Picture a small mass orbiting around a large mass

[Jonny_trigonometry]

Picture a small mass orbiting around a large mass. Now picture both masses at rest, but the large one is spinning. They are both the same system but viewed from different frames, Newtonian gravity says nothing about the angular momentum of the larger mass, so in the second referance frame it would still predict that the two masses will get pulled together over time, but we know that that will not happen, because the smaller mass is orbiting around the larger one. I'm betting GR would predict differently because the second referance frame is an accelerated frame. How does this relate with the predictions of GR and the machian interpretation of centrifugal force? Is there such thing as an unaccelerated referance frame, since there is always something in the universe that is spinning?

 

[turbo]

Is there such thing as an unaccelerated referance frame, since there is always something in the universe that is spinning?

Einstein thought so, and in this 1924 paper, he was working hard to incorporate this ether into GR.

In Newton's science of motion, space has a physical reality, and this is in strict contrast to geometry and kinematics. We are going to call this physical reality, which enters into Newton's law of motion alongside the observable ponderable bodies, the 'ether of mechanics'. The fact that centrifugal effects arise in a (rotating) body, the material points of which do not change their distances from one another, shows that this ether is not supposed a phantasy of the Netonian theory, but that there corresponds to the concept a certain reality in nature.

... Ernst Mach, who was the first person after Newton to subject Newtonian mechanics to a deep and searching analysis, understood this quite clearly. He sought to escape the hypothesis of the 'ether of mechanics' by explaining inertia in terms of the immediate interaction between the piece of matter under investigation and all other matter in the universe. This idea is logically possible, but, as a theory involving action-at-a-distance, it does not today merit serious consideration.

This paper is abolutely must-read material, but the English translation is not widely available. It is Chapter 1 in Saunders and Brown's 1991 book "The Philosophy of Vacuum".

 

[Kurdt]

The second picture you propose is an accelerated reference frame because you are accelerating with the smaller mass. While its true that Newtonian mechanics prefers inertial frames, its relatively easy to derive from Newtons equations the tools to deal with spinning reference frames. Thats where the centrifugal force was first predicted/explained.

 

[pervect]

GR is going to predict the same thing as Newtonian gravity in the weak field limit, i.e. if the spinning mass is the Earth, GR predicts that a geosynchronus satellites will orbit the Earth the same way that Newtonian theory does.

GR as currently formulated would predict the exact same thing if the Earth and the satellite were the only two objects in the universe. (Technically this sidesteps some cosmological issues, I'm assuming an infinite universe that's spatially nearly flat). Of course, this particular case hasn't been and can't be experimentally tested.

Basically, your "bet" or intuition is not correct about what GR predicts. I would guess that your intuition has been colored by Mach's principle, which may have influenced Einstein in his formulation of the theory, but is not a part of the theory. There is currently a lot of debate about what Mach's principle actually is, but other than to point out that Mach's principle (whatever it is) is not a part of GR, I'm not going to say much more.

 

[Jonny_trigonometry]

hmm, thanks a bunch. I didn't know that Newtonian gravity dealt with accelerated frames. I suppose it's not the Newtonian theory that deals with them, but the person using Newtonian theory which deals with them. If I had spent the time and effort I would've been able to solve this with Newtonian gravity.

suppose the second referance frame is unaccelerated, then the two masses will fall towards each other right?

 

[Jonny_trigonometry]

to turbo-1
This kind of stuff really gets me confused. We can either referance all the matter surrounding a particular body, or we can referance all matter to space itself. Are matter and space mutually inclusive, or mutually exclusive? Can space permeate matter, or is matter a point of discontinuity of space?

 

[turbo]

This kind of stuff really gets me confused. We can either referance all the matter surrounding a particular body, or we can referance all matter to space itself. Are matter and space mutually inclusive, or mutually exclusive? Can space permeate matter, or is matter a point of discontinuity of space?

Einstein firmly rejected Machian action-at-a-distance. Gravitation, inertia, centrifugal effects, etc all arise from matter's interaction with local space. From "On the Ether" and Einstein's Leyden Address, the properties of space are conditioned by the matter embedded in it, and the intrinsic properties of matter such as mass, gravitation, inertia, etc result from the interaction of matter with space.

 

[pervect]

The laws of physics do not depend on the coordinate system, much as the terrain does not depend on the map.

