Mass of Earth/Sun: Comparing in Different Frames

[Gear300]

I just bumped into something funny...it might be trivial...but I'm not sure. Let us say you had an Earth-Sun system with nothing else. In a frame at rest with respect to the Sun, the Earth is observed to have an orbital speed of v. In a frame at rest with respect to the Earth, the Sun is observed to have the same orbital speed v. So then, using the same law of gravity in both frames, wouldn't one frame find that the mass of the Sun is larger than the mass of the Earth and the other frame find that the mass of the Earth is larger than the mass of the Sun (I guess it would be different if you took into account that the two rotate around a center of mass closer to the sun, but aside from that)?

 

[PeterDonis]

If there are only the two bodies and nothing else, then how do you distinguish a frame at rest with respect to one from a frame at rest with respect to the other? There's no third body to serve as an outside reference. In other words, with only two bodies, the only real observation you can make is the separation between them and how it changes with time; you can't even define an "orbital speed" because you have no outside reference to measure when an "orbit" is completed (or to define a second "dimension" for the plane in which the orbit would lie). So you can't even measure either body's mass (or even the combined mass of the two).

(You can, if you like, arbitrarily declare that one body is "at rest" and say that the change in separation between the two is entirely due to the second body's motion. But that still won't get you anywhere, because the separation will follow the same cyclic pattern regardless of which body you pick to be "at rest". So in so far as you can assign any "properties" like mass to the bodies at all, they would have to be the same. In "reality", of course, what you're observing is that the Newtonian gravity force between the bodies is proportional to the *product* of their masses, which is the same regardless of which one you declared to be "at rest".)

 

[bcrowell]

I don't think PeterDonis's solution in #2 quite works. You can use the Sagnac effect to determine the angular velocity, but you can't use the Sagnac effect to determine the center of mass.

This seems to me like a variation on Einstein's Machian thought-experiment of the two planets, from "The foundation of the general theory of relativity."

The cheap answer is that the paradox depends on the assumption that GR is totally Machian, but it isn't. A more satisfying answer might explain why the masses of the Earth and sun can be distinguished even in a more Machian theory, such as Brans-Dicke gravity.

 

[Gear300]

If there are only the two bodies and nothing else, then how do you distinguish a frame at rest with respect to one from a frame at rest with respect to the other? There's no third body to serve as an outside reference. In other words, with only two bodies, the only real observation you can make is the separation between them and how it changes with time; you can't even define an "orbital speed" because you have no outside reference to measure when an "orbit" is completed (or to define a second "dimension" for the plane in which the orbit would lie). So you can't even measure either body's mass (or even the combined mass of the two).

(You can, if you like, arbitrarily declare that one body is "at rest" and say that the change in separation between the two is entirely due to the second body's motion. But that still won't get you anywhere, because the separation will follow the same cyclic pattern regardless of which body you pick to be "at rest". So in so far as you can assign any "properties" like mass to the bodies at all, they would have to be the same. In "reality", of course, what you're observing is that the Newtonian gravity force between the bodies is proportional to the *product* of their masses, which is the same regardless of which one you declared to be "at rest".)

What if an observer was moving with the Earth and holds his finger in front of him. If he were to measure the time between the event of the sun being behind his finger, wouldn't he be able to measure the orbital period (and continue from there)?

I don't think PeterDonis's solution in #2 quite works. You can use the Sagnac effect to determine the angular velocity, but you can't use the Sagnac effect to determine the center of mass.

This seems to me like a variation on Einstein's Machian thought-experiment of the two planets, from "The foundation of the general theory of relativity."

The cheap answer is that the paradox depends on the assumption that GR is totally Machian, but it isn't. A more satisfying answer might explain why the masses of the Earth and sun can be distinguished even in a more Machian theory, such as Brans-Dicke gravity.

So how is the paradox answered in a Machian context? From what I take, wouldn't it still be necessary to have a third point of reference (or would it be in the difference between inertial and non-inertial mass)?

