Gravitational Orbits: How the Earth Knows Where to Go

[InkTide]

How does the Earth's reference frame "know" to experience the gravitational pull of where the sun/Earth barycenter will be in 8 minutes, rather than orbit around the spot where the barycenter was 8 minutes ago? The latter case seems required by the finite speed of gravitation, but coplanar orbits imply instantaneous gravitation - or at least an effect of gravitation that somehow confers the velocity effects, and somehow increases that velocity effect as distance increases (because the coplanar "maintenance" should become more difficult by the square of the orbital radius as a result of the inverse square law).

Even if we assume all the objects are comoving, any angular momentum (i.e. an orbit) should represent a constantly changing reference frame and linear momentum relative to the central body - in other words, I am really struggling to see how non-conical/dome-shaped orbits are even possible if the central mass is moving and the distances between masses are appreciable to c.

 

[PeroK]

I can't follow your arguments at all. Perhaps because there is no mathematics involved.

The Sun-Earth system is dominated by the Sun and can be closely approximated by a static Schwarzschild solution to the EFE (Einstein Field Equations). You can use the symmetry in the metric to deduce that the motion remains in a plane.

 

[InkTide]

It's less a question of the Sun-Earth system and more a question of why each solar-dominated orbital system is coplanar (or in the case of high inclination orbits, why they are stable), seemingly independent of the radius of that orbital system.

Basically, the symmetry you're talking about, unless I'm misunderstanding something, is why the orbits would remain relatively stable/circular and parallel even if as you moved to further orbiting bodies they "lagged" behind closer ones relative to the central body.

static Schwarzschild solution

Doesn't any spherical solution to EFE require that the mass be unmoving, otherwise the point of the sphere that the mass' velocity vector points towards should be closer to the mass by c-v where v = the magnitude of the velocity vector? Sorry if I'm misunderstanding Schwarzschild metrics.

We know the sun is moving relative to the galactic center, and in a direction that puts the solar system plane nearly perpendicular to the galactic plane IIRC.

 

[PeroK]

Doesn't any spherical solution to EFE require that the mass be unmoving, otherwise the point of the sphere that the mass' velocity vector points towards should be closer to the mass by c-v where v = the velocity of the mass? Sorry if I'm misunderstanding Schwarzschild metrics.

Motion is relative. We may take (Schwarzschild) coordinates centred on the Sun. What's being neglected is the gravitation of the Earth. That's why it's a good approximation.

We know the sun is moving relative to the galactic center, and in a direction that puts the solar system plane nearly perpendicular to the galactic plane IIRC.

That's irrelevant; as it is in Newtonian physics. Motion is relative and we may study the solar system in a frame of reference where it is at rest. The solution to both the Newtonian equations and the EFE give the local motion of the Earth-Sun system. I.e. the motion of the Earth relative to the Sun.

That whole system is then moving relative to the rest of the galaxy; which entails a negligible tidal gravitational correction to the solar system dynamics. Similarly, there is relative motion of the Milky Way and Andromeda galaxies, which does not affect the local motion of the Earth-Sun system to any significant extent.

 

[InkTide]

Motion is relative. We may take (Schwarzschild) coordinates centred on the Sun. What's being neglected is the gravitation of the Earth. That's why it's a good approximation.

I see - so, if, hypothetically, a mass was moving towards me from outside my observable universe bubble, would I feel gravitation from it (however tiny) before I could see it? My intuition based on the speed of gravity suggests the answer should be no, but I'm having difficulty reconciling that intuition with a spherical solution to gravitation.

In my mind, gravity moving at c should mean that a given spacetime curvature around a mass should form not a sphere but a sort of "teardrop" surface, with the rear point exactly opposite the velocity vector of the mass relative to any observer that isn't the mass itself, and with the furthest possible extent of that surface ahead of the object being given by c-v, because otherwise gravitation becomes superluminal.

Basically, adding a travel time to Newtonian gravity as constrained by relativity. Does that make sense?

 

[PeroK]

In my mind, gravity moving at c should mean that a given spacetime curvature around a mass should form not a sphere but a sort of "teardrop" surface, with the rear point exactly opposite the velocity vector of the mass relative to any observer that isn't the mass itself, and with the furthest possible extent of that surface ahead of the object being given by c-v, because otherwise gravitation becomes superluminal.

