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A.T.

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Well it doesn't affect the velocity vector much. The squirrel doesn't reach anywhere close to light speed.Nevertheless, I find it hard to believe that a squirrel falls from a tree to the ground due to gravitational time dilation between its head and its feet. The amount is so small I wouldn't think the velocity vector would be effected enough to cause any detectable motion,

You have to consider the natural scaling in space-time, where 1s in time corresponds to 1ls (~300000 km) in space. So even small distortions in time have what you perceive as large of an effect in space.

Yes, the trampoline analogy is bad. See here why:The shopworn analogy of a marble's path being deflected by a bowling ball on a trampoline makes sense to me for objects in motion. This video, however, makes me realize that that analogy seems to break down when dealing with restrained objects that are released (the marble would fall into the depression, but only because the trampoline is on Earth and subject to its gravity); why DOES the squirrel fall?

https://www.physicsforums.com/threa...-visualization-of-gravity.726837/post-4597121

Below is a better picture. But note that the scaling here is not the natural one mentioned above. The time axis is compressed to show the effect better.

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- #3

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Something that this type of video doesn't say is that you need some sort of law of nature to determine the motion of a particle. In Newtonian physics that law is, of course, ##F = ma##. Then, you have a law of gravity in terms of a force and the downward accelerated motion follows from Newton's law of gravitywhy DOES the squirrel fall?

In GR, you can describe spacetime as curved (that replaces the Newtonian law of gravity), but what replaces Newton's second law? If there is no force, then what law of nature determines the precise motion of a particle through curved spacetime?

There is, in fact, an alternative formulation of Newton's laws based on the

https://en.wikipedia.org/wiki/Hamilton's_principle

It can be shown that the Lagrangian approach is equivalent to Newton's laws, but is based on a deeper principle or law of nature than the force laws.

Note: is is forbidden when making a popular science video or writing a popular science book to mention the "L" or "H" words. You may talk about the genius of Newton or Einstein, but you must never mention Lagrange or Hamilton!

In GR, where there is no force of gravity, you also need a suitable Lagrangian/Hamiltonian principle, as your law of nature to determine the motion of objects. Merely saying "spacetime is curved" is not enough. A particle in curved spacetime could just say "well, so what?" The particle needs an additional law of nature that it must follow to tell it how to move in curved spacetime.

The particular law turns out to be the maximisation of a particle's

There's a thread about

https://www.physicsforums.com/threads/trying-to-understand-what-extremal-proper-time-means.488751/

That's essentially the reason the squirrel falls precisely as it does: it's following the law of maximal proper time through curved spacetime.

- #4

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I think we discussed this before. It isn't at all clear that the explanation works for point particles (I think the presenter makes some vague claims otherwise at the end, but any non-mathematical claim about the behaviour of infinitesimal things should be treated with extreme caution). It seems to me that vertical extent would affect how an object falls in a non-uniform field, which is true, but I'm not sure if this makes the right predictions. That goes double for extensible objects. It's also completely unworkable for non-static gravitational fields (or is it non-stationary? I can never remember which is which - the ones that don't have a timelike Killing vector field anyway), where time dilation isn't defined but gravitational effects still happen.

That said, the part of the metric that is responsible for time dilation is the part that does most of the work in every day circumstances, where spatial curvature (which is depicted by the rubber sheet) is a minor correction. So, arguably, this is better than that. A.T.'s video is better still.

That said, the part of the metric that is responsible for time dilation is the part that does most of the work in every day circumstances, where spatial curvature (which is depicted by the rubber sheet) is a minor correction. So, arguably, this is better than that. A.T.'s video is better still.

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- #5

A.T.

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The correct part of the explanation in the OP-video is that the gradient of gravitational time dilation is directly related to gravitational acceleration. This is already way better than the rubber sheet analogy.I think we discussed this before. It isn't at all clear that the explanation works for point particles (I think the presenter makes some vague claims otherwise at the end, but any non-mathematical claim about the behaviour of infinitesimal things should be treated with extreme caution).

Regarding the ways they visualize the relationship: There are different ways to visualize intrinsic curvature and geodesics on curved manifolds by analogy:

1) Embed the curved manifold in higher dimensional flat space, and explain geodesic as locally straightest possible lines on the embedded curved subspace. This is what the video in post #2 does.