 

[PeterDonis]

What if an observer was moving with the Earth and holds his finger in front of him. If he were to measure the time between the event of the sun being behind his finger, wouldn't he be able to measure the orbital period (and continue from there)?

Holds his finger "in front of him" with respect to what? Suppose I claim that, unless his finger always points straight at the Sun, he isn't "holding it in front of him" correctly (i.e., he's "moving" it and thereby cheating)? With no other bodies in the universe to serve as a reference, how do you tell whether I'm right or wrong?

 

[PeterDonis]

I don't think PeterDonis's solution in #2 quite works. You can use the Sagnac effect to determine the angular velocity, but you can't use the Sagnac effect to determine the center of mass.

If we just look at the results of the Sagnac effect experiment, without considering how we would have to conduct it, the "angular velocity" you determine via the Sagnac effect can't be attributed to one or the other body in isolation; it can only be attributed to the combined effect of both bodies, so whatever property is responsible for the Sagnac effect is "the same" regardless of which body you say is "rotating" with the angular velocity you determine.

However, if we look at what you would actually have to do to *measure* the Sagnac effect due to the Earth revolving around the Sun, you would have to build a huge waveguide with a radius equal to that of the Earth's orbit (let's pretend for a moment that the orbit is exactly circular to make the analysis easier) and send light rays around it in both directions. This waveguide then serves as a third body in the system--an external reference that let's you distinguish "Earth moving around Sun" from "Sun moving around Earth" and thereby establish a background reference frame in which the Sun, not the Earth, is the central mass (since there's no way to build a corresponding waveguide to measure a Sagnac effect showing the Sun "revolving" around the Earth). This violates the conditions specified in the OP, so the conclusion does not constitute the sort of "paradox" the OP is asking about.

 

[Gear300]

Your answer seems to imply that mass is not uniquely defined in a two-body stable/bounded system with gravity as the only interaction. If that is what you're saying, then...that seems kind of odd.

 

[PeterDonis]

Your answer seems to imply that mass is not uniquely defined in a two-body stable/bounded system with gravity as the only interaction? If that is what you're saying, then...that seems kind of odd.

I think a better way to put what I'm saying is that what you specified in your OP was *not* a "stable/bounded system with gravity as the only interaction", because you specifically said "an Earth-Sun system with *nothing else*" (my emphasis). In order to define a "bounded system" at all in GR, let alone the mass of one, you need to have some kind of boundary condition, and physically, that requires a "third body" of some kind--some external physical thing that explains, physically, why the boundary condition is the way it is. (This is the "Machian" issue that bcrowell referred to.)

The boundary condition adopted for studying the solar system (and every other isolated bound system I'm aware of in GR) is an asymptotically flat spacetime outside the central mass--meaning that as you get farther from the mass, the influence of its "gravity" gets smaller, going to zero at infinity, but *also* meaning that what is "left" after the influence of the mass goes to zero is the ordinary flat Minkowski spacetime of SR. But physically, we can't just wave our hands and say that: there has to be a physical *reason* why the spacetime far away from the mass becomes Minkowski in the limit--or else we have to substitute some other condition that is equally consistent with actual observations.

In the case of the actual solar system, our actual physical assumption is that the background spacetime is *approximately* Minkowski, because the universe is so large that its large-scale spacetime curvature is negligible on the scale of the solar system--in other words, we can't detect whatever very small departures from true asymptotic flatness might exist in the spacetime in our vicinity. This, of course, requires a "third body" in the system--the rest of the universe. If you think about it, you'll see that any other physical answer to the question of why spacetime around the solar system is (at least to a very good approximation) asymptotically flat, allowing us to pick out the Sun as the "central mass" rather than the Earth, requires the same sort of "third body"--some external physical thing that explains where the (approximate) asymptotic flatness comes from.

 

[Gear300]

I see. What you said seems to suggest that for a system to be physically realizable, it needs an outside observer or some observation outside of it to impose a boundary condition. Is the system I proposed even physically realizable?

