General Relativity

[Hellbound]

You're correct that in four space dimensions you can spin independently along two axes simultaneously - the one perpendicular to the xy plane, and the one perpendicular to the zw plane. But that kind of rotation still breaks 3D rotation invariance (i.e. isotropy).

Oh well. Thanks for the info...even if you did ruin my brilliant idea

 

[Perpetual Student]

ETA: we can even take it a step further with a concrete example. Let's say we take a big patch of empty, flat space-time. In the middle of it, we stick a really massive spherical shell. Around the shell, spacetime becomes curved, and "fictitious forces" appear. Inside the shell, though, spacetime remains flat. Very far away from the shell, spacetime also remains close to flat.

What do you mean "fictitious forces appear"? Are you not talking about gravity? Why would the inside of the shell be different? It would seem the only flat place would be the center.

Now let's spin our spherical shell. What happens inside? Well, inside, if we use coordinates which are not rotating with respect to distant objects, then we actually get forces. And those forces look like the forces one introduces in Newtonian mechanics when one adopts a rotating reference frame.

Why? Frame dragging?

Except we're not rotating. But to remove these forces from our equations of motion, we would actually need to adopt a reference frame which rotated. But if we do that, then we re-introduce those fictitious forces for distant objects. So it turns out that it isn't even possible to remove all these fictitious forces from any reference frame.

These conclusions are all quite mysterious. Forces appear and disappear because of our choice of coordinates? My understanding of GR has been that the universe is the same (forces and all) regardless of the choice of coordinates. Either a force is effecting something or it is not.
If I use Princeton, NJ as a frame of reference, the moon stays in orbit because of the "force" of gravity from the earth. If I use the sun, the center of the galaxy or the CMB as my frame I see the same force. How does a force change because of a change in frame of reference?

Addendum: I know gravity if not a force under GR, but it's a distortion of spacetime caused by a massive object. But, I don't what choice of words might be better than "force" in order to state my point.

 

[sol invictus]

What do you mean "fictitious forces appear"? Are you not talking about gravity?

In GR, gravity is a "fictitious force".

These conclusions are all quite mysterious. Forces appear and disappear because of our choice of coordinates?

Of course - that happens even in Newtonian mechanics. In a rotating frame there are centrifugal and Coriolis forces.

My understanding of GR has been that the universe is the same (forces and all) regardless of the choice of coordinates. Either a force is effecting something or it is not.

You have to be more careful than that. Forces cause acceleration (F=ma). But acceleration obviously varies depending on your (non-inertial) choice of coordinates. Therefore forces must change when you change coordinates.

Nevertheless, all physical predictions must be identical. How can that be? One way is that if everything accelerates all at the same time and by the same amount, no one will notice. For that to work, these forces all have to be proportional to inertial mass, so that a=F/m is the same for all objects - but that's precisely the characteristic of both Newtonian "fictitious forces" and gravitational "forces" in GR.

Under a general coordinate transformation it's more complex, but the same basic fact (that all these forces are exactly proportional to the inertial mass) - known as the equivalence principle - is what allows it all to work.

 

[Farsight]

What you're saying there could have an analogy drawn to linear rather than angular velocities...

It does. You can change your speed quickly and see that some apparent distance changes in step with your change of speed. Hence you know that it's an observer effect rather than something real. If it was real the change could not occur faster than light.

...and yet you wouldn't argue that velocity wasn't relative would you?

No. It's relative. But in the end it's relative to the universe as determined by the CMB rest frame. And then you don't say that the universe is moving whilst you're standing still. Do you.

ETA:

The standard gravitational field would cancel but there's a gravitomagnetic contribution which doesn't. I'm not sure of a terribly intuitive way to think of it but I am looking.

I know about gravitomagnetism in an intuitive way that really brings out the relationship with electromagnetism. Maybe we should have a thread on that sometime.

 

[Farsight]

What does that have to do with anything?

Everything. You can alter the rotation or change linear speed quickly, and your view of the world changes just as quickly. That tells you it's your view that's changed rather than the world, because the world can't change faster than light can propagate. Relativity tells you that.

The question I was addressing is whether you can tell if some small object in any apparent state of rotation is "really" rotating or not. You can't, at least not unless you can be sure there isn't some huge shell of mass out near infinity that's rotating. Your experiment can't distinguish between the two.

