not relativistic mechanics.

It typically takes quite a bit of time before some new scientific theory that turns some branch of science upside down become accepted. Newtonian mechanics and relativity are two exceptions to this rule. Newtonian mechanics was accepted quickly in part because it explained what we already knew: planets follow elliptical orbits.

The planets do not truly follow elliptical orbits because they interact gravitationally with one another as well as with the Sun. For example, Mercury's orbit is not perfectly elliptical (it "precesses") because of the influence of Jupiter, Saturn, and all of the other planets. Newtonian mechanics could only explain a part of this precession. By the end of the 19th century, it was clear that the Newtonian explanation of the precession did not agree with the observed value. Something was missing. One reason general relativity was quickly accepted in the scientific community was that it provided that missing something.

 

[robphy]

To add to D H's post...
in the classical limit, the Schwarzschild solution in GR leads to the Newtonian explanation of planetary motion.

 

[niin]

So relativity can't explain the elliptical orbits of the planets?
Didn't relativity replace Newtonian gravity?
How can you explain elliptical orbits without gravity?

 

[ZapperZ]

So relativity can't explain the elliptical orbits of the planets?
Didn't relativity replace Newtonian gravity?
How can you explain elliptical orbits without gravity?

Did you not read, or not understand, D H's post?

"elliptical orbit" is a solution of any inverse-square central force problem. If you can use Newtonian classical law to get such a solution, then you can also get it from general relativity, because Newtonian mechanics can be derived from relativity.

BTW, do you also have issues with circular orbits? If you don't, then I don't understand why you are asking about elliptical orbits, since circular orbits is only a special case of elliptical orbits.

Zz.

 

[niin]

I read D H's post, but I don't think it answered my question.
In my mind I find it easy to picture a circular orbit. If i think about the classic diagram with a 2d surface pressed down by the sun and the Earth following a line in this depression of spacetime, it seem to result in a circular orbit. But in reality its elliptical and i just wondered why.
I don't really care about the mathematics, but more about the physical cause, if that make any sense.
Thanks for trying to help me.

 

[D H]

Your question is incorrect. (You don't need relativity to explain elliptical orbits.)

All that is needed to explain elliptical orbits (or hyperbolic orbits) is good old Newtonian mechanics. That inverse square central force systems result in conic sections falls right out of the math. Gravity is an inverse square central force in Newtonian mechanics.

 

[lbrits]

General relativity explains why orbits are NOT elliptical. See: precession of the perihelion of Mercury.

 

[niin]

How can my question be incorrect? It's my question.
Either relativity can explain elliptical orbits or it can't. Which is it?
I'm sure that you could explain it with Newtonian physics but that was not what is asked. I don't find it relevant that we don't need relativity to explain it. I want to know if relativity can explain it or not.
I'm sorry if this is all basic stuff for you guys, but i find relativity hard to understand. I'm just trying to learn a little more about this.

 

[lbrits]

I think I answered your question. But anyway, here goes: Circular orbits are solutions to the geodesic equations for objects moving in the Schwarzschild spacetime (around a star, heavy body, for example). Elliptical orbits are only approximate solutions, and the true solutions resemble precessing elliptical orbits.

 

[robphy]

Amplifying my first response above...
Suppose that Newton had not formulated his theory of gravity...
Einstein's GR with the Schwarzschild solution would have derived (in the classical limit when its so-called "relativistic-corrections" are ignored) what would have been Newton's theory of gravity and its interpretation of elliptical orbits [to explain Kepler's observations].

 

[Fredrik]

Really short answer: Yes.

Slightly longer answer: Newton's law of gravity says that the gravitational potential around a star as a function of the distance "r" to the star has the form 1/r. A little math shows that this leads to exactly elliptical orbits. General relativity says that the potential has the form 1/r + a/r²+b/r³+..., where the a, b, etc, are pretty small numbers (which can also be calculated). So GR says that the orbits are very close to being elliptical, but they aren't exactly elliptical. This prediction of GR has been confirmed by astronomers.

 

[niin]

Guys, thanks for trying to help me, but I think i may be misunderstood a little.
Descriptive math is fine, but that was not what i was looking for. I'm sure you would agree that planets don't move because we have a few equations that are good at describing reality. There must be a physical cause. Right?
I'm guessing that the cause is gravity and now i want to know how the relativity version of gravity explain elliptical orbits. Maybe, this is impossible for relativity or no one has though it up yet, but I would still like to know that.
Maybe if someone would explain the cause for a normal circular orbit in relativity and then we could try to work our way to elliptical orbits. I would appreciate any help. Thanks.