 

[DaveC426913]

Holds his finger "in front of him" with respect to what? Suppose I claim that, unless his finger always points straight at the Sun, he isn't "holding it in front of him" correctly (i.e., he's "moving" it and thereby cheating)? With no other bodies in the universe to serve as a reference, how do you tell whether I'm right or wrong?

You would be able to determine if the Earth (or whichever body you were on) is rotating using a pendulum. Knowing that, you could determine what the orbit is doing. (I think).

[EDIT] No. you couldn't. Axial rotation would not tell you about orbital revolution.

 

[yuiop]

I just bumped into something funny...it might be trivial...but I'm not sure. Let us say you had an Earth-Sun system with nothing else. In a frame at rest with respect to the Sun, the Earth is observed to have an orbital speed of v. In a frame at rest with respect to the Earth, the Sun is observed to have the same orbital speed v. So then, using the same law of gravity in both frames, wouldn't one frame find that the mass of the Sun is larger than the mass of the Earth and the other frame find that the mass of the Earth is larger than the mass of the Sun (I guess it would be different if you took into account that the two rotate around a center of mass closer to the sun, but aside from that)?

A Sagnac device no bigger than a shoe box on the surface of the Earth would tell you whether the Earth is rotating or not in absolute terms and you would be in no doubt that the Earth is orbiting around the Sun and not vice versa, without reference to any other bodies, if you accept Newtonian principles and the relativistic extensions.

 

[bcrowell]

However, if we look at what you would actually have to do to *measure* the Sagnac effect due to the Earth revolving around the Sun, you would have to build a huge waveguide with a radius equal to that of the Earth's orbit (let's pretend for a moment that the orbit is exactly circular to make the analysis easier) and send light rays around it in both directions.

Huh?? No, not at all. You simply measure the Sagnac effect with an ordinary-size apparatus, and it tells you the state of rotation of the apparatus. You know the apparatus's state of rotation relative to the earth-sun system, so you can infer the state of rotation of the earth-sun system.

So how is the paradox answered in a Machian context?

I don't know.

 

[PeterDonis]

You simply measure the Sagnac effect with an ordinary-size apparatus, and it tells you the state of rotation of the apparatus. You know the apparatus's state of rotation relative to the earth-sun system, so you can infer the state of rotation of the earth-sun system.

You're right, I hadn't considered this method when I made my previous post. If you first postulate an asymptotically flat background spacetime, then I agree you can do this. But everything I said about having to physically account for that boundary condition still applies.

 

[Gear300]

What would someone on the Sun observe from the Sagnac effect when compared to someone on the Earth?

 

[bcrowell]

You're right, I hadn't considered this method when I made my previous post. If you first postulate an asymptotically flat background spacetime, then I agree you can do this. But everything I said about having to physically account for that boundary condition still applies.

I don't think your #8 is correct. In GR you can define asymptotic flatness without having to have distant objects, and in any case I don't think asymptotic flatness is relevant here. You don't need asymptotic flatness in order to determine your state of rotation. For example, Sciama figured out in 1967 that the CMB could be used to put an upper limit on the universe's rate of rotation, but the universe isn't asymptotically flat.

What would someone on the Sun observe from the Sagnac effect when compared to someone on the Earth?

Each would observe an indication of his own state of rotation.

I'll take a shot at analyzing the earth-sun thing in a theory more Machian than GR, but I don't know if this is right. First off, I think you can get at the fundamental issue without even worrying about the asymmetry in mass. Suppose you have two planets, A and B, with equal masses, orbiting one another in circular orbits about their common center of mass.

In GR, rotational motion is not relative, so it's easy to explain why the planets don't fall into one another despite their gravitational attraction, even if the rest of the universe doesn't exist so that you have no distant reference points to compare against. You can also check the angular velocity using the Sagnac effect, and everything comes out consistent.

In Brans-Dicke gravity with a small (i.e., Machian) value of $\omega$, if A and B are alone in their universe, then the value of the field $\phi$ is very small, and this is equivalent to having a very large local value of G. Therefore the two planets collide rapidly, and this is exactly what we expect in a Machian theory, where rotation is unobservable without distant bodies to compare against.

