elfmotat

It's just a label. You could call it the "zero'th" dimension (indeed, as some do), or the first, second, third (though it doesn't really make sense to not group the spacial dimensions together) or fourth.

Even more simply: By the work-energy theorem, the energy required to bring a mass to a certain velocity is equal to the change in its kinetic energy. Kinetic energy in Special Relativity is given by: [math]E_k=\frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2[/math] (m refers to the rest mass.) As you can see, as v grows larger the energy required to increase v grows without bound. If you wanted to bring something to the speed of light, the change in kinetic energy becomes infinite.

I couldn't really tell what the article meant by "spinning universe." If it's implying that the universe is anything like Godel's rotating universe model, then we definitely have significant reason to doubt it.

For FTL transmissions? Considering neutrinos don't travel FTL, no. For slower-than-light transmissions I don't really see the point, but I suppose yes.

That's a pretty terrible attitude. Why do you assume he is the only person who can answer your question?

Special Relativity applies whenever the metric is Minkowskian (flat). This means that SR still works as long as there are no nearby sources of gravitation. Accelerated worldlines are easily handled, because they only need be analyzed from an inertial frame. Accelerated frames are trickier, but they can still be handled within the domain of SR.

Cool stuff, thanks for the info.

Assuming the hole was drilled right down the center of the Earth, there's no air resistance, rotational effects, etc. you would oscillate in simple harmonic motion (the same motion that a mass hanging on a bobbing spring would follow) with a period (time it would take to fall all the way through and come back to your initial position) of about: [math]T=\sqrt{\frac{3\pi}{G\rho}}[/math] , where ρ is the average density of the Earth. If you plug in the numbers, you get about 84 minutes. So it would take about 42 minutes to reach the other side. Interestingly enough, this is the same period that a satellite orbiting very close to the Earth's surface would have. Your maximum speed (speed at the very center of the Earth) would be: [math]v_{max}=2R\sqrt{\frac{\pi G\rho}{3}}[/math] , where R is the radius of Earth. This works out to be ~ 17,700 mph.

Fair enough. I suppose "axiom" is a bit of a strong word to use. Even so, the SEP was definitely still a core assumption in the formulation of GR.

Also, for anyone unfamiliar with this, I think it's interesting enough to warrant some explanation. If you have a metric given by (in units where c=1, Ω=constant for simplicity): [math]ds^2=-(1-\frac{2GM}{r})dt^2+dr^2[/math] it's easy to show that geodesics reduce to Newtonian gravity. Using the principle of extremal proper time: [math]\delta \int d\tau =\delta \int \sqrt{-g_{\mu \nu}\dot{x}^\mu \dot{x}^\nu}dt=0[/math] and the appropriately chosen Lagrangian: [math]L=(1-\frac{2GM}{r})-\dot{r}^2[/math] , then: [math]\frac{\partial L}{\partial r}=\frac{\partial }{\partial t}\frac{\partial L}{\partial \dot{r}}\Rightarrow \ddot{r}=-\frac{GM}{r^2}[/math] Incidentally, [math]\Delta \tau =\Delta t \sqrt{1-\frac{2GM}{r}}[/math] is the gravitational time dilation predicted by the Schwarzschild metric. Geometric Newtonian gravity can therefore be viewed as an approximation to GR which neglects spacial curvature.

The link is broken. The actual link is here.

I wasn't very clear here. I'll elaborate: The equivalence principle is a local principle. A gravitational field is locally indistinguishable from an accelerated reference frame. Einstein used Newtonian gravity along with the EP to calculate the deflection of light relative to local straight lines, lines that could be established by rigid rulers. This is fine, but only locally. When applied to a global system, like it was, it fails. Straight lines are bent near sources of gravitation, and by just enough to yield the factor of two difference between the general relativistic and Newtonian predictions. This doesn't indicate a failure of the equivalence principle, it's just a misapplication of it. Like I said, there is tons of experimental support for the EP.

To avoid confusion, I would like to amend the above to the following: The "diag()" would imply that the trace of the SET is still a rank 2 tensor, which it is not. T is a scalar - a rank 0 tensor.

They are a good enough approximation that it would be well worth your time to give them some thought.

The error comes from the fact that the equivalence principle applies only locally, whereas Einstein used it in coordination with Newtonian gravity to predict an incorrect value for the deflection of light. If you think the equivalence principle is in error, by all means come up with a explanation for why experiment agrees with it so well. Would you agree with the following(?): "Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." That is the strong equivalence principle, also called the "comma-goes-to-semicolon rule" because of the switch from partial to covariant derivatives. If this doesn't count as an axiom, I honestly don't know what does.

No, he used the accelerating elevator though-experiment to demonstrate that light would, in fact, fall due to gravity (something that wasn't known at the time). He made no such calculations. After demonstrating that light should fall due to gravity, he then calculated by how much using Newtonian gravity. I've seen Newtonian gravity formulated geometrically using a time dilation factor in the tt component of the metric, but I wouldn't go so far as to say that Newtonian gravity "takes into account the warping of time." The reason he obtained an incorrect value the first time 'round was because he was using an incorrect theory of gravitation. This says nothing about the validity of the EP. The EP is actually one of the axioms of GR, so if it weren't true then GR wouldn't be either. There have been numerous tests of the EP over the years, and they have all confirmed it to high accuracy - some even up to 13 decimal places. A few of them are listed on wikipedia: en.wikipedia.org/wiki/Equivalence_principle .

This is patently false. A spaceship with the appropriate amount of acceleration is indistinguishable from a local gravitational field. I'm guessing this is coming from the factor of 2 difference between the Newtonian and GR predictions about the deflection of light around the sun. I assure you this has nothing to do with a failure of the equivalence principle. In fact, it gives more credence to it.

Yes, but that's not my point. My point is that there isn't only one now. Two events that happen "now" in your frame will not happen at the same time for my friend.

You're using the word "now" extremely loosely. Too loosely, I would argue, for it to have any physical meaning.

My friend moving with respect to you would disagree.

"Why" questions can't really be answered in physics. "Why is G/ħ/e/etc. the value that it is?" It depends on what kind of answer satisfies you. The constancy of c was implied by Maxwell's equations and later postulated by Einstein. Essentially, he said "let's see what it would imply if c was the same to all inertial observers." We know his postulate is correct because we've experimentally determined it to be the case.

4 times as far way to yield 8 times the period is correct. This makes me afraid about what else it could say in there. [math]T^2 \propto r^3[/math]

Are you familiar with Gauss' Law for electric fields? There's also a Gauss' law for gravitational fields: [math]\oint\boldsymbol{g}\cdot d\boldsymbol{A}=-4\pi GM[/math] **Hint: In a typical E&M class you'll eventually be asked to solve for the electric field in the interior of a sphere with constant charge density (E as a function of the distance from the center).

A spin-2 field on flat spacetime is equivalent to curved spacetime. http://arxiv.org/abs/astro-ph/0006423

Only one person will actually feel an acceleration, while the other feels nothing. Acceleration is not relative in the same way that velocity is.