elfmotat

If you keep making claims without evidence, yes. I don't really understand what you're asking for. The nearest thought to my mind would be the following example: [math]y = x^2[/math] If we introduce a displacement [math]\Delta x[/math], then we have: [math]\Delta y = (x+\Delta x)^2 - x^2 = 2x \Delta x + (\Delta x)^2[/math] or: [math]\frac{\Delta y}{\Delta x} = 2x + \Delta x[/math] This is well-defined for all [math]\Delta x \neq 0[/math]. But, the whole point of calculus is the concept of the limit. If we take the limit as [math]\Delta x \to 0[/math], then: [math]\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} 2x + \Delta x = 2x[/math] We simply define: [math]\frac{dy}{dy} := \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = 2x[/math]. Is there something wrong with what I just did? If so, why/how? That's calculus, at its most basic. Alternatively (and more hand-wavy), we can introduce an infinitesimal displacement [math]\Delta x = \epsilon[/math] such that [math]\epsilon^2 = 0[/math]: [math]\Delta y = (x+\epsilon)^2 - x^2 = 2x \epsilon + \epsilon^2 = 2x \epsilon[/math] or: [math]\frac{\Delta y}{\Delta x} = 2x[/math]. Is this what you're taking issue with? The introduction of a number such that its square is zero? Both of these methods can be made much more rigorous and well-defined. There is nothing logically inconsistent here, unless you'd care to point out the flaw?

I agree that "force of gravity" is poorly defined in GR, but there are a few meaningful ways to interpret what it means for a Schwarzschild BH. For example, the "force required to hold a particle stationary at a particular r-value." This force does indeed go to infinity as r->2M.

The observer could measure the distance between the objects shrinking at >c. Note, however, that the rate of distance shrinkage isn't actually the velocity of anything. It's not even a velocity, though it does have the units of velocity. The objects themselves would measure each other to be moving at <c.

Yes!

I'm still not sure why it matters if they make their measurements simultaneously or not. Could you explain why this would impact the experiment? You can assume whatever you'd like. Relativity tells us that if two events happen simultaneously in one frame, they do not and cannot happen simultaneously in any other frames. So you can set up the experiment so that Bob and Alice are at rest w.r.t. each other, and so that they make their measurements simultaneously in their rest frame. The fact that they made their measurements simultaneously in this frame is physically irrelevant, because there are an infinite number of equally valid frames in which they don't make their measurements simultaneously. The point is that there's no way for Bob to tell if the wave-function has collapsed until he makes his measurement. And once Bob makes his measurement, he is also causing wave-function collapse.

1) Why do you think this would make a difference? 2) If they measure at "the same time" in their rest frame, they won't measure at the same time in a frame moving w.r.t. them.

If he observes/measures it then he is also causing wave-function collapse!

Obesity is strongly linked to low T. Society has gotten much fatter, so I'm not surprised that T-levels have also dropped. It would be interesting to see a study on how much of the T-level drop can be explained by increased obesity rates.

What do you mean? Are you under the impression that spacetime curvature is not dependent on your distance from a gravitational source? Because it is. While this is true, I'm not sure how it's relevant. I never said anything of the sort. No, not speculation, fact. This is what we observe.

Newtonian gravity is an approximation of General Relativity when gravity is weak and objects are moving slowly compared to light. So, obviously, they need to agree on the range of gravity. If Newtonian gravity's range is infinite, GR's range must also be infinite. Whether or not you think it's true is irrelevant to whether or not it's actually true. What speculation? Electromagnetism is described by 1) field equations which describe how the EM-field behaves, and 2) an equation of motion which describes how objects behave when they interact with the field. Similarly, GR is described by field equations and an equation of motion. There is no speculation -- observation fits these models.

An increasing function is a function where [math]f'(x)>0[/math] for all [math]x[/math]. So you need to find a way to prove that the derivative of the function is always greater than zero. (Hint: this will be easier to see if you combine those fraction terms into a single fraction.)

True. I think the key (for Eldad and others) is to recognize when things are being simplified to make them easier to explain and visualize without the JHM, but that at the end of the day the JHM is what defines the theory. Many tend to take analogies, mathematical objects with physical-sounding names, etc., too seriously and derive meaningless predictions by extrapolation. Fair enough. I haven't been around for a while, so I'm unfortunately no longer "in the loop" in regard to many users' habits.

I understand the point you're making (that this results in Hawking radiation), but really you're oversimplifying things a lot. It's not nearly so simple as "virtual particles are created, one falls in and the other is ejected." That's a nice picture that humans have an easy time visualizing, but again, speaking in terms of virtual particles only makes sense in perturbation theory. The full, non-perturbative model of interacting quantum fields on a Schwarzschild background would describe what's going on better, and with no such reference to internal lines on Feynman graphs.

I don't think you guys are being very fair. While I agree 'virtual particles' are well-defined (namely, internal lines in Feynman diagrams), and that it would be hard/impossible to change the term, I also think the term is confusing. See e.g. the link Strange provided. Perhaps the biggest thing to note about virtual particles is that they only make sense in perturbative QFT. They are mathematical objects (propagators) that arise from a Taylor series approximation of the full theory. People (i.e. popsci documentaries) attribute much more meaning to them than they should really have.

What steps have you made to solve it? Are you given the equation for a catenary (/are you allowed to look it up), or are you supposed to derive it via calculus of variations? I remember the latter being pretty tedious.

I saw this recently:

Really it depends on what you mean by "destroy the moon." If you make your question more specific then you'll recieve more specific answers.

The issue is that there isn't a way to test it, even in principle. De Broglie-Bohm's predictions are completely equivalent to those of any other interpretation of QM.

White.

If the new frame with its origin at the event [math](t,x)=(T,X)[/math] in the first frame is moving with velocity [math]v[/math] w.r.t. the first frame, then the new coordinates are given by: [math]t' = \frac{t-T-v(x-X)/c^2}{\sqrt{1-v^2/c^2}}[/math] [math]x' = \frac{x-X-v(t-T)}{\sqrt{1-v^2/c^2}}[/math] If they are stationary w.r.t. each other then [math]v=0[/math] and this reduces to: [math]t' = t-T[/math] [math]x' = x-X[/math]

When the gravitational constant is promoted to a field, the result is Brans-Dicke Theory. The new field must obey a field equation that constrains it to behave according to basic physical principles like Lorentz symmetry, locality, etc. Modified gravity theories that attempt to reproduce observed galaxy behavior without dark matter like TeVeS are even more complicated, introducing a new type of vector field, a non-dynamical scalar field, and a dynamical scalar field like the one in Brans-Dicke Theory. Gravity theories of this sort have largely been ruled out as they fail to explain behavior like the gravitational lensing in the Bullet Cluster.

They are all well-tested fields of physics. Keep in mind that different regimes require different physics. For example, you'd never use Newtonian mechanics to describe how electron orbitals behave - you need quantum mechanics for that, because Newtonian mechanics doesn't apply to systems that small. That doesn't mean NM is 'wrong' or 'less tested.'