I think the asymmetric, Machian case is then a straightforward generalization of this. Planets A and B, with unequal masses, come closer and closer together. Since only relative motion is observable in a purely Machian theory, the only observable quantity is then the function d(t), where t is time and d is the distance between the two planets. We can't tell, and don't care, whether there's rotation, and we also can't tell whether one planet is accelerating more than the other.

Now suppose we have n>2 bodies, all with unequal masses, alone in a Machian universe. I think we can now tell that some bodies are accelerating more than others.

 

[Gear300]

Each would observe an indication of his own state of rotation.

I'll take a shot at analyzing the earth-sun thing in a theory more Machian than GR, but I don't know if this is right. First off, I think you can get at the fundamental issue without even worrying about the asymmetry in mass. Suppose you have two planets, A and B, with equal masses, orbiting one another in circular orbits about their common center of mass.

In GR, rotational motion is not relative, so it's easy to explain why the planets don't fall into one another despite their gravitational attraction, even if the rest of the universe doesn't exist so that you have no distant reference points to compare against. You can also check the angular velocity using the Sagnac effect, and everything comes out consistent.

In Brans-Dicke gravity with a small (i.e., Machian) value of $\omega$, if A and B are alone in their universe, then the value of the field $\phi$ is very small, and this is equivalent to having a very large local value of G. Therefore the two planets collide rapidly, and this is exactly what we expect in a Machian theory, where rotation is unobservable without distant bodies to compare against.

I think the asymmetric, Machian case is then a straightforward generalization of this. Planets A and B, with unequal masses, come closer and closer together. Since only relative motion is observable in a purely Machian theory, the only observable quantity is then the function d(t), where t is time and d is the distance between the two planets. We can't tell, and don't care, whether there's rotation, and we also can't tell whether one planet is accelerating more than the other.

Now suppose we have n>2 bodies, all with unequal masses, alone in a Machian universe. I think we can now tell that some bodies are accelerating more than others.

Thanks for the reply. I may not have understood your explanation well enough, but all in all...as an observer moving with the Earth in the proposed system, would one be able to appropriately determine the mass of the Sun using only the gravitational interaction?

 

[bcrowell]

Thanks for the reply. I may not have understood your explanation well enough, but all in all...as an observer moving with the Earth in the proposed system, would one be able to appropriately determine the mass of the Sun using only the gravitational interaction?

In the case of general relativity, or Brans-Dicke gravity?

 

[yuiop]

Thanks for the reply. I may not have understood your explanation well enough, but all in all...as an observer moving with the Earth in the proposed system, would I be able to appropriately determine the mass of the Sun using only the gravitational interaction?

In the Earth-Sun system the relativistic effects are very small so the Newtonian equations are good enough. The force of gravity acting on the Earth is F = GMm/r^2 where M is the mass of the Sun and m is the mass of the Earth. If this is expressed as a centripetal force then the equation is F = mv^2/r where v is the instantaneous tangential velocity. Equating these two expressions and solving for M gives M = v^2*r or in terms of angular velocity (where w = v/r) the result is M = w^2/r. Since the angular velocity of the Earth around the Sun is the same as the apparent angular velocity of the Sun around the Earth it is more convenient to measure the latter.

 

[bcrowell]

In the Earth-Sun system the relativistic effects are very small so the Newtonian equations are good enough.

Right, but this only applies to GR, not Brans-Dicke gravity.

Equating these two expressions and solving for M gives M = v^2*r or in terms of angular velocity (where w = v/r) the result is M = w^2/r. Since the angular velocity of the Earth around the Sun is the same as the apparent angular velocity of the Sun around the Earth it is more convenient to measure the latter.

I don't think this is quite right, since it clearly couldn't give the right answer in the case M=m. I think what you can get in terms of $\omega$ and r is the reduced mass, Mm/(M+m). The r appearing in $v=\omega r$ is not the same as the r appearing in Newton's law of gravity.