Yes it can. And I don't have to lean forward to walk towards that spinning wheel. There is no huge shell of rotating mass "out near infinity". This is just like the waterfall, you're peddling garbage under the banner of relativity.

In other words, rotation in GR is relative - it's relative to infinity, or to distant matter if there is any.

Try adding a second spinning wheel. But oh, are there now two huge shells of mass out near infinity rotating? In different directions? How about a third wheel? And a fourth? And a fifth? How about a zillion, all frame-dragging away?

...You're correct that in four space dimensions you can spin independently along two axes simultaneously - the one perpendicular to the xy plane, and the one perpendicular to the zw plane...

Four space dimensions? What four space dimensions? Besides, you can spin along two axes simultaneously in three dimensions. Like this. Only then λ = 4π / nc^1½ applies where n=1 and a lot of people don't recognise you.

 

[Farsight]

...If the universe is homogeneous and isotropic, would not the net gravitational influence on earth be zero? Of course the sun, galaxy, etc are nearby, but that's not mentioned above as a factor in this reasoning? What am I missing?

Nothing. You're in line with Einstein. See his 1920 Leyden Address. This is the bit:

"According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν), has, I think, finally disposed of the view that space is physically empty".

That's Einstein telling us that where there's a gravitational field, space is not homogeneous. So if space is homogeneous, there's no gravitational field.

 

[Reality Check]

See his 1920 Leyden Address

This is nothing to do with the question which is about the matter & energy in the universe - the stuff that exerts the "net gravitational influence on earth".

It is quite obvious that if the stuff that creates the net gravitational influence on the Earth is homogeneous and isotropic then that net influence is zero.

 

[ynot]

It's either, in the sense that either one can alter the locally inertial frame. But in the real universe we live it, there's real mass out there, at least as far out as we can see.

And that real mass out there somehow forms a sphere that is concentric and coaxial with a particular top on Earth? And at the same time that mass has exactly the same relationship with every other top on Earth plus everything that is capable of being spun in the Universe? And these infinite number of spheres of mass that are simultaneously spinning in infinite directions and infinite speeds have an effect on tops and all spinning things in the Universe but they don’t have any effect on each other? I make no apology for thinking this is more the stuff of fantasy entertainment than science.

You're assuming that it wasn't spinning to begin with. But you don't know that.

Would appreciate an explanation of what you mean by “to begin with”. Do you mean before a person spun the top and it was lying down? Or when it was stood on it’s end but before it was spun? Or something different?

 

[Vorpal]

First, homogeneity and isotropy of space requires constant curvature, not zero curvature. So this still gives less-than-completely-trival geometry. Second, Einstein defined the Christoffel symbols as the components of the gravitational field, from the very first GTR paper. That means:

• The presense or absense gravitational force somewhere is completely coordinate dependent. Gravitational force is an inertial force and all inertial forces are gravity. (A less sympathetic name for an inertial force is "fictitious" force.) Gravitational freefall is inertial! This is the significance of the equivalence principle and Einstein's realization that freefalling objects don't feel their own weight.
• A completely homogeneous gravitational force field just describes normal flat spacetime, since such a field would have no tidal forces, i.e., zero curvature. It's flat spacetime in our old friend the Rindler coordinate chart, actually--and in this case the equivalence principle relating acceleration and gravity turns into a globally correct statement!

Einstein calls g_μν as the gravitational potentials, as he does elsewhere. If that's the potential, what's the force? The Christoffel symbols. In fact it's almost eerie the extent to which the relationship between
(1) gravitational potential g_μν, Christoffel symbols, geodesic equation
is just like the relationship between
(2) electromagnetic potential A_μ, electromagnetic field F_μν, electromagnetic (Lorentz) force equation.
If one compares them without context, it's almost like someone decided to slap on an extra index and symmetrize.

 

[Ziggurat]

What do you mean "fictitious forces appear"? Are you not talking about gravity?

Yes. What do you think a fictitious force is? A fictitious force is a force that's proportional to mass. Why do we call that a fictitious force? Because it's always an acceleration, and by adopting the right reference frame, we can make it disappear. Other forces (electromagnetism, nuclear forces) are not proportional to mass, and so no change of reference frame can make them disappear.