2) Represent the intrinsic curvature of the manifold by postulating a variable "density", that affects how fast things advance in the manifold. This is similar to how a gradient in optical density bends light rays. In this analogy the free fall world lines bend due to the gradient of gravitational time dilation. I think the OP-video is supposed to work along these lines.

Both methods are just analogies that do not necessarily capture all aspects of GR. Method 1) explains the concept of geodesics much better, but it requires to reduce the space-time dimensions to 2 (1 time + 1 space). So you cannot visualize orbits which require 2 spatial dimensions.

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- #6

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I haven't watched the video in question, but from your remarks about the squirrel, it sounds like it's explaining an approximation to General relativity, known as the post-Newtonian approximation, rather than the full theory.

The approximation is good for weak fields like we have here in the solar system, but won't give correct answers in the strong field case, so for example it would be useless for analyzing or understanding black holes.

In its realm of applicability, it's not a bad approximation, though.

Some related material. "A Call to Action", by E.F. Taylor. A guest editorial for the American Journal of Physics, it's available on the autrhor's website at http://www.eftaylor.com/pub/call_action.html. Also at https://aapt.scitation.org/doi/10.1119/1.1555874. I'll give a short excerpt below.

Would you like to begin the study of Newtonian mechanics using no vectors and noF=ma? How about starting quantum mechanics with no complex numbers and no Schrödinger equation? Would you and your students enjoy exploring general relativity with no tensors and no field equations?

Suppose, further, that along the way your students learn concepts and methods central to contemporary physics research. Finally, what if particle motions described by Newtonian mechanics, general relativity, and quantum mechanics were summarized in three brief but powerful commands of nature that turn out to be variations of the same command?

Physics is already being taught this way, and students respond with enthusiasm and understanding. Moreover, the theoretical background for this curriculum has been around for a long time and is well developed and deeply understood by the physics and mathematics communities.

Here are nature's commands to the stone and electron: At the stone moving with nonrelativistic speed in a region of small space–time curvature, nature shouts:Follow the path of least action!

At the stone moving with any possible speed in a region of any finite space–time curvature, nature shouts:Follow the path of maximal aging!

At the electron, nature shouts:Explore all paths! That's it. We now examine each of these commands in turn.

So the squirrel, being a large object where we can ignore quantum effects, follows the principle Taylor describes above, the principle of maximal aging. Some authors, with good reason, call it the principle of extremal aging, what this gains in accuracy is offset by the loss in comprehensibility.

In terms of time dilation, the squirrel stayed where it was, it would age less, due to "gravitational time dilation".

[edit]

The path that maximizes the squirrels aging is a path that is locally in free fall. So if we compare a squirrel stationary at a constant height, and a squirrel that moves upwards at some velocity, slows down due to "gravity", then returns to its initial position, all in local free fall, it is the later squirrel is the one with the maximum possible age.

In flat space-time the principle of maximal aging always works fine, in curved space-time one may under some circumstances need to apply the more robust principle of extremal aging.

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- #7

Sagittarius A-Star

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In the OP-video, a law of motion is described at 03:00 by referring to Newtons laws.Something that this type of video doesn't say is that you need some sort of law of nature to determine the motion of a particle.

I think you don't need the Lagrangian/Hamiltonian principle in GR to describe gravitational effects.In GR, where there is no force of gravity, you also need a suitable Lagrangian/Hamiltonian principle, as your law of nature to determine the motion of objects. Merely saying "spacetime is curved" is not enough.

You can also make use of the fact, that you can define in every point a local inertial frame and use as law of motion, that in there, locally the 4-momentum of an isolated particle or system (for example a collision of 2 particles) remains constant, together with spacetime curvature. That means: An inertial particle moves locally rectilinear/uniformly though 4-dimensional curved spacetime.

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- #8

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You mean you can generate the correct geodesic equation from the existence of local inertial reference frames and instantaneous inertial motion therein?I think you don't need the Lagrangian/Hamiltonian principle in GR to describe gravitational effects.

You can also make use of the fact, that you can define in every point a local inertial frame and use as law of motion, that in there, locally the 4-momentum of an isolated particle or system (for example a collision of 2 particles) remains constant, together with spacetime curvature. That means: An inertial particle moves locally rectilinear/uniformly though 4-dimensional curved spacetime.

I haven't seen that done.

I'm sceptical because the LIF has no information on the spacetime curvature at that point.

