elfmotat

I probably should have said that this property appears to be unique to metric theories of gravity. The equations of motion for Brans-Dicke theory, Nordstrom's (second) theory of gravity, etc. can all be derived from ∇aTab = 0, which holds in general for theories with diffeomorphism invariant actions. LITG doesn't have the property that the equations of motion can be found "in" the field equations by analogy with Maxwell's equations, which also don't display this property. LITG is essentially the same theory as classical electromagnetism.

Even though gravitational acceleration cancels so that you do get m1L1=m2L2, I would still regard this as a torque problem. It's more meaningful in this context to talk about (force)*(length) than (mass)*(length). I agree. I'm having trouble coming up with instances where (mass)*(length) is used at all. I suppose if you wrote the center of mass equation as: [math]M \bar{x}= \sum_i m_i x_i[/math] Even then though, the actual quantity [math]M \bar{x}[/math] isn't really physically meaningful.

"Position" is not a coordinate-independent (covariant) quantity. There's no such thing as a coordinate-independent "position vector," so position vectors by themselves are not physically meaningful. For example, let's say we have a position vector given by [math]\mathbf{x}=(x,y)[/math]. If we shift the origin of our coordinate system by a distance k in the +x direction, the vector (x,y) no longer corresponds to the point that it did before. Instead, our previous point now corresponds to the vector (x-k,y). Consequently, the product of mass and a position vector is also not physically meaningful. Positions only become meaningful in relation to each other (i.e. when we take a difference of positions) because they represent the "distance" from one place to another. Distances obviously don't depend on your choice of coordinates. "Velocity" is just a measure of change in position over time, and is therefore a coordinate-independent quantity. The product of mass and velocity is therefore physically meaningful (momentum). Similarly, acceleration and force are also physically meaningful, as are higher derivatives of position and their product with mass. The Joule is defined as 1 kg*m2/s2 = 1 N*m. Physicists simply decided to give the unit kg*m2/s2 its own name, and that name is "Joule." I think you mean L=kM. What you have now implies a longer wire must have smaller mass. A wire with constant cross-sectional area will have mass proportional to length, not inversely proportional.

Any thrust at all is going to yield better MPG, even if only marginally. There's no cut-off you can point to and say "the car starts getting significantly better mileage at x lbs. thrust."

I didn't know that the geodesic equation fell out the field equations until recently, and Einstein didn't realize it himself when he first formulated his theory. That's essentially the topic of the this thread. It appears to be a feature unique to GR, because the equations of motion for a theory are usually postulated separately from its field equations. For example, Maxwell's equations tell us how electromagnetic fields are generated and how they behave, but they don't tell us how matter placed in an electromagnetic field will behave. The Lorentz Force Law is a separate postulate from Maxwell's equations, and it is this equation (the equation of motion) which tells us how matter behaves in the presence of an EM field. In (the field theory formulation of) Newtonian gravitation, the Poisson equation tells us how the gravitational field is generated and how it behaves, but it doesn't tell us how matter behaves in the presence of a gravitational field. Postulated separately is the equation of motion: [math]d^2x^i / dt^2=-\partial_i \phi[/math], where [math]\phi[/math] is the Newtonian potential. Likewise, the Einstein Field Equations tell us how the gravitational field is generated and how it behaves. There's no immediately obvious reason to assume the field equations for this theory should contain the equations of motion, or Einstein wouldn't have bothered postulating geodesic motion separately. In fact, all experience tells us that the EFE's probably shouldn't contain the equations of motion. What are "motions in the metric?" I fail to see how this follows from anything you just said. This is the geodesic equation. It's the same equation I derived in my original post, and it's the equation Einstein postulated separately from the Field Equations in original papers. I'm not sure what your link has to do with anything. The Field Equations were developed to explain how the gravitational field is related to the energy-momentum distribution in spacetime, not to "deal with relative motions in non-inertial frames of reference." Of course GR can deal with such scenarios, but they really have minimal to do with the Field Equations. SR and GR aren't consistent with Galileo! And I'm not really sure what point you're trying to make here anyway - what do low-energy/velocity limits have to do with the equations of motion in GR or its field equations. I honestly have no idea what that means.

Again, I'm really not sure what any of this has to do with anything. I was certainly surprised. It seems like something unique to GR. There's no electromagnetic analogy, for example; the equations of motion are a separate postulate from its field equations.

Special relativity has absolutely nothing to do with "perception." It makes concrete quantitative predictions about how the universe works.

What is this nonsense supposed to mean?