If we are 100 miles away from Paris, and we get out a new map, we are still 100 miles away from Paris. The same thing holds true about rotating vs non-rotating coordinate systems. Describing the physics in a non-rotating coordinate system will give the same answer as describing the physics in a rotating coordinate system (when we translate the coordiantes from one map to the other).

This is sometimes called "general covariance", and is an important property of physics.

There are several methods of doing Newtonian physics in arbitrary coordinate systems, the Lagrangian formalism is one of them. In introductory courses students are advised not to attempt this, because the same result can always be gotten by doing the problem in a non-rotating coordinate system and the answer will be the same. For harder problems and more advanced students, this restriction is usually eased.

[clarify]
The basic point is very simple the question of whether a satellite falls down into the primary or orbits always has one physical solution, one that depends on the initial conditions.

We can compute a theoretical solution to this physical problem with Newtonian mechanics (which will give an accurate solution in the realm where it applies), or General Relativity (the best theoretical explanation we have).

And in the case of reasonably weak fields, these two soluitons will be essentially identical (this is not the case in general, but it requires very strong fields for the GR solution to diverge significantly from the Newtonian solution.)

Because physics actually works, the theoretical solutions match the physical solutions to within experimental accuracy.

The numbers we use to describe these initial conditions change as we change our coordinate system, but the underlying physical reality does not change, and always acts in the same manner.

In short, changing the "map" (the coordinates we use to describe reality) does not change the underlying reality itself (the behavior of physics). I.e. "the map is not the territory".

 

[Jonny_trigonometry]

hmm... I understand your analog of maps and terrains, but I still have a problem with this question, because viewing a large rotating sphere at the origin of a coordinate system with smaller body lying a distance r from it (either rotating or not) makes me think that they will "fall" towards each other, but they won't if the large body is rotating sufficiently fast (ie, the smaller body is moving sufficiently fast around the larger one to not fall towards it, but orbit around it) so as to provide a momentum to the other body capable of keeping it in orbit around the large body. It's almost like the spin of the large body affects the amount of attraction between the two bodies when viewing the "terrain" with this "map" but when you view the "terrain" with a different "map" where the large body isn't rotating, but the smaller body is orbiting around the large body, you see that it's not the angular momentum of the large body that causes the smaller object not to fall towards the larger object, but it's the momentum of the smaller object that keeps it in orbit around the larger object. So you've got two different maps viewing the same terrain, but there can a different interpretation of physical laws when viewing the terrain from different maps. The only difference between the coordinate systems is that one sees the stars circling around the origin at a very fast speed, and the other sees them circling at a much slower speed, possibly even stationary. If you don't take into account the background stars, you may develop a different explanation of physical laws ("terrain") when viewing from different coordinate systems ("maps").

So there is only one solution that can be arrived at based on initial conditions. You are then taking into account the background stars' motion with the initial conditions? Or is that superflous, and unnecessary? I would hope that is isn't necessary... I'm probably just missing something obvious, I hope. Because if what you say is true, and you don't need to account for the stars' motion, then the paper I'm working on has more validity, but if you do need to take into account the motion of the background stars, then my paper is going in the trash.

 

[Jonny_trigonometry]

in GR, is mass defined as separate from space, or as a consequence of how space is configured, or both? Does matter exist in space, or is a point particle a point of discontinuity of space, ie a singularity at it's center, but curved space around that center?

 

[pervect]

hmm... I understand your analog of maps and terrains, but I still have a problem with this question, because viewing a large rotating sphere at the origin of a coordinate system with smaller body lying a distance r from it (either rotating or not) makes me think that they will "fall" towards each other, but they won't if the large body is rotating sufficiently fast (ie, the smaller body is moving sufficiently fast around the larger one to not fall towards it, but orbit around it) so as to provide a momentum to the other body capable of keeping it in orbit around the large body.

We seem to have some sort of communication problem here.

If the smaller body is moving sufficiently fast around the larger one it, the smaller body will orbit in a circle.

If the smaller body is not moving sufficently fast around the larger one, it will orbit in an ellipse. If the ellipse isn't wide enough to avoid the larger body and its atmosphere, the smaller body will hit the larger body.

There are some extremely tiny relativistic effects, but basically the spin of the large body is almost totally irrelevant to how the smaller body falls.