Basically, adding a travel time to Newtonian gravity as constrained by relativity. Does that make sense?

I'm not sure. If we look at a light source in SR, then the shape of the wave-front is spherical in all inertial reference frames. This follows from the 2nd postulate of SR that the speed of light (in vacuum) is independent of the motion of the source. It's not tear-shaped or elliptical, as one would expect from classical physics.

That said, the Sun-Earth system may be studied most simply using a coordinate system where the Sun is at rest. There is no need to consider the motion of the Sun relative to the galactic centre. If you do want to study the Sun-Earth system using coordinates where the Sun is not at rest, then it will get mathematically more complicated. Technically, you must get the same physical solution - in terms of the proper time of the Earth's orbit, say - although the mathematical solution would look different. (For example, in a frame where the Sun is moving, then the Earth's orbit is no longer an ellipse - it's only an ellipse relative to the Sun.) In fact, in a reference frame where the Earth is at rest, it has no orbit at all!

The only thing that matters is that by studing the Earth-Sun system using coordinates where the Sun is at rest, we get an orbit that remains in a given plane. You can then move and spin that solution any way you want. It would look very different from Jupiter's frame of reference, say, although it's the same underlying physical system.

 

[Ibix]

There's a paper by Carlip that explains this. https://arxiv.org/abs/gr-qc/9909087

Newtonian gravity with a delay bolted on is an unhelpful mental picture. Several attempts along those lines were made after it became apparent that SR and Newtonian gravity were incompatible, but it doesn't work.

 

[PeroK]

There's a paper by Carlip that explains this. https://arxiv.org/abs/gr-qc/9909087

Newtonian gravity with a delay bolted on is an unhelpful mental picture. Several attempts along those lines were made after it became apparent that SR and Newtonian gravity were incompatible, but it doesn't work.

That's a B-level bazooka!

 

[Orodruin]

That's a B-level bazooka!

The paper, yes. The statement that bolting a time-delay on Newtonian gravity is an unhelpful mental image, no.

This also goes for the EM field of a constant velocity charged particle. It is not ”lagging behind” as you would obtain if you just tried to do electrostatics with a time delay.

 

[InkTide]

There's a paper by Carlip that explains this. https://arxiv.org/abs/gr-qc/9909087

Newtonian gravity with a delay bolted on is an unhelpful mental picture. Several attempts along those lines were made after it became apparent that SR and Newtonian gravity were incompatible, but it doesn't work.

Thank you very much for this, that paper is almost exactly dealing with the question I was trying to ask. If I'm reading it correctly, the observed effect does seem to be related to the effect of velocity on gravitation, in that the gravitation includes the relevant velocity effects to yield basically an extrapolated gravitation towards the location of the mass that it would be in based on that velocity.

I believe that supports what I suggested as a possibility in the original post:

or at least an effect of gravitation that somehow confers the velocity effects, and somehow increases that velocity effect as distance increases

I do wonder if that's a bit too convenient, though. Is there more recent work on this? Though 1999 is pretty recent on the scale of the history of cosmological theory, to be fair.

 

[PeterDonis]

If I'm reading it correctly, the observed effect does seem to be related to the effect of velocity on gravitation, in that the gravitation includes the relevant velocity effects to yield basically an extrapolated gravitation towards the location of the mass that it would be in based on that velocity.

Yes, that's basically what Carlip's analysis shows. The difference between gravity and electromagnetism here is that gravity is spin-2 while electromagnetism is spin-1; the higher spin means the extrapolation has to be higher order, and therefore the aberration--what is left over after the velocity-dependent terms have canceled out as much as they can of the effects of light travel time delay in the interaction--has to be much smaller.

I believe that supports what I suggested as a possibility in the original post

Sort of. The "velocity effect" doesn't really "increase as the distance increases". (In fact, since planets further from the Sun move slower, the "velocity effect" is smaller for them.)

I do wonder if that's a bit too convenient, though.

IIRC, Carlip discusses this in his paper, going to some trouble to emphasize the point that the kind of cancellation of light travel time delay that occurs here must happen in any field theory that obeys the usual conservation laws.

 

[InkTide]

Sort of. The "velocity effect" doesn't really "increase as the distance increases". (In fact, since planets further from the Sun move slower, the "velocity effect" is smaller for them.)