- #9

Sagittarius A-Star

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You mean you can generate the correct geodesic equation from the existence of local inertial reference frames and instantaneous inertial motion therein?

The example of the OP-video is a description in an accelerated reference frame in flat spacetime (local scenario). You can describe the relative motions also in the locally inertial frame for a nearby free falling elevator cabin.

I mean, that you don't need the Lagrangian/Hamiltonian principle as a separate "law of motion", that would be missing in the video. You can even derive this principle from the postulates of Relativity plus the conservation of 4-momentum:I haven't seen that done.

I'm sceptical because the LIF has no information on the spacetime curvature at that point.

That the inertial worldline between two events is that with maximum proper time, can be easily seen at the Minkowski metric / spacetime interval formula. From that principle of maximum proper time together with the conservation of 4-momentum, the principle of least action can be derived.

- #10

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I thought the principle of maximal proper time is what you claimed we don't need?I mean, that you don't need the Lagrangian/Hamiltonian principle as a separate "law of motion", that would be missing in the video. You can even derive this principle from the postulates of Relativity plus the conservation of 4-momentum:

That the inertial worldline between two events is that with maximum proper time, can be easily seen at the Minkowski metric / spacetime interval formula. From that principle of maximum proper time together with the conservation of 4-momentum, the principle of least action can be derived.

What do you mean by conservation of 4-momentum in curved spacetime?

- #11

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Of course you can do this, but it's very tedious of course. I think Weinberg, Gravitation and Cosmology derives in this way the generally covariant formalism. What you get of course is the usual geodesic equation for the motion of a test particle in an electromagnetic field,You mean you can generate the correct geodesic equation from the existence of local inertial reference frames and instantaneous inertial motion therein?

I haven't seen that done.

I'm sceptical because the LIF has no information on the spacetime curvature at that point.

$$\ddot{x}^{\mu} + {\Gamma^{\mu}}_{\alpha \beta} \dot{x}^{\alpha} \dot{x}^{\beta}=0.$$

The dot denotes derivatives wrt. proper time. That the affine parameter is proper time leads to the condition

$$g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=c^2.$$

If you consider a massless particle ("naive photon" for the derivation of light bending in gravitational fields), you have an arbitrary affine parameter ##\lambda## instead of the proper time (because there's no proper time for massless particles) and the additional constraint

$$g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=0.$$

Formally the derivation of the general covariant tensor formalism given the equivalence principle, i.e., the existence of a local inertial frame in any space-time point goes as follows. Let the local coordinates in one specific point, where the particle is momentarily located, be ##\xi^{\mu}## and ##\tau## the proper time, such that ##\dot{\xi}^{\mu} \dot{\xi}^{\nu}\eta_{\mu \nu}=c^2##. The equation of motion for a free particle in this point (i.e., a particle being under no other influence than the graviational field) reads as in special relativity simply

$$\dot{u}^{\mu}=\ddot{\xi}^{\mu}=0.$$

Now how looks this same equation in arbitrary coordinates ##x^{\mu}##? Then you get

$$\dot{u}^{\mu}=\mathrm{d}_{\tau} \left ( \frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \dot{x}^{\alpha} \right) = \frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \ddot{x}^{\alpha} + \frac{\partial^2 \xi^{\mu}}{\partial x^{\alpha} \partial x^{\beta}} \dot{x}^{\alpha} \dot{x}^{\beta}=0.$$

Just multiplying this by ##\partial x^{\gamma}/\partial \xi^{\mu}## (implying of course the summation over ##\mu##) leads to

$$\ddot{x}^{\gamma} + \frac{\partial x^{\gamma}}{\partial \xi^{\mu}} \frac{\partial^2 \xi^{\mu}}{\partial x^{\alpha} \partial x^{\beta}} \dot{x}^{\alpha} \dot{x}^{\beta}=0.$$

In this way the Christoffel symbols are given by

$${\Gamma^{\gamma}}_{\alpha \beta}=\frac{\partial x^{\gamma}}{\partial \xi^{\mu}} \frac{\partial^2 \xi^{\mu}}{\partial x^{\alpha} \partial x^{\beta}}.$$