(Exception: rotating black holes. But you seem to be talking about things that happen on the scale of satellites orbiting the Earth).

What counts is the motion of the smaller body, which is totally different from the spin of the large body.

It's almost like the spin of the large body affects the amount of attraction between the two bodies when viewing the "terrain" with this "map" but when you view the "terrain" with a different "map" where the large body isn't rotating, but the smaller body is orbiting around the large body, you see that it's not the angular momentum of the large body that causes the smaller object not to fall towards the larger object, but it's the momentum of the smaller object that keeps it in orbit around the larger object.

It is the angular momentum of the smaller object that keeps it in orbit around the larger object.

I'm not quite sure what your problem is, I suspect you are imagining that general relativity incoroporates Mach's principle. It does not - while GR allows one to work in rotating coordinates, it does so by incorporating Christoffel symbols that essentially duplicate the Newtonian ideas of "centrifugal force" and "coriolis force" in any rotating coordinate system.

 

[pervect]

I see this is a duplicate post. Perhaps the best home for it could be determined, and the threads merged?

 

[pervect]

in GR, is mass defined as separate from space, or as a consequence of how space is configured, or both? Does matter exist in space, or is a point particle a point of discontinuity of space, ie a singularity at it's center, but curved space around that center?

GR does not deal gracefully with point particles. Mass appears in GR as a "mass density", i.e. mass per unit volume. (Actually, energy is what's important for GR, so it's really an energy density). Energy density is only one parameter in a larger entity, the "stress-energy tensor".

The complete stress-energy tensor in GR contains the energy density, the momentum density, and the pressure. Knowledge of all of these components allows one to compute the stress-energy tensor in any coordinate system one desires if it is specified in any coordinate system.

Note that knowing only the energy density in one coordinate system is not sufficient information to determine the energy density an arbitrary coordinate system - the energy density alone is not a complete description of the properties of matter at some point in space. The stress-energy tensor, however, is a complete description of the properties of matter at some point in space.

Space is separate from mass. Space is defined with a metric, which tells you how to compute the distance between any two points.

However, there is an equation that relates the curvature of space (as defined by the metric) to the stress-energy tensor (which includes mass densities, momentum densities, pressures, etc). This is the Einstein field equation

$$G_{uv} = 8 \pi T_{uv}$$

The quantity on the left, $G_{uv}$ depends only on the curvature of space, and is derived from the metric. The quantity on the right, $T_{uv}$, depends only on matter and is derived from (actuall it is equal to) the stress-energy tensor. This is GR in a nutshell.

A popular description of GR from John Wheeler

"Matter tells space how to curve, and space tells matter how to move".

 

[turbo]

in GR, is mass defined as separate from space, or as a consequence of how space is configured, or both? Does matter exist in space, or is a point particle a point of discontinuity of space, ie a singularity at it's center, but curved space around that center?

Singularities (point particles) are tough, so let's take a simple case:

Imagine you have a lump of matter floating out in space, and it is rotating. The matter is subjected to centrifugal effects due to its rotation. In cases where the lump of matter is large and sufficiently plastic (capable of deformation) and the rotation is sufficiently rapid, the centrifugal effects will cause the lump to assume the shape of an oblate spheroid (larger around the equator, flattened at the poles).

How does the centrifugal force arise? The matter comprising the lump is relatively fixed and the particles making up the lump are not rotating with respect to one another - they are moving in unison, so absent a background, the body cannot be said to be rotating and cannot experience centrifugal effects. The rotation of the body has to be expressed against some background, else how does centrifugal force arise? Newton would have explained that the body was rotating with respect to a fixed mechanical ether. Ernst Mach was uncomfortable with a mechanical Newtonian ether, so he postulated that centrifugal force, inertia, etc, arose from the body's rotation or acceleration with respect to the totality of all the other massive bodies in the universe. Einstein rejected this notion because it required instantaneous action-at-a-distance and he had already convinced himself that nothing can travel faster than the speed of light. This left him with one inescapable truth - inertia, centrifugal force, and indeed gravitational attraction and mass itself are the result of matter's interaction with the local space-time in which it is embedded. These basic properties of matter are not intrinsic, but emergent, arising from matter's relationship to local space-time.

