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31

I just read your blog post and it's clear to me where you've gone wrong. The equivalence principle only allows you to transform to an inertial frame locally. This means that if your spacetime is curved, then the falling observer can only choose Minkowski coordinates for an infinitesimal region around her. Think of a curved surface and having to choose a ...


21

While everyone agrees that jumping in a falling elevator doesn't help much, I think it is very instructive to do the calculation. General Remarks The general nature of the problem is the following: while jumping, the human injects muscle energy into the system. Of course, the human doesn't want to gain even more energy himself, instead he hopes to transfer ...


16

The interpretation of gravity as curvature of spacetime is model-dependent. You already mentioned the teleparallel equivalent of general relativity, modelling gravity by torsion. Another possibility are bi-metric theories, where the metric is a more ordinary field on a fixed background (this should be more in line with how string theorists tend to think of ...


14

You could think of it this way: 1) Take a free particle, put it at some spacetime point, and leave it evolve. 2) Imagine the motion is not geodesic, that is $a_\mu\equiv v^\nu v_{\mu;\nu}\neq 0$, or in other words the acceleration is not zero. Note: We know that $a_\mu v ^\mu = 0$, or the 4-acceleration is normal to 4-velocity. 3) Imagine you are that ...


14

The fact that gravitational field can be simulated/canceled by inertial forces relies upon the following elementary but fundamental fact. The gravitational coupling constant of a given body, i.e. its gravitational mass,$M$, coincides with the other universal constant associated with that body, appearing in the general law of motion, i.e. the inertial mass ...


11

These are two different effects. Satellites don't fall down because they are moving on a circular orbit. Actually, they are falling down all the time, since circular motion is accelerated (though the velocity doesn't change absolute value, it changes direction!), so it is kind of "falling around the earth". The second question is, why doesn't an astronaut ...


11

That this works for a test mass is essentially a postulate, which, as indicated by Alexey Bobrick's answer, is related to the equivalence principle. On the other hand, it is hypothesized that this behaviour can actually be demonstrated to be a direct consequence of Einstein's equations for physical masses. To prove this, however, requires actually solving ...


11

As an addition to already posted answers and while realising that experiments on Mythbusters don't really have the required rigour of physics experiments, the Mythbusters have tested this theory and concluded that: The jumping power of a human being cannot cancel out the falling velocity of the elevator. The best speculative advice from an elevator ...


10

The reason that jumping can make a relatively large difference is that the kinetic energy is proportional to the square of the velocity. Thus relatively small changes to the velocity can result in relatively large changes to the kinetic energy. In addition, the velocity which a human can achieve in jumping is a substantial percentage of the velocity of fatal ...


10

A derivation of Einstein's equation isn't why the Equivalence principle is central to GR. The reason that the equivalence principle is central to GR is in the fact that you can represent the gravitational field with a metric tensor at all--you can replace a force equation with a geodesic equation for a test mass precisely due to the fact that the geodesic ...


9

indeed there would be a (very small) and homogenous pressure within the blob, coming from surface tension. This pressure is calculated by the Kelvin Equation and is significant in small droplets (reason for small droplets to have a higher vapour pressure than bulk liquid) In Your 100 m blob, this extra pressure is negligible of course. There is another ...


9

His starting point was to realize that Newton's gravity didn't satisfy his principles of the (special) theory of relativity because it wasn't Lorentz-invariant and it included action at a distance, faster-than-light effects of gravity that could spread immediately. So he was looking for a better theory that would be compatible with the principles of ...


9

Imagine you live in a universe governed by extremely simple rules, like Conway's Game of Life, for example. Once you discovered those rules, you might wonder, "Why do cells come alive if they have three living neighbors? Why do they die if they have one? How does that work?" (By "how" here I am referring to "what underlying mechanism makes it work?", which ...


8

Dbrane's answer contains the essential points. However I should point out that General Relativity is more sophisticated than your models suggest. The Inertial Frame concept (as used in the Equivalence Principle) is really only valid infinitesimally (whence it matches Minkowski space and "idealised gravity-free universe"). Some authors have critized the EP ...