I wasn't clear on what I meant by that, sorry - I meant that the velocity effect would have to become a larger proportion of the overall effect because though the mass-related effect would decrease with the inverse square law, the further away any planet was the greater the amount of future motion that would need to be extrapolated from the parent body's velocity. Is this what happens?

Also, I'm still curious about how this interacts with the speed of information:

First, I imagine myself at the center of an observable universe sphere of radius c * t where t is the age of the earliest photons visible to me in that universe, but that universe is larger than c * t (i.e. inflationary cosmology), so as t increases the radius expands. Now, say there is an object at a distance of (c * t) + d, and that said object has another object orbiting it at an orbital radius of r, on a plane that is horizontal to me. If the orbiting second object is directly between me and its orbital parent when t becomes large enough to include points at a distance of (c * t) + d - r but crucially not yet points at a distance of (c * t) + d, does the velocity effect of the parent object still confer that information to the path of the orbiting second object? If so, how does that avoid me getting information about an object that is potentially r too far for me to get information about without that information being superluminal?

 

[PeroK]

If so, how does that avoid me getting information about an object that is potentially r too far for me to get information about without that information being superluminal?

Imagine you have planet A one light year away. And another planet B a further light year away. Planet B sends a signal to planet A, who reacts in some way (waves a flag). You don't see the flag wave for a further year.

Whatever gravitational effects planet A displays based on the presence of planet B are not instantaneously observable by you. There will be a further year before you observe them.

 

[InkTide]

Whatever gravitational effects planet A displays based on the presence of planet B are not instantaneously observable by you. There will be a further year before you observe them.

Doesn't this rule out the orbit of a planet including information about the motion of the thing it is orbiting, i.e. the velocity effect? I'm not talking about a change in the orbit, or any actual change other than that of the size of my observable universe - I'm talking about what I can learn about the mass and/or motion of the central object from the orbit of an object that is just enough closer to me than the object it is orbiting to be within my observable universe .

Perhaps a better example would be a binary star system - whichever star is closer to me has a past orbital motion that is partially determined by the velocity extrapolation of its partner's future motion, but if the distance between each partner was a light year, I initially see from the motion of the closer star that it is orbiting something that I should have no information about because it is beyond my observable universe. If we tune the distance to have space expansion take the entire binary system out of my observable universe after I see the closer star but before I see the binary partner, I should have no way to know that the partner exists - but I would know from the orbital motion something about an object that is entirely outside my light cone, hence superluminal information.

 

[PeroK]

Perhaps a better example would be a binary star system - whichever star is closer to me has a past orbital motion that is partially determined by the velocity extrapolation of its partner's future motion, but if the distance between each partner was a light year, I initially see from the motion of the closer star that it is orbiting something that I should have no information about because it is beyond my observable universe.

If we ignore expansion for the moment, what constitutes the observable universe increases in size according to the speed of light. It's not like Jupiter could be within the observable universe and one of its moons be outside the observable universe. The edge of the observable universe is not like a permanent line in the sand.

 

[InkTide]

It's not like Jupiter could be within the observable universe and one of its moons be outside the observable universe.

Why not? Doesn't the expansion at the speed of light (of the spherical volume of observable points in space) allow for light from Jupiter to reach me before light from one of its moons does (if the moon is at a point in its orbit that places it further away from me than Jupiter when I first detect light from Jupiter), thereby necessitating a period of time (even if vanishingly small) where Jupiter would be within the observable universe and any of its moons further away from me than Jupiter would be outside of it?

 

[PeroK]

Why not? Doesn't the expansion at the speed of light (of the spherical volume of observable points in space) allow for light from Jupiter to reach me before light from one of its moons does (if the moon is at a point in its orbit that places it further away from me than Jupiter when I first detect light from Jupiter), thereby necessitating a period of time (even if vanishingly small) where Jupiter would be within the observable universe and any of its moons further away from me than Jupiter would be outside of it?

And what are you going to learn about the moon in that vanishingly small time? There's not enough time to infer the existence of a moon.

 

[InkTide]

The existence of the vanishingly small time at all is the thing that makes this superluminal information, because it means that we can scale the effect in distance and therefore time for systems such as solar systems, binary stellar systems, and entire galaxies where the time between seeing an orbiting object between us and the object it orbits allows enough time of observation to see an orbital path and thereby infer the existence, mass, and if gravitation confers an extrapolation of velocity, possibly even motion of a parent body before light from the parent body reaches us. I don't see a way out of making this superluminal if 3 or more bodies (including ourselves as the observer body) are involved.