 

[Gear300]

In the case of general relativity, or Brans-Dicke gravity?

In GR.

Since the angular velocity of the Earth around the Sun is the same as the apparent angular velocity of the Sun around the Earth it is more convenient to measure the latter.

From the principle of relativity, wouldn't the two masses of the Sun be different?

 

[bcrowell]

Thanks for the reply. I may not have understood your explanation well enough, but all in all...as an observer moving with the Earth in the proposed system, would one be able to appropriately determine the mass of the Sun using only the gravitational interaction?

In GR.

In GR, if all you can measure is the angular frequency $\omega$ and the separation r, then in the weak-field limit, $\omega^2r^3=(...)\mu$, where (...) is a constant and $\mu=Mm/(M+m)$ is the reduced mass. Knowing $\mu$ doesn't tell you M or m. In order to find M and m I think you need a third test body.

In Brans-Dicke gravity with n isolated bodies existing in an otherwise empty universe, the weak-field limit doesn't apply.

 

[Gear300]

I see...does that mean that in either case, you would need a third test body?

 

[bcrowell]

I see...does that mean that in either case, you would need a third test body?

I think so, although I'm not certain that my analysis of the Brans-Dicke case is correct. I've often come to wrong conclusions by reasoning qualitatively about Brans-Dicke gravity.

In either theory, you clearly need *some* way of measuring something other than the separation and the frequency, because if all you can measure is those two quantities, there is nothing to tell you which mass is greater and which is smaller.

 

[PeterDonis]

I don't think your #8 is correct. In GR you can define asymptotic flatness without having to have distant objects...

Mathematically, yes, you can define it by just saying the metric is Minkowski at spatial infinity. But physically, IMO, that in itself is not a good enough argument. I should probably clarify that in this thread I'm talking about the *physical* aspects of the models we use, not what's mathematically possible per se.

In our current cosmological models of the universe as a whole, we do not (as you note later in your post) postulate any asymptotically flat region of spacetime, so all we can say about an "isolated" system like the solar system is that it is *approximately* asymptotically flat--the actual boundary condition is how the local metric from the Sun's field merges into the overall background metric for the universe as a whole, as I noted in an earlier post. (Of course this starts getting into quasi-Machian territory--for example, does *any* boundary condition make sense by the "physical" criterion I'm implicitly using here, unless the universe as a whole is spatially closed?)

...and in any case I don't think asymptotic flatness is relevant here. You don't need asymptotic flatness in order to determine your state of rotation. For example, Sciama figured out in 1967 that the CMB could be used to put an upper limit on the universe's rate of rotation, but the universe isn't asymptotically flat.

That's correct, it isn't. But there's still *some* boundary condition for the universe as a whole, which still has to have *some* physical reason. (IIRC, wasn't Sciama's argument dependent on assuming that the universe is spatially closed?)

 

[bcrowell]

That's correct, it isn't. But there's still *some* boundary condition for the universe as a whole, which still has to have *some* physical reason.

What kind of boundary condition do you have in mind? I don't think cosmological solutions really have boundary conditions in the sense that, say, the Schwarzschild solution had boundary conditions.

(IIRC, wasn't Sciama's argument dependent on assuming that the universe is spatially closed?)

The limit is much tighter for a closed universe than for an open one, but a limit can be determined for either case: http://adsabs.harvard.edu/full/1969MNRAS.142..129H

 

[PeterDonis]

What kind of boundary condition do you have in mind? I don't think cosmological solutions really have boundary conditions in the sense that, say, the Schwarzschild solution had boundary conditions.

Perhaps not quite in that sense, but they do, for example, have to specify what kind of initial state they're using (since the "initial singularity" that arises mathematically in the models is not physically realistic, so something has to be put in its place); normally one would call that an "initial condition", true, but in cases like, for example, Hawking's "no boundary" proposal for which quantum "histories" should count in determining the wave function of the universe, the distinction gets blurred.