 

[shalayka]

As mentioned above, the Schwarzschild solution for a static, spherically symmetric spacetime can be approximated in the weak-field region -- that is, far away from the gravitating mass' Schwarzschild radius.

One particularly important aspect of the Schwarzschild solution is that as one moves closer to a massive body, one experiences stronger and stronger gravitational time dilation. It is this increase (or gradient, technically) in this strength of gravitational time dilation that results in greater acceleration as one approaches closer to the massive body.

Simplifying the gravitational source as a pressureless perfect fluid at rest, we consider solely energy density (in units of Joules per metre cubed). Since we're assuming a spherically symmetric body, it may be approximated as a point object thanks to Newton's shell theorem. And so we just lump the whole thing together as:

$${\it T_{\rm 00}} = \frac{E}{1^3}{\;}{\rm J^1}{\rm m}^{-3}$$

Curvature of spacetime due to a single point source can be "fudged" as:

$${\it G_{\rm 00}} = \frac{8\pi G}{c^4} {\it T_{\rm 00}}{\;} {\rm m}^{-2}$$

The Schwarzschild radius of a spherically symmetric body of energy (again, approximated as a point source), or also in terms of mass-energy:

$${\rm R_{\rm S}} = \frac{{\it G_{\rm 00}} \cdot 1^3}{4\pi} = \frac{2GM}{c^2}{\;}{\rm m}^{1}$$

Here is the formula that describes how one's rate of time diminishes. At the Schwarzschild radius it is seen that one's rate of time is zero and so velocity is practically that of light. However, we'll assume that we're far away from the Schwarzschild radius, and that one's rate of time is practically identical to that in the absence of a gravitational source (Newton didn't know that time was variable):

$$\tau = {\rm t}{\;}\sqrt{1 - \frac{\rm R_{\rm S}}{\rm r}} \approx 1{\;}{\rm s^1}$$

The derivative of the previous gravitational time dilation formula, which describes the gradient of time dilation, or how fast the rate of time changes with a change in distance, simplifies to:

$$\frac{\partial \tau}{\partial \rm r} \approx \frac{\rm R_{\rm S}}{2 {\rm r^2}}{\;}{\rm m^{-1}}$$

Which gives acceleration based on distance (the second version is Newton's):

$${\rm a} = \frac{\partial \tau}{\partial \rm r} {c^2} = {\frac{GM}{\rm r^2}}{\;}{\rm m^1 s^{-2}}$$

Which gives orbit velocity based on distance (the second version is Newton's):

$${v} = \sqrt{{\rm a} \rm r} = \sqrt{\frac{GM}{\rm r}}{\;}{\rm m^1 s^{-1}}$$

The direct answer to your questions is then: the gradient of gravitational time dilation. For another example of how time dilation is related to velocity, see special relativity. It's not quite the same principle, but you'll get the idea that one's rate of time reduces due to velocity. In general relativity it's almost the other way around... One's velocity increases because one's rate of time reduces.

 

[chronon]

I read D H's post, but I don't think it answered my question.
In my mind I find it easy to picture a circular orbit. If i think about the classic diagram with a 2d surface pressed down by the sun and the Earth following a line in this depression of spacetime, it seem to result in a circular orbit. But in reality its elliptical and i just wondered why.

I can see why you're having problems trying to think in terms of the 'marbles on a rubber sheet' model because, despite the fact that it's used so often, it's actually a very poor model of what's going on in general relativity

I don't really care about the mathematics, but more about the physical cause, if that make any sense.

I think you're likely to be disappointed. Going back to the Newtonian case, if it were possible to say 'there is an inverse square force therefore the orbits are obviously ellipses' then why did Halley have to pester Newton for a mathematical demonstration of this fact.

 

[Dale]

I'm guessing that the cause is gravity and now i want to know how the relativity version of gravity explain elliptical orbits. Maybe, this is impossible for relativity or no one has though it up yet, but I would still like to know that.

You are already aware of the explanation. GR explanation for gravitation is curved spacetime.