 

[PeroK]

And what are you going to learn about the moon in that vanishingly small time? There's not enough time to infer the existence of a moon.

PS It's more fundamental than that. If Jupiter were a distant star, say, that we observe close to the limit of the observable universe, then we see that star as it was in the distant past. If that star has existed for some time, then we should have seen the star before then. Or, at least, the light from that earlier time has had time to reach us. A fully-formed star or galaxy cannot be literally at the limit of the observable universe. That's where your paradox arises.

 

[PeroK]

The existence of the vanishingly small time at all is the thing that makes this superluminal information, because it means that we can scale the effect in distance and therefore time for systems such as solar systems, binary stellar systems, and entire galaxies where the time between seeing an orbiting object between us and the object it orbits allows enough time of observation to see an orbital path and thereby infer the existence, mass, and if gravitation confers an extrapolation of velocity, possibly even motion of a parent body before light from the parent body reaches us. I don't see a way out of making this superluminal if 3 or more bodies (including ourselves as the observer body) are involved.

See above.

 

[InkTide]

Isn't that what inflationary cosmology requires? A superluminal pre-expansion to space that means even now light from ever older and more distant objects still continues to reach us. Otherwise the observable universe would seem to end with a void rather than progressively more redshifted objects.

 

[PeroK]

Isn't that what inflationary cosmology requires?

No. If you can see a star one million years after its birth, then the light from the star when it was only 500,000 years old must have been able to reach us.

 

[Orodruin]

I suggest that you first try to understand the electromagnetic case for a moving charge using retarded potentials and understand how that relates to the corresponding field’s symmetries.

https://en.wikipedia.org/wiki/Retarded_potential

Note that the field is symmetric relative to the forward/backward direction of motion and that it depends on the charges along the light cone of the relevant spatial point.

 

[PeroK]

... and, the light from as far back as the last scattering surface must have had time to reach us. We can't go further back detecting light, but can potentially go further back detecting gravitational waves:

https://news.mit.edu/2020/universe-first-gravitational-waves-1209

 

[PeterDonis]

I meant that the velocity effect would have to become a larger proportion of the overall effect because though the mass-related effect would decrease with the inverse square law, the further away any planet was the greater the amount of future motion that would need to be extrapolated from the parent body's velocity. Is this what happens?

No. This is one of those cases where you can't just wave your hands and use intuition, particularly if your intuition is untrained; you have to look at the math. Look at the equations in Carlip's paper (equation 2.3 is the best one to use) and look at the size of the various terms for, say, the Earth orbiting the Sun. The first thing that should jump out at you is that all of the terms have the Newtonian inverse square factor out front, so to compare the Newtonian term (the "mass-related effect") with the corrections (the "velocity effect"), you're just comparing $1$, the relative magnitude of the Newtonian term, with various powers of the velocity $v$ in units where $c = 1$. Since $v << 1$ in those units, all of the velocity-dependent terms are obviously much smaller than the Newtonian term. And since the magnitude of $v$ gets smaller as the planet gets further from the Sun, the magnitude of the velocity-dependent terms relative to the Newtonian term gets smaller for planets further from the Sun.

 

[PeterDonis]

@InkTide, your intuition is being misled by the fact that the light travel time from Sun to planet gets longer as the planet gets further from the Sun. But the magnitude of the correction for aberration is not the same as the light travel time. Look at Carlip's equation 2.5. That is an equation for the "corrected" direction of the force, $\eta^i$. On the RHS of that equation, we see $\left( t - z^0 (s_R) \right)$, which is the light travel time, but it is multiplied by derivatives of $n^i$. Those derivatives are roughly of order $(v / R)^n$, where $n$ is the derivative order and $R$ is the light travel time, which is the same as the retarded spatial distance (note that this is how Carlip defines it earlier in the paper). So the correction terms to $n^i$ in Carlip's equation 2.5, which are of order the $n$th derivative times $R^n$, are roughly of order $v$ and $v^2$, which again are much smaller than $n^i$, which is of order $1$.

 

[InkTide]

@InkTide, your intuition is being misled by the fact that the light travel time from Sun to planet gets longer as the planet gets further from the Sun.

