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If you bring an object from infinity to a distance $R$ then the potential energy change is: $$ \Delta U = -\frac{GMm}{R} $$ Assuming your object starts at rest, the potential energy change is equal to the change in kinetic energy, so we have: $$ \frac{GMm}{R} = \tfrac{1}{2}mv^2 $$ so: $$ v^2 = \frac{2GM}{R} $$ You want $v \ge c$, so: $$ \frac{2GM}{R} ...


3

You are asking about time dilation. Using $$\Delta t' = \gamma \Delta t$$ where $$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ We can rearrange for $v$ to get: $$v = \sqrt{c^2 - \frac{\Delta t^2}{\Delta t'^2} c^2 }$$ or maybe better $$\frac{v}{c} = \sqrt{1-\frac{\Delta t^2}{\Delta t'^2}}$$ to get the speed as a fraction of the speed of light. Plugging in ...


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This question is quite broad, but I'll try my best. When making the jump from special relativity, we have two good rules of thumb: Wherever there is a Minkowski metric in SR, put a general metric in GR. Wherever there is a partial derivative in SR, put a covariant derivative in GR. Take, for instance, mass-energy equivalence. In SR, we have ...


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You may feel the acceleration, but if you're not accelerating you won't notice anything. Other people may see you moving and observe you contracting, however, from your reference frame, you are not moving at all, and hence you won't notice any length contraction of yourself.


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It is not correct saying that no force is applied. A photon carries momentum see PE here so on reflection there is momentum transfer. This is the idea behind laser propulsion discussed here. Concerning the speed it is even more complicated. The fact that light gets reflected usually requires an abrupt change in the index of refraction. To get reflected, ...


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Your question is so confused that it's hard to give a meaningful answer... if matter were to travel the speed of light, would it become energy? Anything that has a rest mass $m_0$ has an energy associated with that mass of $E=m_0 c^2$. If this mass is also moving with some momentum $p$, it has a kinetic energy associated with the motion. The total ...


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At hyperrelativistic speeds, a massive object has the bulk of its energy in kenetic form so the invariant mass is small residue. It acts like a very low mass particle at meerly relativistic speeds, or a almost massless neutrino at any speed. A 7Tev proton is travelling at $0.999999991c$ and its mass is only contributing on the order of one one hundredth of ...


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Since no one else has mentioned it ... If you want to have a better conceptual understanding of the apparent slowing of light (and other electromagnetic waves) in materials, I strongly suggest reading Richard Feynman's lectures, especially Chapter 31 of volume I. That will give you much more explanation than is possible in this forum. All the Feynman ...


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Any time you read a report of superluminal whatever, it's another illustration of Phase vs Group velocity. The pattern indeed moves, but by pre-arrangement not by communication at that speed. A wonderful illustration is from Greg Egan: http://gregegan.customer.netspace.net.au/APPLETS/20/20.html So playing with that applet is the way to explain it.


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Relative speed can be greater than "c". This does not mean that in an inertial frame fixed on one ship the other ship looks like it is going faster than "c". In an inertial frame fixed on one ship the other ship looks like it is going slower than "c", so light from one could in principle reach the other. In an inertial frame fixed on one ship the other ship ...


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Basically, no, you wouldn't experience length contraction or time dilation or increased mass or any effects like that. You'd feel acceleration, like you feel when a plane takes off, but acceleration and effects from close to light speed velocity aren't related. At .86 of the speed of light - time dilation would be 50%, So, lets say, You're traveling to ...


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Time is what an ideal clock measures. So what's an ideal clock? It's something that measures time. In other words, physicists don't quite know what time is. That's okay. They don't quite know what space is, either. What they do know, and know very, very well, is how to measure both, and how the two (time and space) relate to one another. That the speed of ...


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Yes, you have very likely done what the teacher wanted you to do. But you should maybe mention to teacher that he has asked a poor question (actually... maybe you shouldn't mention it... or at least be diplomatic about it...) The reason the question is dumb is because, as pointed out in the comments, you could (reasonably) use the relativistic expression ...


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Has this kind of phenomena ever been used to measure SR properties ? Yes. A similar apparatus, called a Fizeau apparatus, has been around for a long time... It uses a rotating cog rather than a falling box (for obvious reasons). http://en.wikipedia.org/wiki/Fizeau%E2%80%93Foucault_apparatus EDIT (re comment): Yes, lasers can also be used to measure ...


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First, I highly suggest reading up on the concept of Locality. The issue is where you're measuring the speed from... Remember that it isn't so much that light can't escape due to the escape velocity, as it is that space itself is being dragged into the black hole (and anything residing in it), which happens to be falling in at the speed of light where you ...



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