# Do moving massive objects drag curved spacetime with them?

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PeterDonis
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the trajectory of the particles is slightly bent in the direction of the rotation.
"Bent" might convey a wrong impression. What happens is that, instead of falling straight in towards the mass, the infalling particle acquires a gradually increasing angular velocity in the direction of rotation of the mass. So its trajectory in space looks like an ingoing spiral.

timmdeeg and vanhees71
a) not rotating: the trajectory of the particles is a straight line towards the center of the mass.
b) rotating: the trajectory of the particles is slightly bent in the direction of the rotation.
“Rotating” or “not rotating” against what? What is the frame of reference with respect to which the mass is rotating (or not)? Other stars? Microwave back ground? The space time itself?

PeterDonis
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“Rotating” or “not rotating” against what?
With respect to an observer at infinity who is at rest with respect to the center of mass of the massive object.

With respect to an observer at infinity who is at rest with respect to the center of mass of the massive object.
Any observer at rest with respect with the center of mass of the massive object by definition is rotating (moving) with that object unless it is non-rigidly connected with that massive object.

Any observer at rest with respect with the center of mass of the massive object by definition is rotating (moving) with that object unless it is non-rigidly connected with that massive object.
That rest frame observer will not be able to determine the state of motion of the massive object from within its rest frame. However , a stationary observer (not moving with the center of the mass of the massive object hence one who has a relative velocity with respect to the massive object) will be able to determine a relative velocity. Which bring us to the original question. Do moving object carry/ drag the curved spacetime in their vicinity?

Nugatory
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Which bring us to the original question. Do moving object carry/ drag the curved spacetime in their vicinity?
Do not confuse rotating and moving. Rotation is not relative - it can be measured with an accelerometer without reference to anything external or requirement to choose any particular reference frame. Frame dragging will happen in the vicinity of a rotating object.

But a massive body moving past you is a different problem. That motion is relative - we could just as reasonably say that the body is at rest while you are moving past it. So we have two descriptions of the same physical situation, and because it’s the same physical situation the spacetime curvature has to be the same. So no, the moving body will not produce any sort of frame-dragging effect. (This is assuming that your mass is negligible - two massive bodies moving past one another is a completely different problem and not what you started this thread with).

Do not confuse rotating and moving
I put moving in the bracket to indicate that a rest frame observer with respect to the center of the mass of the massive object, moves by definition with the center of the mass whether it’s rotating or linearly moving.

I put moving in the bracket to indicate that a rest frame observer with respect to the center of the mass of the massive object, moves by definition with the center of the mass whether it’s rotating or linearly moving.
I feel that my original question is analogous to the question that started it all in the 19th century and led to Michelson and Morley experiment of course with different terminology. They asked if there was a relative motion between the Earth and the ether. Gravity probe B asked if there was a frame dragging (linear or rotational) by massive objects. And I am not suggesting that ether exists or that curved spacetime is ether. But are we not trying to find the same thing in reverse ? Namely the rotation of curved spacetime with respect to earth from within the rest frame of the earth?

PeterDonis
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Any observer at rest with respect with the center of mass of the massive object by definition is rotating (moving) with that object unless it is non-rigidly connected with that massive object.
No, it's not that simple.

We are up against the limitations of ordinary language when talking about physics. So I'll have to take a step back and give a more technical explanation. Hopefully it will still be close enough to ordinary language to make sense.

The technical explanation starts with the concept of an asymptotically flat spacetime: this is a spacetime which, as you go out to infinity, becomes flatter and flatter, so that it is flat "at infinity". However, this is still not the same as flat Minkowski spacetime, because the massive object at the center is still there, and that object defines a unique "rest frame" with respect to the asymptotically flat spacetime, which is a frame that is inertial at infinity and in which the center of mass of the massive object is at rest. Or, to put it another way, timelike worldlines at rest in this unique rest frame, not just at infinity but anywhere, are purely "vertical"--they don't wind around the massive object like a helix. (The still more technical way of expressing this is that the worldlines "at rest" in this unique rest frame are the integral curves of a timelike Killing vector field.)

The above is true whether the massive object is rotating or not. But we can still tell whether the massive object is rotating by looking at whether the worldlines "at rest" in the unique rest frame above are orthogonal to the spacelike 3-surfaces of constant time in the frame. If they are, the massive object is not rotating; if they aren't, it is. The physical effects associated with "frame dragging" near rotating objects are fundamentally due to the worldlines "at rest" in the unique rest frame above not being orthogonal to the surfaces of constant time in that frame. But an easier way to picture the "rotating" vs. "non-rotating" distinction is to simply imagine the observers who are at rest in the unique rest frame above watching the massive object beneath them: if they see the object rotating, it's rotating; if they see it not rotating, it's not rotating. That is why I said that "rotating" is defined with respect to such an observer (the one at infinity).