The usual relation between the pseudo-metric components and Christoffel symbols follow from

$$\mathrm{d} s^2=\eta_{\mu \nu} \mathrm{d} \xi^{\mu} \mathrm{d} \xi^{\nu} =g_{\alpha \beta} \mathrm{d} x^{\alpha} \mathrm{d} x^{\beta},$$

from which

$$g_{\alpha \beta}=\eta_{\mu \nu} \frac{\partial \xi^{\mu}}{\partial x^{\alpha}} \frac{\partial \xi^{\nu}}{\partial x^{\beta}}.$$

Taking the derivative of this wrt. to ##x^{\gamma}## and writing ##\partial_{\gamma} = \partial/\partial x^{\gamma}## from now on gives

$$\partial_{\gamma} g_{\alpha \beta} = \eta_{\mu \nu} [(\partial_{\gamma} \partial_{\alpha} \xi^{\mu}) \partial_{\beta} \xi^{\nu}+(\partial_{\alpha} \xi^{\mu}) \partial_{\gamma} \partial_{\beta} \xi^{\nu}] = \eta_{\mu \nu} [{\Gamma^{\delta}}_{\alpha \gamma} (\partial_{\delta} \xi_{\mu}) \partial_{\beta} \xi^{\nu} + {\Gamma^{\delta}}_{\beta \gamma} (\partial_{\alpha} \xi^{\mu}) \partial_{\delta} \xi^{\nu}] $$

or

$$\partial_{\gamma} g_{\alpha \beta}= {\Gamma^{\delta}}_{\alpha \gamma} g_{\beta \delta} + {\Gamma^{\delta}}_{\beta \gamma} g_{\alpha \delta}.$$

Now we just cyclically change the free indices of this equation to get

$$\partial_{\alpha} g_{\beta \gamma}={\Gamma^{\delta}}_{\beta \alpha} g_{\gamma \delta} + {\Gamma^{\delta}}_{\gamma \alpha} g_{\beta \delta}$$

$$\partial_{\beta} g_{\gamma \alpha}={\Gamma^{\delta}}_{\gamma \beta} g_{\alpha \delta} + {\Gamma^{\delta}}_{\alpha \beta} g_{\gamma \delta}.$$

Then adding the first two and subtracting the third of these equations leads to

$$\partial_{\gamma} g_{\alpha \beta} + \partial_{\alpha} g_{\beta \gamma} - \partial_{\beta} g_{\gamma \alpha}=2{\Gamma^{\delta}}_{\alpha \gamma} g_{\beta \delta} .$$

Eliminating the ##g_{\beta \delta}## factor and dividing by 2 finally leads to the well-known expression

$${\Gamma^{\delta}}_{\alpha \gamma}=\frac{1}{2} g^{\beta \delta} (\partial_{\gamma} g_{\alpha \beta} + \partial_{\alpha} g_{\beta \gamma} - \partial_{\beta} g_{\alpha \gamma}).$$

So we find the geodesic equation from the equivalence principle, i.e., the existence of a local inertial reference frame in terms of arbitrary coordinates.

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- #12

DrGreg

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For a free-falling object that can be considered a particle of constant mass ##m## it has a 4-momentum ##P^\mu = m \dot{x}^{\mu}##, so if you multiply Vanhees71's geodesic equationWhat do you mean by conservation of 4-momentum in curved spacetime?

by ##m##, that is your conservation of 4-momentum equation ##\mathrm{D} P^\mu / \mathrm{d}\tau = 0## (where ##\mathrm{D} / \mathrm{d}\tau## denotes absolute or intrinsic derivative, not coordinate derivative ##\mathrm{d} / \mathrm{d}\tau##).$$\ddot{x}^{\mu} + {\Gamma^{\mu}}_{\alpha \beta} \dot{x}^{\alpha} \dot{x}^{\beta}=0$$

- #13

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We don't have the geodesic equation (yet) because we aren't assuming the principle of proper time. The task is to generate the geodesic equation from local inertial motion.For a free-falling object that can be considered a particle of constant mass ##m## it has a 4-momentum ##P^\mu = m \dot{x}^{\mu}##, so if you multiply Vanhees71's geodesic equation

- #14

DrGreg

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But can't you reverse the argument and start with conservation of momentum to derive the geodesic equation?We don't have the geodesic equation (yet) because we aren't assuming the principle of proper time. The task is to generate the geodesic equation from local inertial motion.