Why would the inside of the shell be different? It would seem the only flat place would be the center.

No. The shell theorem works in GR too when you aren't rotating. IIRC, the derivation is a bit different but the result is the same for our purposes.

Why? Frame dragging?

Yes.

These conclusions are all quite mysterious. Forces appear and disappear because of our choice of coordinates?

Yes. And it's not as mysterious as you're making it. Passengers on the International Space Station don't feel gravity in their orbiting reference frame.

My understanding of GR has been that the universe is the same (forces and all) regardless of the choice of coordinates. Either a force is effecting something or it is not.

Non-fictitious forces, yes. Fictitious forces (including gravity) are very much reference-frame dependent. But the way they vary between reference frames is such that all predictions of motion will agree.

If I use Princeton, NJ as a frame of reference, the moon stays in orbit because of the "force" of gravity from the earth. If I use the sun, the center of the galaxy or the CMB as my frame I see the same force. How does a force change because of a change in frame of reference?

You don't actually see the same force. The gravitational field of a moving object is not the same as the gravitational field of a stationary object. So when we change reference frames, we don't actually calculate the same fields, if we're using GR. If the differences in velocity between frames is small, then the difference in forces will be small too, but it's not actually zero.

 

[Roboramma]

No. I take great pride in my honesty and sincerity. And I don't disagree with everything that's presented in university courses and textbooks, just those elements that contradict experimental evidence and what Einstein actually said.

So, you agree with some parts of what's taught in university physics and not other parts, but when presenting your viewpoint you present it as though all of it were in agreement with what's taught in university physics. I'd say that's misleading.

You'll note I say "as though": I'm not saying you explicitly make that claim, I'm saying that to a casual reader, that's what it sounds like.

I'm saying that the way you choose to phrase your statements is misleading to casual readers in a specific way - they'll tend to think that you are claiming to be representing physics as it is understood by the scientific community, when some of the ideas you are presented are not in agreement with physics as it's understood by the scientific community.

 

[sol invictus]

Everything. You can alter the rotation or change linear speed quickly, and your view of the world changes just as quickly. That tells you it's your view that's changed rather than the world, because the world can't change faster than light can propagate. Relativity tells you that.

No one is talking about changing the rotation speed. You've completely misunderstood the conversation.

Four space dimensions? What four space dimensions?

Why keep butting into conversations that are obviously over your head?

 

[Perpetual Student]

I hope the physicists here will tolerate a little garbling of the language and concepts of GR, so here goes: There have been a few discussions above about "ficticious forces," including, to my surprise, gravity. Perhaps this is an opportunity for me to get a handle on this.
When I hold something in my hand and I feel a force from that object, is that a ficticious force? If so, is it ficticious because that object's inertia (or place?) along the spacetime geodesic (warped by the earth's gravity) is being impeded by my hand? So, then my hand must be exerting a force (Why isn't that force "real"?) on that object? Does this question make any sense or am I just adding another convoluted layer of confusion?

 

[Perpetual Student]

Wouldn't it be nice if the world were Newtonian?

 

[Vorpal]

When I hold something in my hand and I feel a force from that object, is that a ficticious force? If so, is it ficticious because that object's inertia (or place?) along the spacetime geodesic (warped by the earth's gravity) is being impeded by my hand? So, then my hand must be exerting a force (Why isn't that force "real"?) on that object? Does this question make any sense or am I just adding another convoluted layer of confusion?

The question makes sense, yes. You feel the stress in your hand as it exerts on the object. That force is not fictional because it accelerates the object: as you say, it impedes the object from going along a geodesic in spacetime.

Einstein's insight was that gravity is an inertial (fictitious) force: as you stand on the surface on the Earth, you are not inertial, but rather accelerated by the forces put on you by the ground, which causes you to feel weight. On the other hand, if you in freefall, you don't feel any weight, and your trajectory is spacetime is a geodesic (has no curvature, i.e., unaccelerated).

Wouldn't it be nice if the world were Newtonian?

I don't know about that... I've seen coordinate-independent formulation of Newtonian gravity in Galilean spacetime, and it is ugly as sin.