Gravity probe B asked if there was a frame dragging (linear or rotational) by massive objects.
No, only rotational. Gravity probe B did not investigate "linear frame dragging" at all.

alantheastronomer and hnaghieh
vanhees71
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but a moving charge creates a magnetic or electric field when it moves through a medium. If the medium moves with the charge no effect can be found.
No, a moving charge in vacuum has always its electromagnetic field around it as described by the "microscopic" Maxwell equations.

timmdeeg
pervect
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In the weak field, one can make an analogy between electromagnetism and gravity with a few other assumptions also necessary such as low velocities and others. This can be formalized as "gravitoelectromagnetism" or GEM for short. There's a wiki overview at https://en.wikipedia.org/w/index.php?title=Gravitoelectromagnetism&oldid=970487534

If we use this weak field analogy, we can regard gravity as a force, the coulomb force of electromagnetism between charges is, except for a necessary minus sign, is analogous to the gravitational force between masses. The sign issue comes into play because like charges repel, but "gravitational charge" is always positive, and two objects with positive masses, positive "gravitatioanl charge", attract each other, they do not repel each other.

If we have two moving charges, covariance demands that in the rest frame of said particles, there is only an electric force, while in a frame moving relative to the two charges, there is both an electric component and a magnetic component to the force.

Similar reasoning applies to gravity in the weak field using the GEM approximations.

So if we consider two stationary masses, we have only the "electric" GEM components betwen them. If we go to a frame in which the masses are moving, we find that there is both an "electric" force and a "magnetic" force. This is necessary for covariance, for our choice of frame not to matter to things we can observe.

In the strong field, where GEM doesn't work things are not so easy. There is an approach I like involving the Bel decomposition of the Riemann tensor, but I think what I'd write on that would not be helpful without a detailed knowledge of the Riemann tensor and tensor mathematics. I'm guessing that this is not a background you share, so it wouldn't be productive to go into it.

I would concentrate on understanding the origin of the magnetic force in special relativity first, and why it is needed for covariance. Then you can consider linear frame dragging to be a similar phenomenon - something that we need to make our theory covariant.

As far as experiment goes, though, we've directly confirmed frame-dragging from the rotating earth.

vanhees71 and alantheastronomer
if they see the object rotating, it's rotating; if they see it not rotating, it's not rotating.
Any observer who “see an object not rotating” could be at relative rest with respect to rotating object’s rest frame (rotating with them), therefore their “observations “ can not be considered as privileged and absolute. This was indeed one of the starting points of General relativity. The need for extension of the principle of relativity. I feel there is still a prevailing notion of spacetime as a “container” where “asymptotically flat space time”, a lorentzian manifold, “killing vector space”, “virtual particles”, “ world-lines”, “massive objects” etc. live, and move, which conflicts with everything STR and GR have said. That is what I am trying to clarify for myself.

weirdoguy
I'm guessing that this is not a background you share, so it wouldn't be productive to go into it.
I am not an expert but I am familiar with the mentioned topics. I would appreciate if you do include it in a step by step fashion so even I could follow it. Thank you.

PeterDonis
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Any observer who “see an object not rotating” could be at relative rest with respect to rotating object’s rest frame (rotating with them), therefore their “observations “ can not be considered as privileged and absolute.
The "rest frame" I described is a physical property of the spacetime. In other words, while it is true that the laws of GR do not privilege any particular choice of frame, particular spacetimes that are solutions of those laws can have particular frames that are "privileged" in the sense that they match up with properties of the spacetime.

However, if you don't like my description in terms of this "rest frame", it is easy to restate it without using frames at all: An asymptotically flat spacetime that contains a non-rotating massive object has a property (hypersurface orthogonality of the timelike Killing vector field) that an asymptotically flat spacetime that contains a rotating massive object does not have. The presence or absence of this property is a geometric fact about the spacetime that is independent of any choice of reference frame.

I feel there is still a prevailing notion of spacetime as a “container” where “asymptotically flat space time”, a lorentzian manifold, “killing vector space”, “virtual particles”, “ world-lines”, “massive objects” etc. live, and move, which conflicts with everything STR and GR have said.
You are mistaken.

Spacetime is a dynamic geometric entity in GR--its geometry is determined by the distribution of stress-energy via the Einstein Field Equation. Properties like "asymptotically flat" and "timelike Killing vector field" are properties of particular solutions of the Einstein Field Equation--particular spacetime geometries that are determined by particular distributions of stress-energy (which is what the terms "non-rotating massive object" and "rotating massive object" that I used above describe--distributions of stress-energy that determine the spacetime geometry via the Einstein Field Equation).

"Worldlines" do not "move"; they are curves in the 4-dimensional spacetime geometry.