Einstein called this local background the "ether" of general relativity, which was unfortunate. He should have had a public-relations person think up a snazzy name for the local background that would catch the public interest. "Ether" (or "aether") had grown entirely unfashionable by the 1920s, and the words carried connotations from earlier theories that were entirely inappropriate for Einstein's GR.

The short answer to your question is that matter does not exist apart from space, and in fact derives its basic properties from its relationship to space.

 

[Jonny_trigonometry]

to turbo-1
thanks a lot, this helps a bunch with my understanding of GR.

Would it be safe to say that GR is incomplete, and has some problems? It seems that maybe there IS a quantum gravity theory that emerges into GR with a similar type correspondance principle as QM with classical physics. GR deals with an energy density, so I take that as a summation of a bunch of point particles within a given volume, but each point particle is (in my mind) a singularity (if it doesn't have a radius), so they are all playing "tug of war" but since you can't know the position and momentum at the same time, you have to take the average position and momentum of the group of particles within a specified amount of space, then you end up with a gaussian type energy-density of all those particles... I would guess that on smaller and smaller scales, the gaussian gets a smaller and smaller standard deviation (higher and higher energy density for a smaller and smaller volume) until it turns into a delta function centered on a single point particle? (ya I know, I kind of went off on a tangent, this is merely for conversation)

If gravity were instantaneous, then there wouldn't be an uncertainty principle, and you could model a system of particles as singularities, but then GR wouldn't be needed since between them all is a flat space-time. Then again, all the particles are moving at fast speeds with respect to each other, so SR comes into play, come to think of it, can you interpret GR as the average effects from SR acting on all particles (rather than using the principle of equivalence)? (this is "out there" even more I know)

 

[Jonny_trigonometry]

If the smaller body is moving sufficiently fast around the larger one it, the smaller body will orbit in a circle.
If the smaller body is not moving sufficently fast around the larger one, it will orbit in an ellipse. If the ellipse isn't wide enough to avoid the larger body and its atmosphere, the smaller body will hit the larger body.

Yes, I totally follow you here, this is exactly how I understand orbits.

There are some extremely tiny relativistic effects, but basically the spin of the large body is almost totally irrelevant to how the smaller body falls.
(Exception: rotating black holes. But you seem to be talking about things that happen on the scale of satellites orbiting the Earth).
What counts is the motion of the smaller body, which is totally different from the spin of the large body.

Yes, I agree. This all fine and good. My problem emerges when you view only the two bodies, and not anything else, as if they are the only two things in the universe. Then when I pick a frame where the large body is at the origin, and the orbiting body is (lets say in a pervect circular orbit) sitting at a distance r on the x-axis (seemingly), so the large body is must be spinning with it's axis of spin perpendicular to the x-axis. This is my problem, this is the same system but viewed in a frame that is orbiting at the same speed as the orbiting body (or spinning at the same rate as the angular velocity of the orbiting body, and looking down on the origin). When you have a non-circular orbit, in this frame it looks like the orbiting body is occilating back and forth along the x-axis while the larger body has a non-constant spin rate which occilates in resonance with the orbiting body.

I think this problem is resolved though, thanks to your help, because you've mentioned enogh for me to put together into a reason why this isn't a problem. Since GR referances spin to space-time as if space-time is an ether (in the same sense as Newton referanced spin to "absolute space"), there is no need for a Machian principle. So when you see that in that frame, space-time is rotating, then there is no problem.

 

[Phobos]

jonny - please don't duplicate topics in different forums.

I have merged the two topics & tried to fix some of the discussion continuity. But I'll leave the rest for you to figure out.

 

[Jonny_trigonometry]

ok, my bad. I just kinda wanted to keep them going since they were going in different directions. I admit, I created the second one just because I didn't want to see the first one turn out being a dead thread... sorry 'bout that.:shy:

 

[pervect]

My problem emerges when you view only the two bodies, and not anything else, as if they are the only two things in the universe. Then when I pick a frame where the large body is at the origin, and the orbiting body is (lets say in a pervect circular orbit) sitting at a distance r on the x-axis (seemingly), so the large body is must be spinning with it's axis of spin perpendicular to the x-axis. This is my problem, this is the same system but viewed in a frame that is orbiting at the same speed as the orbiting body (or spinning at the same rate as the angular velocity of the orbiting body, and looking down on the origin). When you have a non-circular orbit, in this frame it looks like the orbiting body is occilating back and forth along the x-axis while the larger body has a non-constant spin rate which occilates in resonance with the orbiting body.
I think this problem is resolved though, thanks to your help, because you've mentioned enogh for me to put together into a reason why this isn't a problem. Since GR referances spin to space-time as if space-time is an ether (in the same sense as Newton referanced spin to "absolute space"), there is no need for a Machian principle. So when you see that in that frame, space-time is rotating, then there is no problem.