Maybe a better way of saying what I'm trying to say is simply that the fact that a particular mathematical solution to the EFE exists does not, in and of itself, mean that the spacetime based on that solution is "physically realistic"--one has to actually look at what would be physically required for that spacetime to exist. In the case of the Schwarzschild solution, the asymptotically flat region at infinity is the sort of thing that is not "physically realistic" as an exact solution--one needs to either merge the solution (at some very large radius) into a larger-scale background, or account in some other way for why the asymptotic flatness assumption works to a sufficient approximation.

(Another example would be the "eternal black hole" spacetime that includes the entire maximal analytic extension, with the "white hole" as well as the "black hole" present. I don't think anyone claims that this solution is physically realistic, since there is no way to account for why the "white hole" exists.)

The limit is much tighter for a closed universe than for an open one, but a limit can be determined for either case: http://adsabs.harvard.edu/full/1969MNRAS.142..129H

Thanks for the link, I wasn't aware of this paper.

 

[bcrowell]

Perhaps not quite in that sense, but they do, for example, have to specify what kind of initial state they're using (since the "initial singularity" that arises mathematically in the models is not physically realistic, so something has to be put in its place); normally one would call that an "initial condition", true, but in cases like, for example, Hawking's "no boundary" proposal for which quantum "histories" should count in determining the wave function of the universe, the distinction gets blurred.

Hmm...I'm not trying to talk about anything quantum-mechanical. In GR, which is a classical field theory, cosmological solutions don't have an initial boundary. They're solutions that exist on a manifold, not a manifold-with-boundary, and to find them you don't need to postulate initial conditions -- all you need to do is specify some symmetries and an equation of state.

 

[PeterDonis]

Hmm...I'm not trying to talk about anything quantum-mechanical. In GR, which is a classical field theory, cosmological solutions don't have an initial boundary. They're solutions that exist on a manifold, not a manifold-with-boundary, and to find them you don't need to postulate initial conditions -- all you need to do is specify some symmetries and an equation of state.

Ok, but the mathematical solutions, as they stand, are not physically reasonable because they contain an initial singularity. For a physically reasonable solution you need to remove the singularity and replace it with some kind of reasonable initial state or equivalent "starting" condition--for example, an "eternal inflation" type of model where the overall "initial state" is the eternally inflating "false vacuum" and the initial state of the FRW model that we see as the "universe" is what comes out of the phase transition to the "true vacuum".

However, I will agree with you that there may not be many useful similarities between these types of conditions and a boundary condition like asymptotic flatness for the Schwarzschild solution. I would only observe, in relation to the question in the OP, that in situations of the former type--like the cosmological models--there is no good way of defining concepts like "the total mass of the universe" in any sense analogous to the sense in which we can define the "central mass" of an asymptotically flat spacetime.

 

[bcrowell]

Ok, but the mathematical solutions, as they stand, are not physically reasonable because they contain an initial singularity. For a physically reasonable solution you need to remove the singularity and replace it with some kind of reasonable initial state or equivalent "starting" condition--for example, an "eternal inflation" type of model where the overall "initial state" is the eternally inflating "false vacuum" and the initial state of the FRW model that we see as the "universe" is what comes out of the phase transition to the "true vacuum".

I guess it's a matter of taste whether something is physically reasonable. I don't see an initial singularity as being physically unreasonable.

 

[PeterDonis]

I guess it's a matter of taste whether something is physically reasonable. I don't see an initial singularity as being physically unreasonable.

Even though the density and curvature are infinite there?

 

[bcrowell]

Even though the density and curvature are infinite there?

Yes.

 

[PeterDonis]

Yes.

Hmm...okay, at least I'm clear about your position. I don't think it's a very common position to take (as I understand it, the "standard" position in GR is that singularities, such as the initial singularity of the FRW models and the r = 0 singularity of the Schwarzschild interior, are a sign that the theory breaks down, and a better theory is needed to cover such cases--presumably some kind of quantum gravity theory), but in the absence of any actual experimental evidence concerning singularities, I can't say it's ruled out.