Gravity

Originally posted by sonhouse
I don't know if it would be 'breaking down', it would really just lower the number of digits of accuracy, for instance, the flight time close to c would be different from Newtonian physics but close to Earthy velocities, they would read about the same out to a sufficient number of decimal places that there would be no usable difference, unless extremely acc ...[text shortened]... clock is raised a meter from it's previous position, now in a different gravitational strength.

Actually, Newton's Laws of motion are still rigorously true in relativity. Things keeps travelling in a constant state of motion unless a force is applied (constant state of motion is understood as travelling in a straight line in classical physics and along a geodesic in GR). Force is proportional to rate of change of momentum (*). And action and reaction are still equal and opposite. The change to relativity is in an underlying framework connected with assumptions about the space we live in.

To translate between relativistic mechanics and classical mechanics one simply chooses a frame of reference and then treats space time as a fibre bundle (with time the base space and the 3 space-like dimensions the typical fibre). One can then either have a velocity dependant mass (m = m_0/sqrt(1 - v^/c^2)) or simply take the limit that the speed of light goes to infinity.

(*) Force is a more or less deprecated concept in both relativity and quantum mechanics - we tend to use potentials and momentum. Even so, there is a four dimensional analogue of force, which is rate of change of 4 momentum with proper time.

Originally posted by DeepThought
Actually, Newton's Laws of motion are still rigorously true in relativity. Things keeps travelling in a constant state of motion unless a force is applied (constant state of motion is understood as travelling in a straight line in classical physics and along a geodesic in GR).

Are you taking Gravity to be a force in that description?

Originally posted by twhitehead
Are you taking Gravity to be a force in that description?

I wasn't considering gravity, just the laws of motion.

Originally posted by DeepThought
I wasn't considering gravity, just the laws of motion.

So you were talking about special relativity not general relativity? Because if Gravity is seen as a bending of space time and not a force, then I am not certain that objects do follow geodesics.

Originally posted by twhitehead
So you were talking about special relativity not general relativity? Because if Gravity is seen as a bending of space time and not a force, then I am not certain that objects do follow geodesics.

Newton's laws of motion are still valid in General Relativity, but with GR you have to be careful. Things follow geodesics, but the object that is moving curves space-time itself so the object's motion has to be analysed in the light of its own curving of space-time. The normal approach is to ignore this as a vanishingly small correction, but it would matter, for example, for two neutron stars that are about to merge.

The problem with your statement is that it doesn't make sense to treat gravity as anything other than geometrical in origin in General Relativity, that's the point of the theory. However, if one is going to take the Einstein Field equations as a piece of technology for finding the orbits of objects in the classical picture of 3 space with time as a parameter, then one is treating gravity as a conventional force. But then we are treating space as flat and so all the geodesics are straight lines and naturally the body doesn't go along the Newtonian geodesic.

Originally posted by DeepThought
Newton's laws of motion are still valid in General Relativity, but with GR you have to be careful. Things follow geodesics, but the object that is moving curves space-time itself so the object's motion has to be analysed in the light of its own curving of space-time.

Imagine two objects going past earth with no force other than gravity acting on them. One is travelling a little bit faster than the other. The faster one will travel a 'straighter' line than the slower one (largely independent of mass).
Are you saying that both paths are geodesics? Does the slower one bend space time more?

Originally posted by twhitehead
Imagine two objects going past earth with no force other than gravity acting on them. One is travelling a little bit faster than the other. The faster one will travel a 'straighter' line than the slower one (largely independent of mass).
Are you saying that both paths are geodesics? Does the slower one bend space time more?

I may be interpreting this wrong, but are you implying that velocity alone bends space-time?

Originally posted by joe shmo
I may be interpreting this wrong, but are you implying that velocity alone bends space-time?

No, I am asking. Clearly, under gravity, velocity affects the path through space that an object takes. If objects follow geodesics, then the geodesics change depending on velocity, or I am missing something.

Originally posted by twhitehead
No, I am asking. Clearly, under gravity, velocity affects the path through space that an object takes. If objects follow geodesics, then the geodesics change depending on velocity, or I am missing something.