You may run into trouble further down the line if you take the idea of space-time as an ether too seriously, but it is correct to note that GR can distinguish rotating coordinate systems from non-rotating ones. This shouldn't be too surprising, since experiments can distinguish between a rotating coordinate system and a non-rotating one (i.e. you can put a laser gyroscope in a box, and tell if the box is rotating).

The idea of space-time as some sort of ether is what allows one to talk about space-time "swirling" around a rapidly rotating balck hole in many popular analogies.

Places where the ether idea may mislead you are in thinking that linear motion can be detected (it can't be), and in understanding what people mean by "expanding space time".

 

[Jonny_trigonometry]

it (linear motion) can only be detected relative to other frames right? If you look at an object moving past you, you can tell it has linear motion w/respect to your space-time coordinates, so there is a space flux moving through the object when you use your coordinates. This space flux causes it to contract and move through time slower.

What if you could stay at a specific altitude from the sun without orbiting it, and there was a detector much closer to the sun in the same radial ray, and a detector much further away on the same radial ray. All three positions require a constant thrust to keep you in the same position. Suppose the middle position emmits two photons which travel to the positions close to, and far away from the sun, would the position close to the sun see a blue shift whereas the position far from the sun would see a redshift?

If this is true (which I'm betting it is), could you then make a principle of equivalence of space flux near the presence of matter? Since what happened in the experiment is equivalent to moving through space at different speeds as a function of distance from a mass. Then could you say that the sun pulls in space-time at a rate proportional to it's mass and the distance from it? Hence concluding that all particles do the same?

This gives me a picture of space-time as kind of like a fluid which has a viscosity that interacts with matter in such a way that it only puts force on matter in the opposing direction of the matter's acceleration, but somehow is frictionless otherwise. But, when space-time itself is accelerating, towards some "drain" like the sun, then the opposite happens, and it takes force to move at a constant speed through space-time (seemingly contradicting Newton's 1st law), but it's only a problem of viewing the "terrain" with a different "map", because if you had a frame that was attatched to space-time as it acclerates towars the sun, you would see that the the three previously mentioned positions are all accelerating w/respect to space-time. The three positions were moving through space-time at constant rates (in the frame at rest w/respect to the sun), but felt acceleration because space-time itself was accelerating towards the sun at each point, but if they stopped opposing the accelerating flow of space-time, they would flow in the direction of space-time's acceleration.

How different is this than GR?

what if you calibrated the lasar gyroscope in a spinning frame?

 

[pervect]

it (linear motion) can only be detected relative to other frames right? If you look at an object moving past you, you can tell it has linear motion w/respect to your space-time coordinates, so there is a space flux moving through the object when you use your coordinates. This space flux causes it to contract and move through time slower.

You could define a "rest frame" if you like, though there's nothing special that singles it out experimentally, and you can define a vector field that gives the velocity of an object relative to this "rest frame".

It's not very clear what you'd do with this, though. The flux through any 3-d surface will be zero. A 2-d surface perpendicular to the vector field will NOT experience any contraction effects, but WILL have a flux. A 2-d surface that is parallel to the velocity WILL have a reduction in area and have ZERO flux.

So you'd only have contraction in a 2-d surface when there is no flux. This doesn't seem especially promising to me for making this idea work.

Basically, it'd really be better if you tried to learn relativity, rather than making up your own theories.

What if you could stay at a specific altitude from the sun without orbiting it, and there was a detector much closer to the sun in the same radial ray, and a detector much further away on the same radial ray. All three positions require a constant thrust to keep you in the same position. Suppose the middle position emmits two photons which travel to the positions close to, and far away from the sun, would the position close to the sun see a blue shift whereas the position far from the sun would see a redshift?

Yes, the photons falling down the gravity well will be blueshifted, and the photons falling upwards will be redshifted.

If this is true (which I'm betting it is), could you then make a principle of equivalence of space flux near the presence of matter?