An orbiting body travels through the curved spacetime along a special kind of geometric path called a geodesic. Because spacetime is curved these geodesics are also curved. In fact, these geodesics are elipses in the classical limit.

 

[Fredrik]

Descriptive math is fine, but that was not what i was looking for. I'm sure you would agree that planets don't move because we have a few equations that are good at describing reality. There must be a physical cause. Right?

I would say that planets move because reality (in particular gravity) actually behaves as described by those equations. The rest is just math. I understand that you want an easy-to-visualize geometric picture of what's going on, but I don't think that's possible. That doesn't mean that GR doesn't explain elliptical orbits. It does, in exactly the way that we described.

Edit: What DaleSpam just said is probably as close as you're ever going to get to the kind of explanation that you seem to be looking for.

 

[niin]

You are already aware of the explanation. GR explanation for gravitation is curved spacetime.

An orbiting body travels through the curved spacetime along a special kind of geometric path called a geodesic. Because spacetime is curved these geodesics are also curved. In fact, these geodesics are elipses in the classical limit.

Thanks. I hope you would help me clarify some points.
So, would you say that the cause for orbits is that the planets follow geodesic? which is the shortest path between points?
Why are the geodesics ellipses?
When i try to picture it in my mind circles looks like a shorter path than ellipses, but maybe I'm wrong.

 

[Mentz114]

Hi niin,
I think your question has been answered as well as science can answer it. Elliptical orbits are a reality and are solutions of the Euler-Lagrange equations for a body in a gravitational field. This means that elliptical orbits are just as 'short' and graceful as circular ones.

M

 

[MeJennifer]

General relativity explains why orbits are NOT elliptical. See: precession of the perihelion of Mercury.

True, but neither are they under Newton's laws. It looks pretty elliptical if one if the masses is negligible but in the general case two masses in orbit do not follow an elliptical path under Newton's laws.

 

[niin]

This means that elliptical orbits are just as 'short' and graceful as circular ones.

So would you say that a comets orbit which is very elongated ellipse (i think), is just at short as a circular one from...lets say a satellite?

Edit: I wonder if this is a fair way to ask this. maybe it's not, because I'm comparing two different objects...forget this question if it is.

 

[Mentz114]

niin,

So would you say that a comets orbit which is very elongated ellipse (i think), is just as short as a circular one from...lets say a satellite?

It's a fair question. A comet has a great deal more energy than an artificial satellite, but it is still effectively 'falling' around the earth.

MeJennifer,
I think you're right. I once wrote a simple coarse grained simulation of Newtonian gravity and got elliptical orbits. I found I was using a retarded position in my code, so inadvertently adding the finite propagation speed of gravity. This is included in GR I expect, hence the elliptical orbits.

So it looks as if relativity does explain elliptical orbits..

 

[lalbatros]

... I'm sure you would agree that planets don't move because we have a few equations that are good at describing reality. There must be a physical cause. Right?
I'm guessing that the cause is gravity and now i want to know how the relativity version of gravity explain elliptical orbits. Maybe, this is impossible for relativity or no one has though it up yet, but I would still like to know that.

The reason for closed orbits must be analysed before the shape is analysed.
(I assume that it is obvious why it moves in a plane ;>} )
The reason for closed orbit is simply because the radial oscillations and angular motion have exactly the same period for a Newtonian force.
The radial and angular motion have a 1/1 resonance.

However, other forces may behave completely differently.
They may show a rational resonance n/m with n and m integer.
Or they most probably have no resonance at all.

Actually the real motion of a planet is NOT periodic.
That's -as is well known- the first main prediction of general relativity.
Therefore, it is not really elliptic either.

The Einstein field equations lead to the motion equation and fully explain the Newtonian limit and the Newtonian force of gravity. Therefore they alos explain the -nearly- closed and -nearly- elliptical orbits.

If this is not enough, they we are back to an old debate: what is meant by "an explanation?".

But read that too: http://www2.hawaii.edu/~zxu/feynman.pdf [Broken]

 

[Dale]

Thanks. I hope you would help me clarify some points.
So, would you say that the cause for orbits is that the planets follow geodesic?

Yes. Same with any other object in freefall under the influence of gravitation alone.

which is the shortest path between points?