What are the "light equations?" I'm not really sure what you mean here either. Thanks for the response. I did some digging and found this in Wald as well:

You clearly have no idea how GR or black holes work. First of all, it's not true in general that external observers will not observe objects crossing the event horizon. This is only true if you're using Schwarzschild coordinates because there's a coordinate singularity at R=2M. If you use, for example, Kruskal coordinates (which are well-behaved everywhere outside the physical singularity) then you can certainly observe things falling into the horizon. This is a clear example of when knowing the math is necessary. Your wordy descriptions are annoying to read and convey significantly less information than a couple of equations would. Second, you have objections to the concept that matter, once it has crossed the horizon, can never return. You, for some strange reason, think that this is a question of acceleration. In fact, this is true no matter how an object is lowered passed the horizon. For example, if you tied a ball to a rope and slowly lowered it passed the event horizon it would still be impossible for the ball to ever return. The reason this happens is because the light-cone of the ball is "tilted" so drastically that the ball would need to exceed c in order to escape. In fact, the "t" coordinate actually becomes spacial and the "r" coordinate becomes timelike once you've crossed the horizon, so you can no more avoid the singularity than you can avoid growing older. The singularity lies in the future of every geodesic inside r<2M. Also - fun fact - since geodesics are paths that maximize an object's proper time, fighting against falling into the singularity will actually make you arrive there faster.

This is completely untrue. See Special Relativity.

Time dilation has no affect on mass, and the Higgs field has nothing to with time dilation.

Did you read what he said? He said that when considering light signals, picking a value for d or t immediately specifies the other. They aren't independent variables.

The Bianchi identities along with the EFE's with zero CC imply: [math]\nabla_{\nu} G^{\mu \nu} = \nabla_{\nu} T^{\mu \nu}=0[/math] This implies that, for arbitrary [math]\xi_\mu[/math], the following holds true: [math]\xi_\mu \nabla_{\nu} T^{\mu \nu}=0[/math]. Therefore: [math]\nabla_{\nu}(\xi_\mu T^{\mu \nu}) = T^{\mu \nu} \nabla_{\nu} \xi_\mu[/math] Now let's say we're considering a free point particle traveling along some worldline with stress-energy given by: [math]T^{\mu \nu} (s) = m \frac{dx^\mu}{ds} \frac{dx^\nu}{ds}[/math] If we integrate both sides over some region containing the worldline and allow [math]\xi_\mu[/math] to vanish at the boundary, the left side goes to zero by Stokes. The right side must therefore be zero regardless of the size of the region, so we can simply take the integral over the path: [math]\int \nabla_{\nu} \xi_\mu \frac{dx^\mu}{ds} \frac{dx^\nu}{ds} ds=0[/math] Now we recognize that [math]\frac{dx^\nu}{ds}\nabla_{\nu} \xi_\mu = \frac{D \xi_\mu}{ds}[/math]. So if we integrate by parts, we get: [math]\int \frac{D}{ds}\left [ \frac{dx^\mu}{ds} \right ]\xi_\mu ds=0[/math] Since [math]\xi_\mu[/math] is just an arbitrary one-form, we have that the following must be true everywhere: [math]\frac{D}{ds}\left [ \frac{dx^\mu}{ds} \right ]=0[/math]. This is just the geodesic equation. So it seems we can get the equations of motion for a single point particle in GR directly from the field equations. Is this a unique case? Will a similar argument lead to the equations of motion for, say, dust or a perfect fluid? Is there some way to generalize this for all matter distributions? And also, is the fact that the equations of motion come out of the theory's field equations something unique to GR?

That's simply wrong. Who told you this? Photons have momentum given by [math]p^\mu =\hbar k^\mu [/math]. When they are absorbed by a medium, they also transfer their momentum into it. Since force is just the time derivative of momentum, shining a constant stream of photons onto a surface is equivalent to applying a force to that surface. That makes you a crackpot. Electric potential energy has nothing to do with mass or motion.

OP, I believe what you're asking is whether or not "conservation of magnetism" exists. The answer to that question is no. You can create one permanent magnet without significantly (or at all) decreasing the strength of the parent magnet.

Your graph should have changed concavity again (sometime recently) because the expansion of the universe is accelerating at present.

You've pretty much answered your own question then, haven't you? You could use anything. You could launch electrons at 2c/3 for example.

Sure, because in scenario 3 the events are separated by a timelike geodesic, whereas in scenario 2 they're connected by a spacelike geodesic. Events connected by spacelike geodesics can't possibly be causally related because light hasn't had enough time to go from A to B, and information can't travel faster than c.

Events A & B in scenario 3 don't necessarily correspond to the location of the same object. Say an object is traveling along the x-axis with velocity v. We mark event A as the object crossing the origin. Event B, which is at the same location but later in time, will obviously NOT correspond to the location of the object because the object is currently located at the event (v,0,0,1).

What are you talking about? It did't "disappear," it's just older now. You asked how we can observe three year-old objects located two light-years away from us. Of course the light from the object that reached us a year ago fits the bill. Signals made of something other than light would do the trick as well. If you had a signal made of massive particles that traveled at 2/3 light speed, it would reach us in three years from a location two light-years away. Your question is rather strange by the way. It's similar to asking, "I know that when I look at my friend I'm seeing him slightly in the past. How come I can't see him even further in the past."

We observed it last year.

Light doesn't have a rest frame, so talking about how much time passes for photons in meaningless.

Nothing's actually moving - it's just a program.

Did you read a word that swansont said?