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In a rotating reference frame, the coordinate velocity of an object can exceed $c$. However, this doesn't mean that they're moving "faster than light". If we were to look at the light-cones at these distant locations, we would see that the four-velocities of these objects are still confined within the light-cones at those locations. To put this another ...


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What you're missing is that the speed of light is not constant. There's this modern-day myth that says "Einstein told us that the speed of light is constant". But search the Einstein digital papers on "speed of light" or "velocity of light" for examples like this: The speed of light is spatially variable. And that isn't some discarded idea from 1911, see ...


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Let's consider two ships passing each other. When they pass, a rope is thrown from a ship to the other ship. Then then rope is pulled sharply. That causes the ships to collide, the rears of the ships hit each other and the ships start to spin. In the previous scenario part of the energy used to pull the rope became rotational energy of the ships, that ...


3

How does the kinetic energy of a ballerina increase? Conservation of angular momentum: $$L_1=L_2 \implies I_1\omega_1=I_2\omega_2\quad\quad (1)$$ Pulling in your arms reduces moment of inertia $I$, since the same mass is now distributed over a volume closer to the spin centre, $I=\sum mr^2$. As you say, reducing $I$, so $I_2<I_1$, implies ...


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First off, traveling at constant velocity in flat spacetime is not the same as traveling g in a uniform circular motion. Quite the contrary, free falling towards the gravitational source is actually equivalent to moving with constant velocity in flat spacetime. This is so because the objects are following a geodesic path defined by the geodesic equation. I ...


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This shows the situation as viewed by the Schwarzschild observer i.e. an observer far from the black hole: (Note that the angle $\theta$ is not connected to the Schwarzschild $\theta$ coordinate.) The angle $\theta$ is (obviously) given by: $$ \tan\theta = \frac{b}{a} = \frac{b}{r_2 - r_1} $$ But we've calculated the angle using the Schwarzschild $r$ ...


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It is not possible to find a frame of reference where a photon is at rest. I will argument in two different ways: 1. Maxwell equations and electromagnetic argument: From Maxwell it is expected that electromagnetic disturbances propagate in vacuum at a constant speed c~299792458 m/s which is the maximum speed for the propagation of electromagnetic ...


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When 1686 Newton writes "Principia...", the inertial frame concept does not exist yet. However, we can find in it Corollary IV (introducing the center of mass CM concept for any interacting body set), Corollary V (Galileo's Principle of Relativity, applied to any limited body set with CM at any uniform velocity), and the today almost forgot Corollary VI (a ...


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But when I look at it from an inertial frame, I cannot intuitively understand how does the spinning of the Earth makes the mass free falling more slowly than when the Earth is not rotating? In the inertial frame, the mass will have the same radial acceleration whether rotating or not. But on a rotating earth, the mass also has a tangential speed. ...


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(I've read your question yesterday and could not find peace because of it, since I could not answer it to myself satisfactorily. ^^ But I figured it out and I hope the following helps... ) They key point is that, if the gravitational force acts as a centripetal force, the amount of centripetal force needed to let an object go round a circle with angular ...


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The answer from a Newtonian perspective: TL;DR: Objects do feel gravitation, but only if they're very big, or if the gravitational field is very strong. Suppose you are in a spacesuit and are orbiting the Earth. Your feet are pointed toward the Earth, your head into space. Because gravitation is a 1/r2 force, the force on your feet is slightly stronger ...


3

From a Newtonian perspective, the difference between being accelerated by gravity in freefall (which includes orbits) and being accelerated in a car has to do with the fact that you only "feel" accelerations when the external force is only being applied to one part of your body, rather than accelerating every particle equally as with gravity. For example, if ...


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Both observers see the other approaching at 99% of light but since observer 1 is the reference frame that accelerated, and changed it's behavior of motion, it is the reference frame that will have experienced less time than observer 2 during the duration of it's travel. The change in motion (acceleration/deceleration) is what changes the rate at which a ...


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I think the answer must be either the book has got it wrong or that there is some confusion between what the author meant to put in the book and the question before us. If I understand the question correct the point is does an object falling towards a planet undergravity have different acceleration (different dynamics) depending on whether the earth is ...


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Tme dilation refers to time in the accelerated frame (the rocket). So a clock in the rocket will run slowly compared to a clock on earth. It is us who would measure the rocket as moving at 0.5c. The speedometer in the rocket could actually show a speed greater than that. From the POV of the crew in the rocket, their clock is running normally. It is the ...