I don't see any particular reason to try and explain things in this manner.

Sorry, but I'm feeling overdosed on speculation and "what if's" here.

But I can try to say a few things about rotating frames in relativity. The basic tool that describes space-time is the metric. The metric is a tensor. A tensor has no concept of rotation. More specifically, the components of a tensor are unaffected _at the origin of the coordinate system_ by the choice of rotating or non-rotating coordiantes. When you specify a basis of vectors at a point, you specify the values of a tensor quantity. The rotation or lack of rotation of this basis does not affect any of the components of a tensor.

Where rotation first comes into the picture is with the Christoffel symbols, which are not tensors. These Christoffel symbols can tell us whether or not the choice of coordinates we picked is rotating or not rotating, depending on the value of certain components of the symbols at the origin of the coordinates.

Christoffel symbols can be calculated by an appropriate sum of first-order derivatives of the metric tensor.

The presence of rotation in the x-y plane can be identified with non-zero values of the Christoffel symbols $\Gamma^x{}_{ty}$ and $\Gamma^y{}_{tx}$, for instance.

Physically, given the principle that objects follow geodesics (this is true in GR), these Christoffel symbols correspond to coriolis forces. If an object experiences an acceleration in the x direction that's proportional to it's y velocity, and an inverse acceleration in the y direction that's proportional to it's x velocity, we can say that the system has a component of rotation in the x-y plane.

The Riemann curvature tensor can be defined from the Christoffel symbols, but because it is a tensor, the value of all of its components are independent of rotation around the origin (rotation about another point can be detected, but not rotation around the point where the tensor is measured). The Riemann is composed of second derivatives of the metric tensor, and some non-linear terms that involve the products of two first derivatives.

Einsteins equation then expresses one tensor quantity, the Einstein tensor (derived from the Riemann, a contraction), in terms of another tensor quantity, the stress-energy tensor. By definition, these tensor quantites at a point are not affected by rotation around the origin.

 

[Jonny_trigonometry]

Hmm, so only the Christoffel symbols can tell if the frame is rotating. If I only knew what all those tensors were about... I haven't taken riemannian geometry, I don't know if I'll be able to since my next 2.5 years of school are going to be packed, but I know I'd be able to understand it much better if I took a class on it. I guess my question is, is it possible to express an equivalent GR using different relations like the equivalence I'm offering? Rather than viewing space as bent, view it as flowing, and instead of an amount of bending (curvature tensor?), it would be an amount of flux (flux tensor?).

You could define a "rest frame" if you like, though there's nothing special that singles it out experimentally, and you can define a vector field that gives the velocity of an object relative to this "rest frame".

It's not very clear what you'd do with this, though. The flux through any 3-d surface will be zero. A 2-d surface perpendicular to the vector field will NOT experience any contraction effects, but WILL have a flux. A 2-d surface that is parallel to the velocity WILL have a reduction in area and have ZERO flux.

So you'd only have contraction in a 2-d surface when there is no flux. This doesn't seem especially promising to me for making this idea work.

Basically, it'd really be better if you tried to learn relativity, rather than making up your own theories.

Picture this, an object moving through space. Now change your frame of referance, and view it as at rest in this new frame, but space is moving through the object. It's the same "terrain", but different "map". Now find a relation between amount of flux through that object and velocity, it's going to be proportional.

I have formally learned special relativity, but not general relativity. I'm saying the same thing as "object moving through space" when I speak of flux of space moving through an object. You can calculate a 3d flux or a 2d flux, but you'll get the same answer for the amount of area perpendicular to the direction of space flow through the object... or object moving through space. It's easiest to calculate flux with a 2d area perpendicular to the direction of travel (for an approximate answer). The only reason why I'm viewing it this way is for the equivalence I'm offering, of time dilation and length contraction caused by an object at a distance from a mass (in a locally flat space), and time dilation and length contraction caused by an object moving through space.

So back to the main question, can this be done (regardless if it offers nothing more than GR already offers)?

 

[Jonny_trigonometry]

Come to think of it, the way to calculate the relation between space flux and velocity must be independant of the area that you use to calculte flux, so it's not going to be simply space flux, but rather space flux density, and the corresponding tensor relating this with a mass is a space flux density tensor. you know? So am I retarded, or stubborn?