Actually, in spacetime geodesics are the longest path between events. Remember, an event has four dimensions, 3 space and 1 time. So it is not just the longest distance between points A and B, but the longest distance between point A at t0 and point B at t1. This is a very important point to notice. One reason that the "rubber sheet" analogy is so poor is that it only demonstrates curvature in space. In GR it is not just space that is curved, it is spacetime that is curved. And since v<<c in typical scenarios the curvature in the time part is actually very important.

Why are the geodesics ellipses?

If you want an answer for that you need the math. That is just how the differential geometry works out.

When i try to picture it in my mind circles looks like a shorter path than ellipses, but maybe I'm wrong.

When you include the time dimension both circles (helixes) and ellipses (distorted helixes) are geodesics in the same spacetime depending on the initial or boundary conditions.

 

[HallsofIvy]

I think someone should point out that relativity does NOT explain "elliptical orbits" because orbits are NOT elliptical!

While the difference is unmeasurable for most planets, it was precisely the calculation of Mercury's non-elliptical orbit that was the first real test of general relativity.

 

[D H]

I think someone should point out that relativity does NOT explain "elliptical orbits" because orbits are NOT elliptical!

While the difference is unmeasurable for most planets, it was precisely the calculation of Mercury's non-elliptical orbit that was the first real test of general relativity.

See post #2.

 

[DavidWhitbeck]

I think someone should point out that relativity does NOT explain "elliptical orbits" because orbits are NOT elliptical!

While the difference is unmeasurable for most planets, it was precisely the calculation of Mercury's non-elliptical orbit that was the first real test of general relativity.

What's also interesting is that Newtonian gravity could also explain the precession of the perihelion of Mercury... I know that sounds like what!? But it's very simple if the sun is oblate, the higher order multipoles do not vanish (recall that classical gravity is not described by Newton's law of gravitation, it's really given by the more general Gauss' Law just as Coulomb's law is not the final word on classical electrostatics), and the potential could look the same as the one you get from the effective Schwarzschild potential in GR. So even when GR perfectly predicted the funny part of Mercury's orbit, it still wasn't cut and dry. And until there were ways to assess the oblateness of the sun (because it doesn't have to be much) it was still open as to which interpretation was right.

 

[lbrits]

True, but neither are they under Newton's laws. It looks pretty elliptical if one if the masses is negligible but in the general case two masses in orbit do not follow an elliptical path under Newton's laws.

Nope. Elliptical orbits are exact solutions to the motion of two spherically symmetric bodies with different masses moving about their center of mass. This is covered in pretty much any classical mechanics text, e.g., Goldstein. It is easily seen by first defining the reduced mass and changing to generalized coordinates. I'm not sure what you're referring to but your statement is incorrect.

 

[shalayka]

You asked for the physical mechanism causing orbit in a Sun-Mercury type system. It's the gradient of gravitational time dilation. I apologize if the math in my previous post distracted from this point.

 

[RandallB]

So relativity can't explain the elliptical orbits of the planets?
Didn't relativity replace Newtonian gravity?
How can you explain elliptical orbits without gravity?

So relativity can't explain the elliptical orbits of the planets?
Didn't relativity replace Newtonian gravity?
How can you explain elliptical orbits without gravity?

The question your asking is different than most are trying to answer here.
You are trying to understand the difference between the Newtonian or Standard Model (QM) view using gravitons verses the General Relativity view that where there are no gravitons.
Both use the “inverse R squared” principle in different ways to match the elliptical math of Kepler.

You are describing the curved space expansion that looks like those coin drop donation cones you see in the mall. I hope you have seen one where you launch your coin in a near perfect circle (but you will note if you don’t make a good coin launch it speeds up and slows down as it follows an ellipse instead of perfect ellipse (circle). Also as friction slows down a good circular launch the coin falls closer to the hole it will drop through SPEEDS up over the surface it is rolling on to orbit faster and faster. Same thing happens if you slow down Jupiter – it falls to a lower orbit where it speeds up to orbit even faster.

Now the alternate to this GR “inverse R squared” curve in the QM / Standard-Model idea of Gravitons being tosses out of masses and causing thing only to attract, just as has already been show for magnets or electric charges do (attract or repel). Call it Newtonian if you like but to be fair Newton only established the math and insisted on not speculating on who or why gravity works. Obviously the number of gravitons that will hit an object will follow “inverse R squared”, just like a pizza twice the size has four times as much stuff. Move the Earth to half the distance from the sun it sees four times as much light and gravitons.

