How can a square-based lattice turn into a circular-based one?

[DaTario]

Hi All,

I was just watching the video on the Veritassium channel about scientific dissemination of general relativity. And I could see once again that when you try to use animations and/or computer graphics to show the transition from free space geodesics to geodesics around a planet, what you do is simply fade out the free space geodesics (based on square cells) and apply a fade in effect to a typical geodesic system around a mass uniformly distributed in a spherical region.

My question is: Let's consider that initially we have a mass-free space geodesic system (with square or cubic cells) and we choose a point in space (let's be the origin of our coordinate system) such that at this point a small sphere will appear whose mass will grow in time smoothly from zero according to a function something like $m(t) = (t^2)/\alpha$, where $\alpha$ is a positive real parameter. How does the transition occurs from this system of geodesics to the one which is usually shown in didactic expositions, namely, a system of geodesics that is formed by closed curves (typically circles) centered on the location of the mass?

Best wishes,

DaTario

 

[Dale]

at this point a small sphere will appear whose mass will grow in time

This violates GR.

How does the transition from this system of geodesics to the one we hope to be the system shown in these didactic expositions, that is, a system of geodesics that is formed by closed curves (typically circles) centered on the location of the mass?

It doesn’t. This is not a valid way to teach or demonstrate GR.

 

[DaTario]

So this is not a problem because GR has no way to model the appearance of a particle with mass m, as for instance an electron, out of the empty space (even if some EM radiation is present). Is it?

 

[vanhees71]

The Bianchi identity for the Einstein tensor implies that $\nabla_{\mu} T^{\mu \nu}=0$, which excludes a scenario, where some mass is created out of nothing.

 

[DaTario]

But, according to GR (I read something in this respect in a paper published in Am. J. of Phys.), energy densities may produce effects similar to mass distributions. Then if a system of EM waves propagate in a way that produces at time t = 1000 s a constructive interference pattern which is shaped as a highly peaked gaussian package, at this time we expect the geodesics to have suffered the same modification I alluded in the OP, don't we?

 

[vanhees71]

In GR the sources of the gravitational field are the energy-momentum-stress tensors of matter and radiation, not only mass distributions. In #3 it seems you have something in mind like the Schwinger process, i.e., the creation of electron-positron pairs due to the presence of a strong electromagnetic field. Of course here in principle everything is fulfilled, including the local conservation of energy and momentum, which is a necessary integrability condition for solving the Einstein field equations. You have to take into account the energy-momentum tensor of the electromagnetic field as well as the Dirac field describing the electrons and positrons. I don't know, whether such a scenario has somewhere been described.

 

[Dale]

But, according to GR (I read something in this respect in a paper published in Am. J. of Phys) energy densities may produce effects similar to mass distributions

Yes. Energy is locally conserved. So it cannot just appear out of nothing. That is precisely why mass cannot just appear: it has energy.

Then if a system of EM waves propagate in a way that produces at time t = 1000 s a constructive interference pattern which is shaped as a highly peaked gaussian package, at this time we expect the geodesics to have suffered the same modification I alluded in the OP, don't we?

No. This is a very different scenario, not at all the one described in the OP. Here you would have energy flowing in from outside to some place. The spacetime would begin curved and the curvature would change. It would not go from flat to curved.

 

[DaTario]

Thank you Dale and vanhess71.

Yes. Energy is locally conserved. So it cannot just appear out of nothing. That is precisely why mass cannot just appear: it has energy.

Ok with this part.

No. This is a very different scenario, not at all the one described in the OP. Here you would have energy flowing in from outside to some place. The spacetime would begin curved and the curvature would change. It would not go from flat to curved.

But are you sure that no straight line is turned into a circle?The heart of my problem lies in this disruptive transition between two systems of geodesics, one based on straight lines and the other based on circles. At some point there must be a break in continuity, when, from smooth deformations, a line closes in a circle, for example. I see a similar problem with certain representations of the Earth's magnetic field, especially when it reacts to solar winds.

I think what bothers me has to do with topology.

 

[Dale]

But are you sure that no straight line is turned into a circle?

I have never seen such a thing in GR.

I think that you are getting your information from GR from a bad source. Veritasium is good entertainment, not a GR textbook

The heart of my problem lies in this disruptive transition between two systems of geodesics, one based on straight lines and the other based on circles.

I have never seen such a thing in GR.

 

[DaTario]

Thank you, Dale, very much.

I was thinking here. When we look at the reflection of straight and luminous lines on the body of a car like the Volkswagen Beetle, it is common to see open lines becoming closed with the change in point of view. And I notice that I don't see much of a problem with this phenomenon. There are actually no open (one-dimensional) curves becoming closed. I understand it more like mountain ranges (as in a 2D graph of light intensity) that deform and merge, migrating from a linear conformation to a circular conformation.

I think I need to mature the way of expressing this doubt or annoyance. Anyway, thank you very much.

 

[Dale]

I think I need to mature the way of expressing this doubt or annoyance. Anyway, thank you very much.