8

Suppose you and I start on the equator, a kilometre apart, and we both head exactly due North in a straight line, so we head off in exactly parallel directions: Now we know that in Euclidean geometry parallel lines remain the same distance apart. But if you and I measure the distance, $d$, between us we find that $d$ starts off at 1km but decreases as we ...


7

I would say Brian Cox is being too cryptic. He is stating what is known as the Principle of Equivalence. In pure general relativity, gravity is not a force. It is the curvature of spacetime causing objects to obey the geodesic equation. This is a geometrical feature: the geodesic equation has no mass dependence. In free fall, the objects are unaware of their ...


7

There are several qualitative and quantitative differences between gravity and magnetism. When you attract 'neutral' bits of metal with a magnet, or attach it to something like a plate of metal, what's happening is that individual atoms of the metal react to the magnetic force. In a ferromagnetic metal, one with a similar electronic structure to Iron or ...


7

Different regions of a general spacetime that are Minkowskian to $\mathcal{O}(\Delta x^2)$ can have light cones which have null rays that point in different directions. A particle in this spacetime moves from one such region to another by connection coefficients, sometimes called the Christoffel symbols, which patch together these different locally flat ...


7

In 1921 Einstein gave a series of lectures in Princeton, that you can read today under the title "The Meaning of Relativity". It is an early and very special description of General Relativity, where he emphasizes much the concepts and reasoning that lead him to the theory. Nobody is able to know what was really inside Einstein's mind, but in that lectures ...


7

Equivalence principle states (very roughly) that movement of objects doesn't depend on their mass (so long as they are massive, of course). These important observation is what introduces (pseudo)Riemannian geometry into the theory of gravitation, because it essentially tells us that matter that is not acted on by other forces follows the geodesics of the ...


6

Inertial mass describes an object's resistance to change in velocity. The more inertial mass something has, the harder it will be to change its velocity. Gravitational mass describes an object's ability to attract other matter (and under GR, to curve spacetime). The more gravitational mass something has, the more attracted to it other things will be. When ...


6

The Weak Equivalence Principle, or WEP for short, states that under identical initial conditions, the motion of particles of different masses in a given gravitational field is identical. Or in other words, there are no physical effects that depend on the mass of a point particle in an external gravitational field. This is just the equivalence between the ...


6

Well, it depends... If you just made the sun much heavier, so the earth would have to move faster in it's orbit, you wouldn't feel any different. It's just that the year would be shorter and the tides higher. If you just put a rocket behind the earth and somehow put it on rails so it couldn't go to a different orbit, then you'd feel it. You'd be heavier in ...


6

The statement is better in the following form: The Principle of Equivalence implies that there exists a coordinate system such that the metric of spacetime is locally $\eta_{\mu\nu}$. Note the word locally. If there is real curvature, then there exists no coordinate system $x$ such that $g_{\mu\nu}(x)=\eta_{\mu\nu}$. The Principle of Equivalence ...


6

Here's a simple demonstration: Consider flat space (i.e. Minkowski), viewed in a rotating frame (in e.g. cylindrical coordinates one just replaces $\phi$ by $\phi'=\phi+\omega t$). One can calculate (without too much trouble) that, in these coordinates, a spatial line element can be expressed in terms of the canonical cylindrical coordinates as $$ ...


6

To give a short answer: There is a huge geometric framework behind electromagnetism. This framework is gauge theory! The leading idea is that you have electromagnetism as the gauge theory of a $U(1)$ Lie group. To keep the theory invariant under local $U(1)$ transformations, you introduce a connection (the gauge field $A_\mu$ which is identical to the ...


6

Is it because the acceleration is too weak? It is too weak with respect to the four forces we measure. The fact that the four known forces are so much stronger means that agglomerates of particles, up to the scale of galaxies are not internally affected, they keep their structure intact, like the famous raisins in the rising bread. It is only at the ...


5

In general relativity, you're dealing with a 4D spacetime, so the "points" in spacetime are events, and the measures that you can make coordinate-independent statements about are intervals instead of distances. The rule that applies is that the world line with the longest possible proper time between two events is a world line that involves zero proper ...


5

There are two answers to this. The simplest is that the curvature is small if you're far from any masses, so motion will be approximately in a straight line at a constant velocity. The second answer is far more important, but also far harder to explain. Basically it's that we define a straight line as the trajectory followed by a freely moving particle. So ...



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