So the reason GR give the same elliptical orbits (including round elliptical orbits) as gravitons do is the curve on your cone gives the same “inverse R squared” result as counting the number of gravitons that cause gravity. Which is it GR without gravitons or QM with gravitons; a whole differ question you can find in other threads.

Is that a little closer to what you were asking?

And don’t forget to find of these coin drop donation machines if you have not seen one before, they are a lot more fun than dropping coins in a pond and for a pocket full of change you get a good demonstration of two body orbital dynamics. You can also see how circles or ellipses for coin travel orbits are really the same thing, and see the “inverse R squared” in the angle that makes the cone shape very curved and not flat.

 

[Fredrik]

I'd like to point out things that are relevant here:

1. Geodesics are defined as the straightest possible paths, not as the shortest (or longest).
2. It's the path through space-time that has to be a geodesic, not the path through space.

The straightest path connecting two events in space-time is the one that has the longest proper time, so I'd rather say that a geodesic is the longest path than the shortest. Of course, if you draw a space-time diagram representing an inertial frame in SR and draw a geodesic and a non-geodesic curve connecting two events, the geodesic is the shortest path on that piece of paper.

More importantly, the fact that an object in free fall is going to trace out the shortest/longest path between where it is and where it's going to be isn't sufficient to tell you where it's going to be. Let's play a game: You're moving around on the surface of a sphere and you're not allowed to change your speed. At the moment you happen to be at the equator moving north. If you are told that you have to go as straight as possible, you know you have to continue towards the north pole. If instead you're told that you have to take the shortest path, you don't know if you're allowed to make a sharp left turn right away. The rule "go as straight as possible" seems to be equivalent to "take the shortest path and don't make any sharp turns". The rule "don't make any sharp turns" is of course equivalent to "keep going as straight as possible", so we might as well just say that. The instruction to take the shortest/longest path seems completely redundant.

Because of this I prefer to say that a planet in orbit is going as straight as possible through space-time, instead of saying that it takes the shortest/longest path around the sun back to (almost) its original position. Saying that it takes the shortest/longest path leaves the question "...to where?" unanswered.

Let's move on to my second point. If we only talk about geodesics in space rather than in space-time it's impossible to understand how orbits can have different eccentricities. Consider a circle inside an ellipse such that they coincide at the two points on the ellipse that are the closest to the center. It's clear that no matter what the geometry of space is, at most one of them can be a geodesic. However, if we consider the paths through space-time of two objects in orbits that look like what I just described, we can no longer immediately rule out that they are both geodesics.

This is one of the reasons why a bowling ball on a rubber sheet is such a bad analogy. A better mental image is to imagine the sun and the planets as dots on a piece of paper, and imagine time as the "up" direction (out of the paper). The path of the sun is a line straight up. The paths of the planets are spirals around that straight line. Note that if we consider the spirals that correspond to the circle and the ellipse I described, their tangents aren't the same at the points in space where the circle touches the ellipse. They have different slopes relative to the paper. That's why we can't immediately rule out that they are both geodesics.

 

[niin]

I want to thanks everyone who responded.
I think, I will have to think a little more before i fully get it, but i feel i have a better understanding of the subject now.
Thanks for helping me out. =)

 

[Dale]

1. Geodesics are defined as the straightest possible paths, not as the shortest (or longest).

Yes, this is correct, but my understanding is that the two statements are essentially equivalent. You could just as easily define them as the longest path and then derive the fact that they are "straight". The Euclidean analogy is "the shortest distance between two points is a straight line".

If you define your spacetime path with an initial event and a direction (initial conditions) then you can follow the "straight line" rule to construct the geodesic. If you define your spacetime path with an initial and final event (boundary conditions) then you can use a variational approach based on the "longest distance rule" to find a geodesic that connects them. The only difference, other than initial v. boundary conditions, is that there may be more than one geodesic path between two events in a curved spacetime. Each one is a local maximum path length, but one may be a global maximum.

 

[Fredrik]

Agreed. My point is just that the description of a geodesic as the shortest/longest path is more appropriate when the endpoints are known, and the description as the straightest possible path is more appropriate when one endpoint and a velocity (tangent vector) is known. But that's pretty much what you just said, so I don't have to tell you that.