"Virtual particles" are a quantum concept and are off topic in this forum, nor do they have anything to do with what is being discussed in this thread.

particular spacetime geometries that are determined by particular distributions of stress-energy (which is what the terms "non-rotating massive object" and "rotating massive object" that I used above describe--distributions of stress-energy that determine the spacetime geometry via the Einstein Field Equation).
I agree. But may I ask you a question? Isn’ this distribution of stress energy that determines the spacetime geometry via Einstein’s Field Equation reference frame dependent? Aren’t The deviations from orthogonal timelike worldlines a geometric representation of the relative velocity of two respective frames under consideration?

Nugatory
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Isn’ this distribution of stress energy that determines the spacetime geometry via Einstein’s Field Equation reference frame dependent?
No.
You should be thinking of a “frame” as a convention for assigning coordinate values to points in spacetime (not points in space!). Different frames will label points in spacetime differently, but changing the labels changes neither the spacetime nor the distribution of matter across that spacetime. An analogy: Using a different frame is like changing where we draw the zero meridian on the surface of the earth: every feature on the surface of the earth will be at a different longitude, but the relationship between these features will be unchanged.
Aren’t the deviations from orthogonal timelike worldlines a geometric representation of the relative velocity of two respective frames under consideration?
I don’t understand what you’re trying to say here - I know what “relative velocity”, “orthogonal”, “timelike worldline”, “frames”, and “deviation” mean, but you have strung these words together in a way that makes no sense.

You might want to give Taylor and Wheeler’s book “Spacetime Physics” a try; it will help give you a solid grasp of some of the concepts that you’re misunderstanding.

PeterDonis
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Isn’ this distribution of stress energy that determines the spacetime geometry via Einstein’s Field Equation reference frame dependent?
No. The whole point of using 4-vectors and tensors to describe things like the spacetime geometry (the metric tensor) and the distribution of matter and energy (the stress-energy tensor) is that such a description is frame-independent. That allows us to write physical laws in frame-independent form, as the principle of relativity requires.

Aren’t The deviations from orthogonal timelike worldlines a geometric representation of the relative velocity of two respective frames under consideration?
No. As I said, it is a frame-independent geometric property of the spacetime. The frame-independent geometric object that describes it is the metric tensor.

hnaghieh
PeterDonis
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The whole point of using 4-vectors and tensors to describe things like the spacetime geometry (the metric tensor) and the distribution of matter and energy (the stress-energy tensor) is that such a description is frame-independent. That allows us to write physical laws in frame-independent form, as the principle of relativity requires.
Btw, since electromagnetism has also been mentioned in this thread, it is worth noting that the physical laws of EM--Maxwell's Equations and the Lorentz force law--can also be written using 4-vectors and tensors in frame-independent form. The relevant objects are the charge-current 4-vector and the EM field tensor (an antisymmetric 2-index tensor).

You should be thinking of a “frame” as a convention for assigning coordinate values to points in spacetime (not points in space!). Different frames will label points in spacetime differently, but changing the labels changes neither the spacetime nor the distribution of matter across that spacetime
So are you saying( I think you are not) that two observers in two frames K and k with relative motion will agree on simultaneity of two world events? Same as when they use the same frame(no relative motion between them)? My understanding Tensors is that Tesors are invariant object therefore they represent physical laws. They represent “relationships “ between the Compnents of tensor(vectors) while those components are frame dependent (the tensor itself represent invariant transformation relationships). An observer in the rest frame k of a moving object will not agree with observation of another observer at rest in frame K, not moving with frame k (at rest in a different frame)when they compare.

Nugatory
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So are you saying( I think you are not) that two observers in two frames K and k with relative motion will agree on simultaneity of two world events?
You are right that I am not saying that. "Simultaneous" means "has the same time coordinate", so is frame dependent and tells us almost nothing about the relationship between two events in spacetime. The frame-independent concepts that describe this relationship are "spacelike separated", "lightlike separated", and "timelike separated".

(Why did I say "tells us almost nothing"? It turns out that if two events are spacelike separated in flat spacetime, then with a bit of algebra we can show that there exists an inertial frame in which two events have the same time coordinate. But it's the spacelike separation that is the physical fact in this situation, and the simultaneity is a just a happy but limited consequence).

PeterDonis
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the Compnents of tensor(vectors)
Vectors are not components of tensors. Vectors are invariant objects just like tensors; they are best thought of as one-index tensors.

An observer in the rest frame k of a moving object will not agree with observation of another observer at rest in frame K, not moving with frame k (at rest in a different frame)when they compare.
You need to stop thinking in terms of frames; it is only confusing you.

