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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 ...


23

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 ...


23

But can someone please explain why this is without using pure algebra? I will try without a single formula. In Newtonian gravity, the gravitational force on a particle is proportional to the particle's gravitational mass; the more gravitational mass, the more the gravitational force. In Newtonian mechanics, the acceleration of a particle, for a given ...


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

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 ...


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 ...


13

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 ...


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

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 ...


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

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 ...


11

You certainly know that all things fall at the same rate regardless of their mass (neglecting friction). An orbiting body is not different from a falling body in that the only force acting on it is the gravity of the thing it orbits, so there is no reason its mass should influence its orbit.


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 ...


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 ...


9

Why does the mass of the orbitting object have no effect on its revolution at all? It does have an effect! However, the effect is immeasurably small if the orbiting object itself has a very small mass compared to the object it is orbiting. The most massive object we humans have put into orbit is the International Space Station, with a mass of 419.5 ...


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 ...


8

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 ...


8

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 ...


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

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

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 ...


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

If the bus was in a vacuum (both inside and outside), then the passenger would float. However, the effects of air resistance on the two objects (passenger and bus) are probably not negligible in such an instance. The bus will be moving relative to the outside air, and so will be accelerating towards the ground at a rate less than $g$. If we then released ...


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 ...



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