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There are different kinds of frames. A common frame to use is a coordinate frame. For that all you need to imagine is each region of spacetime has a coordinate system that you can use in that region to find and label all the events in that region. An advantage to this is that you can practice using arbitrary coordinate systems even while still doing ...


1

reading special relativity [...] I pictured a frame of reference being grid Of course there is no definitive requirement for the grid constituents to be rigid with respect to each other, or being in any particular way "regularly spaced" or "regularly moving". Required is (only) for the grid constituents to be distinctive, for any two grid ...


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[...] neither country's delegate wants to sign the treaty before the other delegate and thus, a simple system is devised to ensure that both delegates sign the peace treaty simultaneously. The solution involves setting a light bulb at the center of a table in such a way that the light bulb is exactly between the [two] delegate[s ...] the light bulbs ...


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As the other answers point out, there is no "right" or "absolute" frame of reference for measuring time. But that does not answer your question: Is there a way to measure the passing of time for this object? It turns out there is. Oscillations in the orbits of electrons depend only on the material, and in fact our clocks measure the passing of time by ...


1

There is no reference object that transcends all inertial frames of reference. Everything in this universe has an inertial frame of reference, and none of them are privileged. If there were any object that existed independently of the relativistic effects of acceleration/gravity or of observer movement, then theoretically it could provide a reference to ...


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In relativity there is no standard-clock that tells you which time is "right". That's the point about relativity. There is no need for a absolute reference to compare with. Everything is just the way you observe it (that is, relative to you). Things may slightly differ from observer to observer but the qualitative behaviour stays the same just as classical ...


1

You're right that there would be a disagreement over who signed the treaty first, but it would not be between the diplomats on the train; it would be between the people on the train and the people not on the train. The setup initially is that the two diplomats are sitting in the dining car with the curtains drawn, for security. Let's say the light that the ...


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Special relativity itself makes it clear that absolutes cannot be detected, such as being at absolute rest in space, or the inability to detect absolute motion. This therefore prevents one from having an absolute understanding of special relativity, since that which special relativity reveals does not extend to the point of absolutes. Thus the absolute ...


4

Yes, there's a very famous example: muons produced in the upper atmosphere can be detected on the surface of the Earth. Moving at nearly the speed of light, it takes them over 300 microseconds to get down to the Earth's surface, but the average muon decays after 2.2 microseconds. If it were not for time dilation, only a few in every $10^{60}$ muons (so, ...


1

For a classical point charge, the field is divergent at $r=0$, and if you were to take the potential to be zero there, it would be infinite everywhere else. Meanwhile, you can approximate $r=\infty$ as the region with no interaction, so it's reasonably naturally to treat it in the way you would treat ground.


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You evidently understand that any constant can be added to a potential without affecting the physics -- or equivalently, any place can be taken to have zero potential. You also suggest, rightly, that there are really only two "natural" places to define the zero of the potential: either $r=\infty$ or $r=0$. For example, there's no particular reason to ...


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This is accurate, and it ultimately comes down to the fact that we can get arbitrarily close to an electric point charge in classical E&M. That means that the field right up next to the point charge could be arbitrarily large. So you get these huge, singular potentials close to point charges, which is really more-or-less fine. For instance, that huge ...


1

Just to complement John Rennie's answer, one can always perform a Lorentz transformation to a coordinate system such as the particle is at rest for a given time. It's called instantaneous rest frame (IRF). This frame changes point to point, unless the particle's velocity is constant. In such a frame, we have $ ds^2 = -c^2d\tau^2, $ where $\tau$ is the ...


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In general relativity, it just has to be continuously differentiable. If you walk along a grid line, it can't suddenly turn, and the clocks can't suddenly change speed. Beyond that, pretty much anything goes. You don't even have to cover everything with a single map, but you do have to have extra maps to make sure it all gets covered somewhere. For example, ...


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I'm confused what you are trying to reconcile. The magnetic field is given by the Biot-Savart Law: $$ d\vec{B} \sim \frac{Id\vec{s}\times \vec{r}}{r^3} $$ where $$ I = \int \vec{J}\cdot d\vec{A} = \int \rho \;\vec{v}_{\text{drift}}\cdot d\vec{A}$$ So taking the Galilean transformation $\vec{v}_{\text{frame}} = \vec{v}_{\text{drift}}$ leads to: $$ ...


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We need to first ask ourselves what is Space? What is Time? Then we can begin to answer your question after we define what these two are and the relationship between them. According to Geometrical Mathematics and based on Numerical Vector Space is nothing more then an empty construct and has no Dimensions until you give it a coordinate. We can define space ...



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