I think there might be an intuitive way to explain the difference, which I was also missing:

The apparent size in a given planet's sky of the path between the extrapolated future position and the observed position could be thought of as a direct illustration of the maximum effect the velocity could have, and should, for a given amount of time (if we imagine that the Sun has a magical "future path" line ahead of it showing where it will be in that given time), decrease by exactly the same scale as the apparent size of the Sun for further planets, for simple geometric reasons. If that path is extended as we move from the Sun to extrapolate the future position for the time it takes light to reach us, its extension should increase linearly with our distance while its scale should reduce with the inverse square law (for the same geometric reasons the inverse square law exists) - hence the difference in orders you describe.

Well, provided no/insignificant relativistic effects that would change the observed velocity of the Sun significantly between planetary reference frames.

I sincerely apologize for not using the mathematical jargon, but IMO this is an intuitive question that usual treatments of the math don't explain intuitively - I'm ashamed to admit that I'm still trying to wrap my head around some of Carlip's reasoning, especially the role of quadrupole gravity because my understanding of tensors is poor. Thank you for bearing with me.

I think further discussion of the question of observable universe bubbles is something that wouldn't belong in this thread because it's a different question fundamentally from the first one I asked, so let's drop that one for now. Hopefully the rest of the thread is a good resource to explain how gravitation confers velocity information about the gravitating body.

 

[Orodruin]

I sincerely apologize for not using the mathematical jargon, but IMO this is an intuitive question that usual treatments of the math don't explain intuitively

I think you have a slightly warped perception of what intuition is. Intuition is something you gain by experience and if you have not been working with the relevant mathematics of classical field theories, you cannot be expected to have an intuition for them simply because you are lacking that experience. Instead, you are trying to apply your intuition from other things that simply are not applicable.

 

[InkTide]

I think you have a slightly warped perception of what intuition is. Intuition is something you gain by experience and if you have not been working with the relevant mathematics of classical field theories, you cannot be expected to have an intuition for them simply because you are lacking that experience. Instead, you are trying to apply your intuition from other things that simply are not applicable.

This is a conflation of experience with intuition - they are neither synonymous nor implications of each other in any mainstream dictionary. They do, however, both represent descriptors that vary between individuals, and thereby imply a subjectivity that is incompatible with an "objective" perception of them that can be "warped" - in relativistic terms, each individual represents a unique reference frame for the application of "intuition" and "experience". "Intuitive" and "not intuitive" are statements of personal ease of explanation/understanding.

This is also a semantic/linguistic discussion not really relevant here.

 

[PeterDonis]

The apparent size in a given planet's sky of the path between the extrapolated future position and the observed position could be thought of as a direct illustration of the maximum effect the velocity could have

This is somewhat closer to what the actual math is saying.

IMO this is an intuitive question that usual treatments of the math don't explain intuitively

Nobody can explain something "intuitively" to you if you don't understand what the underlying math is saying. That is why I said you can't use your intuition if your intuition is untrained. You have to retrain your intuition for a new domain that it hasn't been trained for before, and the only way to do that with a physical theory is to understand its math.

I'm still trying to wrap my head around some of Carlip's reasoning, especially the role of quadrupole gravity because my understanding of tensors is poor.

For the purpose of this particular question, you don't actually need to understand tensors in full generality. The key point Carlip is making about gravity being quadrupole as opposed to electromagnetism being dipole (or, to put it another way, gravity is a spin-2 interaction while electromagnetism is spin-1) is that quadrupole radiation is much weaker than dipole radiation, so much less energy and angular momentum can be emitted as gravitational radiation by a system with a time-varying quadrupole moment, as compared with a system of charges emitting electromagnetic radiation because it has a time-varying dipole moment.

This point is important because it means the cancellation of aberration effects by velocity-dependent terms in the interaction has to be much more exact with gravity than with electromagnetism, because of conservation of energy and angular momentum. Whatever aberration is not canceled must correspond to radiation emitted by the system through the relevant interaction (gravitational or electromagnetic), because uncancelled aberration means the energy and angular momentum of the system itself, taken in isolation, is not exactly conserved, and the amount of uncancelled aberration corresponds to the amount of non-conservation of energy and angular momentum in the system itself, which must in turn correspond to energy and angular momentum emitted as radiation so that overall conservation is maintained.