 

[ynot]

The question makes sense, yes. You feel the stress in your hand as it exerts on the object. That force is not fictional because it accelerates the object: as you say, it impedes the object from going along a geodesic in spacetime.

Einstein's insight was that gravity is an inertial (fictitious) force: as you stand on the surface on the Earth, you are not inertial, but rather accelerated by the forces put on you by the ground, which causes you to feel weight. On the other hand, if you in freefall, you don't feel any weight, and your trajectory is spacetime is a geodesic (has no curvature, i.e., unaccelerated).

IMO > In freefall gravity hasn’t disappeared and the thing freefalling is being accelerated. The acceleration isn’t “experienced” because it's uniformly equal on every atom simultaneously. To “experience” gravity and acceleration there has to be a part of the “experiencer” that “lags”. It’s the “lag” that’s “experienced”.

 

[Vorpal]

IMO > In freefall gravity hasn’t disappeared and the thing freefalling is being accelerated. The acceleration isn’t “experienced” because it's uniformly equal on every atom simultaneously.

Proper acceleration is what an accelerometer measures, and in freefall, it measures zero. This is unlike coordinate acceleration, which is not objectively measurable and can be made to be just about anything.

In Newtonian physics, unaccelerated motion in an inertial frame has constant velocity. Much the same in STR: accelerating a particle changes the direction of the velocity four-vector and conversely, inertial motion has keeps a constant velocity. And it is also how it works in GTR: geodesics are precisely the curves that do not change four-velocity. The only difference is that it is now a differential, local statement.

To “experience” gravity and acceleration there has to be a part of the “experiencer” that “lags”. It’s the “lag” that’s “experienced”.

If I understand you correctly, you're thinking more of gravitational tidal forces, which in GTR is what spacetime curvature means. Unlike gravitational force, you can't make pick a frame in which they vanish at some point. Which is appropriate, since their existence introduces stresses in extended bodies.

 

[Ziggurat]

I hope the physicists here will tolerate a little garbling of the language and concepts of GR, so here goes: There have been a few discussions above about "ficticious forces," including, to my surprise, gravity. Perhaps this is an opportunity for me to get a handle on this.
When I hold something in my hand and I feel a force from that object, is that a ficticious force?

No. That is a real force, and it's an electromagnetic (mostly electric) force. You never feel fictitious forces. In fact, right now you're not feeling gravity. You're feeling the force of the floor pushing up against you. Jump out of a plane and you won't feel the floor anymore. Nor will you feel gravity. But that's just it: you never felt it to begin with. We commonly mistake the sensation of other forces for gravity itself, but it's never gravity that we actually feel.

So, then my hand must be exerting a force (Why isn't that force "real"?) on that object?

The force of your hand on the object is real. But in your reference frame, the object isn't accelerating, because the force of gravity on the object cancels the force from your hand. But your reference frame is not inertial. In a (locally) inertial reference frame (ie, a free-falling frame), there is no gravitational force, the real force from your hand is unopposed, and the object is accelerating upwards.

In all reference frames, your hand exerts a real force. Whether or not other fictitious forces (such as gravity) are present is reference-frame dependent, and so is acceleration.

 

[Farsight]

...When I hold something in my hand and I feel a force from that object, is that a ficticious force? If so, is it ficticious because that object's inertia (or place?) along the spacetime geodesic (warped by the earth's gravity) is being impeded by my hand? So, then my hand must be exerting a force (Why isn't that force "real"?) on that object? Does this question make any sense or am I just adding another convoluted layer of confusion?

Let's say the object is a brick in your hand. The force you exert on the brick is real, but you aren't moving it a distance, so it doesn't gain any energy. The fictitious force is the one the brick "feels" when you let it go. You can see that it accelerates, but it doesn't actually gain any energy. This is very different to the situation when you strap a rocket to it. The kinetic energy of a falling body comes from the body itself, where it is labelled gravitational potential energy. When you raise a brick you move it a distance, and you add energy to that brick. Its relativistic mass increases, even though it doesn't have any motion relative to you.

 

[Farsight]

First, homogeneity and isotropy of space requires constant curvature, not zero curvature...

I responded to this post on page 24 of the black holes thread. I didn't want to hijack this one.