Under "gravity" the body's trajectory is dependent on its momentum and position in the field via the solution to the second order non-linear DE. I didn't study relativity, but I don't think it's needed to answer your question explicitly as stated. It appears that you might be mixing concepts that shouldn't be mixed between Classical mechanics and Relitivisic Mechanics. But again, Deep Thought ( or other physicists) should rightfully field your question, not me.

Originally posted by twhitehead
No, I am asking. Clearly, under gravity, velocity affects the path through space that an object takes. If objects follow geodesics, then the geodesics change depending on velocity, or I am missing something.

Just to take a pot shot at it; From what Iv'e read in this thread they follow geodesics in GR if they are not accelerated. Those bodies passing by earth would be accelerated.

Originally posted by joe shmo
I may be interpreting this wrong, but are you implying that velocity alone bends space-time?

In relativity theory, "velocity" is relative (to an observer/frame of reference). The name is quite apt, really.

Originally posted by twhitehead
Imagine two objects going past earth with no force other than gravity acting on them. One is travelling a little bit faster than the other. The faster one will travel a 'straighter' line than the slower one (largely independent of mass).
Are you saying that both paths are geodesics? Does the slower one bend space time more?

Both paths are geodesics yes. The two vectors are different so we would expect the integral curves of the vectors (aka geodesics) to be different. In these treatments the masses of the objects are assumed to be small. If we are going to take into account the curvature due to the energy of the moving objects then the one moving faster will tend to produce a larger (but still immeasurably tiny) curvature as it has the higher energy in the reference frame defined by the centre of mass of the Earth, so it will emit more gravitational radiation (in immeasurably small amounts) because of its motion relative to the centre of mass of the earth. We could regard the object as stationary and have the earth zipping along and the earth's gravitational field pull on it. Up to reparameterization we'd get the same answers. Bear in mind that this is a tiny effect - drag from the interplanetary medium is a more important factor by several dozen orders of magnitude.

Originally posted by DeepThought
Both paths are geodesics yes. The two vectors are different so we would expect the integral curves of the vectors (aka geodesics) to be different. In these treatments the masses of the objects are assumed to be small. If we are going to take into account the curvature due to the energy of the moving objects then the one moving faster will tend to prod ...[text shortened]... from the interplanetary medium is a more important factor by several dozen orders of magnitude.

I think you are confusing me by talking about immeasurable effects I have no interest in.
I am saying that my understanding of general relativity is that the shape of space time causes objects to behave as if a force was acting on them (aka gravity). To objects with different velocities passing earth will have dramatically different trajectories.
My questions are:
Are their paths through space time geodesics but spacetime appears bent differently from the perspective of the different objects?
Do the objects think they are moving at constant velocity (or essentially at rest) from their own perspective?

Originally posted by twhitehead
I think you are confusing me by talking about immeasurable effects I have no interest in.
I am saying that my understanding of general relativity is that the shape of space time causes objects to behave as if a force was acting on them (aka gravity). To objects with different velocities passing earth will have dramatically different trajectories.
My que ...[text shortened]... think they are moving at constant velocity (or essentially at rest) from their own perspective?

From the point of view of an observer separate from your two objects they have different initial conditions so their final states will be different. From the third party point of view the curvature is the same in each case, but the velocity vector is different, so they travel along different geodesics. Boosting to the reference frame of either object doesn't particularly change this. Whether the curvature is different depends on which measure of curvature you are using, the scalar curvature is the same in all frames, the Ricci tensor transforms.

An observer co-moving with either object will regard themselves as being in an inertial reference frame. So yes, they are at rest and the earth is moving past them.

Returning to the first point. Imagine you are in England and a sign post says 10 miles to London (on a sphere, apply the vector), if you follow the signpost you'll see a new one 1 yard further on, also pointed to London. Follow those signposts and you are following the integral path of a vector field on a two sphere. In the case of geodesics they are great circles, but there is more than one great circle at each point on the earth. The difference in relativity is that the time like component can be different. This means that the velocity matters, so bodies with different velocities follow different paths in 3 space.

Originally posted by sonhouse
Light and RF follows the curves of space.

Thanks for the post but this is where you lose me.

Curves of space?