What is happening physically is simply that observers in relative motion will make different observations--for example, they will measure light signals coming from the same source to have different frequencies. But all of these observations, including all the differences in the observations made by observers in relative motion, can be described entirely in terms of invariants. There is no need to bring frames into it at all.

Frames are a convenience for calculations, but conceptually they often cause more confusion than they solve.

Nugatory
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An observer in the rest frame k of a moving object will not agree with observation of another observer at rest in frame K, not moving with frame k (at rest in a different frame)when they compare.
That is not correct, although it is a very common misunderstanding (perhaps the root of all relativity misunderstandings). In fact, they will agree about every observation either makes - otherwise we would have hopeless paradoxes in which (for example) whether a bug is squashed or not is frame-dependent. The difference is in how they explain these observations.

For example: I'm sure you've read in a million popular treatments of relativity that if you and I are moving relative to one another, you will "observe" that a meter stick at rest relative to me is length-contracted to the shorter length ##\sqrt{1-v^2/c^2}## where ##v## is our relative velocity. But that's not actually what you observe. What you observe is the time at which each end of the moving meter stick is at various points in space (you can either put a detector taking time-stamped photos at these points, or you can use your eyes and watch the meter stick as moves past you as long as you allow for the time it took the light to get to your eyes). Using these observations, you infer - not "observe"! - the length of the meter stick; it's just the distance between where the ends were at the same time, and it will be less than one meter.

You will explain the results reported by your detectors by saying that the clocks in the detectors are properly synchronized and the detectors are less than one meter apart; the detectors put the same timestamp on their detections because the moving meter stick is length contracted.

I will explain these results by saying that the clocks in your detectors are not synchronized so that despite the identical timestamps the two detections do not represent the positions of the ends of the stick at the same time.

Both explanations are equally valid. And crucially we agree about what is actually observed: the timestamps your detectors produced as the ends of the meter stick passes them.

What happens is that, instead of falling straight in towards the mass, the infalling particle acquires a gradually increasing angular velocity in the direction of rotation of the mass. So its trajectory in space looks like an ingoing spiral.
Similarly, irrespective of rotation, would that also be true for a pair of compact objects in orbit? Does their orbital motion also result in frame-dragging?

hnaghieh
Frames are a convenience for calculations, but conceptually they often cause more confusion than they solve.
In order to make any calculation we need data or measurements. In order to have measurements we need units of measurements(yard sticks and clocks). Yard sticks and clocks are frame dependent in the sense that two reference coordinate systems in relative motion will have different units (due to relativistic effects on the yard sticks and clocks, etc) when they compare each other’s measurements of a world event they will not agree on their measurement or the laws of nature unless they know the transformation relationship connecting their coordinate frames units of measurements ( Lorentz transformations). This is solely because the speed of the signals used (light) to make “time”mearurments or “length” measurements, is finite and constant. Individually they won’t infer that anything (length contraction, time dilation) has taken place in their respective reference coordinate frames. But when they compare their determination their units of “time” and “length” won’t be the same. That’s why we have Lorentz transformation to make the LAWS invariant. That is why the laws are not coordinate dependent, but their components (based on a specific measurement by a specific yard stick etc ) is coordinate dependent. A simple test. Take Two solid discs. Put a pin through its center. This pin can only be rigidly connected to one of the two discs if there is a relative rotational motion between the two discs. If we are rigidly connected to the rotating disc and have no external reference point we will not notice a motion (let’s forget the accelerameter because of our rigidly connected assumption). Same is for linear motion. The motion of a frame can not be determined by any parameter from within the reference frame (from within a ship). Solutions to any invariant law must be a limited case of the invarent law, which is a specific frame dependent component of that law. How do we distinguish between invariant laws and their coordinate dependent components or solutions and do not attribute the same invariant properties to the solutions. My point is that in order to do that we must be clear about our chosen frame of reference. And more importantly how can we ensure that the view point of the observer is actually that of the rest frame observer of the chosen frame?

Nugatory
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The motion of a frame can not be determined by any parameter from within the reference frame (from within a ship).
The “motion of a frame“ cannot be determined for about the same reason that I cannot determine the color of love or the weight of an idea - the notion is meaningless because frames aren’t things that can move. A frame is a mathematical convention for assigning coordinates to points in spacetime (these points are called “events” in the language of relativistic physics) and mathematical conventions aren’t things that move around in space.

Yes, I know you’ve heard people use the term “a moving reference frame” or “a frame moving relative to me” or similar.... but that just shows that natural language isn’t always used precisely. It would be more accurate to say “a reference frame which assigns coordinates to events in such a way that the spatial coordinates of my position are a function of the time coordinate”.

Until you have resolved your confusion about what frames do and don’t do, you will find it very difficult to make sense of relativity. At this point all I can do is repeat my recommendation of the Taylor and Wheeler book.