- #15

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Yes, as per post #11. Which is what I've learned todayBut can't you reverse the argument and start with conservation of momentum to derive the geodesic equation?

The whole argument started, however, over whether it's right and proper that Joseph-Louis Lagrange and William Hamilton remain expurgated from the popular history of science - and that all the credit is given to Isaac Newton!

- #16

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From the equivalence principle we know that there's a local IRF, where the laws of SRT are valid, i.e., we have a proper time and the usual definition of the particle's momentum and mass in that specific frame. As shown above you can derive all the general covariant tensor calculus from this, including the pseudo-Riemannian spacetime structure from it. This is the physical approach to GR as a theory of the gravitational interaction with the usual geometric heuristics being derived from the equivalence principle. As I said, this must be in Weinberg's Gravitation and Cosmology, which I like very much, because the geometrization of the gravitational interaction follows from the physical assumption of the equivalence principle.We don't have the geodesic equation (yet) because we aren't assuming the principle of proper time. The task is to generate the geodesic equation from local inertial motion.

This approach only fails, when you want to introduce spin, where you need to invoke the idea that GR follows from SR by gauging the Lorentz invariance. Then it turns out that the connection and the pseudo-metric are independent fields, but that the connection must also be pseudo-metric compatible, leading to a Einstein-Cartan manifold with torsion. This approach is nicely summarized in P. Ramond, Quantum Field Theory.

- #17

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To the contrary! Lagrange and Hamilton provided the way to formulate the fundamental laws in a way which made quantum theory possible, because it made the application of Lie groups and Lie algebras easy to discover not only Noether's theorems but also the symplectic structure of classical phase space leading to the elegant formulation of quantum theory in terms of observable algebras represented by ray representations on a Hilbert space :-)).Yes, as per post #11. Which is what I've learned today

The whole argument started, however, over whether it's right and proper that Joseph-Louis Lagrange and William Hamilton remain expurgated from the popular history of science - and that all the credit is given to Isaac Newton!

- #18

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Sorry, I posted #11 too quickly. I've extended it by this very derivation.We don't have the geodesic equation (yet) because we aren't assuming the principle of proper time. The task is to generate the geodesic equation from local inertial motion.

- #19

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My point was that they rarely (if ever) appear in popular science accounts and this is a) unfair to them and b) leaves popular accounts of GR, in particular, lacking a key element.To the contrary! Lagrange and Hamilton provided the way to formulate the fundamental laws in a way which made quantum theory possible, because it made the application of Lie groups and Lie algebras easy to discover not only Noether's theorems but also the symplectic structure of classical phase space leading to the elegant formulation of quantum theory in terms of observable algebras represented by ray representations on a Hilbert space :-)).

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- #21

Sagittarius A-Star

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No, to my understanding, the Lagrangian/Hamiltonian principle is only that of least action (maybe I'm wrong). The principle of maximum proper time would exist in GR also without the work of Lagrange/Hamilton. As I said in #9, you can derive the principle of least action from the postulates of Relativity plus the conservation of 4-momentum (from which energy conservation follows).I thought the principle of maximal proper time is what you claimed we don't need?

I mean that in a local inertial frame.What do you mean by conservation of 4-momentum in curved spacetime?

- #22

PeterDonis

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to my understanding, the Lagrangian/Hamiltonian principle is only that of least action (maybe I'm wrong). The principle of maximum proper time would exist in GR also without the work of Lagrange/Hamilton

No. The Lagrangian principle is the general principle for finding an extremum of an integral and using that to derive an Euler-Lagrange equation. The principles of "least action" and "maximum proper time" are just two particular applications of the same general principle.

- #23

Sagittarius A-Star

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In another video (at 04:32), this equation is explained in an easy way.$$\ddot{x}^{\mu} + {\Gamma^{\mu}}_{\alpha \beta} \dot{x}^{\alpha} \dot{x}^{\beta}=0.$$

- #24

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The point is that that is not really an explanation: "the object falls because time curvature puts its future in a different place in space". What does that actually mean? That doesn't give me anything to get a grip on.In another video (at 04:32), this equation is explained in an easy way.

I was just looking at Feynman's

"... the lay reader only wants to have the illusion of understanding and ... buzzwords to throw around at a cocktail party."

That's what I feel about that video.

I prefer the structure of the explanation to follow the structure of the mathematics, so that it's essentially the mathematical details and grunt work that are missing.

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