You are welcome, but I don’t think you need a different way of expressing this doubt. I think you just need to discard what you think you know about GR and start over with some better sources. It seems to me that some bad sources have gotten you confused about something that simply isn’t part of GR

 

[Ibix]

I don't think there's anything wrong with the way you're expressing yourself. The problem is that you are trying to draw conclusions from graphics that are just "artist's impressions" of a change that can't happen in reality.

There is no way to describe a spacetime that is flat (i.e. one containing no matter or energy) at some time transitioning to a curved spacetime later. Any such spacetime implies a violation of local conservation of energy, and the Einstein Field Equations won't work since that local conservation law is built in to the equations.

 

[vanhees71]

It's a "gauge constraint" as is charge conservation in electrodynamics, i.e., from the gauge-field equations (in GR the $g_{\mu \nu}$ in electrodynamics $A_{\mu}$) alone it follows that energy/momentum or electric charge must be (locally) conserved.

Another side effect is that since in relativistic continuum mechanics the equations of motion are given by local conservation of energy and momentum any solution of the Einstein field equations also define a solution of the continuum-mechanical equations.

 

[DaTario]

It's a "gauge constraint" as is charge conservation in electrodynamics, i.e., from the gauge-field equations (in GR the $g_{\mu \nu}$ in electrodynamics $A_{\mu}$) alone it follows that energy/momentum or electric charge must be (locally) conserved.

Ok, I got it. But as I put it in #5 the problem is still 'on' if the system starts from an homogeneous energy distribution, given in terms of EM radiation, and evolves to a, say, highly peaked gaussian distribution centered, for instance, at the origin of our coordinate system.

I think you just need to discard what you think you know about GR and start over with some better sources.

It would be, in fact, very good if I could study a simple problem solved in general relativity theory.
I also think that an earlier stage, even simpler, would be beneficial, which would be analogous to our introductory study of kinematics, where I could study how to operate with matrices called metrics to make basic inferences about aspects of the movement of bodies (effective distances covered, measured of time intervals between events in a given metric, etc). But it seems to be very difficult to stablish these introductory level problems. Otherwise we could find some introductory warm-up exercises in Halliday or Butkov.

Looking up the initial chapter of the book Gravitation from Misner, Thorne and Wheeler, I found it considerably difficult to understand the basic properties of the metric matrix and the machinery as a whole.

 

[PeterDonis]

Looking up the initial chapter of the book Gravitation from Misner, Thorne and Wheeler, I found it considerably difficult to understand the basic properties of the metric matrix and the machinery as a whole.

Sean Carroll's online lecture notes are probably better to start with; MTW is a classic, but it's not necessarily a good introductory textbook, it's more of an advanced comprehensive resource.

 

[DaTario]

Thank you, PeterDonis.

 

[PeterDonis]

I think what bothers me has to do with topology.

In GR, the topology of a given spacetime is fixed. Different solutions can have different topologies, but there is no solution that describes one topology of spacetime changing into another: that wouldn't make sense.

There are speculations in quantum gravity about topology change, but those are speculations and are beyond the scope of classical GR.

 

[DaTario]

In GR, the topology of a given spacetime is fixed. Different solutions can have different topologies, but there is no solution that describes one topology of spacetime changing into another: that wouldn't make sense.

There are speculations in quantum gravity about topology change, but those are speculations and are beyond the scope of classical GR.

Dear Peter, I guess I mentioned topology here with a more primitive sense. I am referring to the fact that a straight line (an element of the initial system of geodesics) will become a circle (an element of the final system of geodesics). As far as I understand the introductory concepts of topology it seems like if a ball becomes a tea cup.

But it also seems to me that we may be talking about the same subject.

 

[PeterDonis]

I am referring to the fact that a straight line (an element of the initial system of geodesics) will become a circle (an element of the final system of geodesics).

This is a topology change. A circle is a closed curve, topology $S^1$. A straight line is an open curve, topology $R^1$. Such a change would require a change in the topology of spacetime.

 

[A.T.]

... what you do is simply fade out the free space geodesics (based on square cells) and apply a fade in effect to a typical geodesic system around a mass uniformly distributed in a spherical region.... namely, a system of geodesics that is formed by closed curves (typically circles) centered on the location of the mass....

Sounds like you are talking about Cartesian vs. polar coordinates. That has nothing to do with mass being there or not.

Also note that the lines in either grid aren't necessarily geodesics. And even if they happen to be geodesics, then they are merely spatial geodesics, not geodesics in space-time, as those grids usual have no time dimension.

 

[DaTario]

Sounds like you are talking about Cartesian vs. polar coordinates.

Hi, A.T., I am sure that I am not referring to Cartesian vc polar coordinates. These are only ways to represent whatever takes place or moves in space (or spacetime).

And even if they happen to be geodesics, then they are merely spatial geodesics, not geodesics in space-time, as those grids usual have no time dimension.

since the OP proposes a discussion about the evolution from a initial system of geodesics to a final one, I guess the time dimension is being considered here as well.

A very enlightening observation that I was able to gather in the article I read in the Am. J. of Phys. is that the parabola that reflects the trajectory of a particle launched from the surface of the Earth is not the geometric structure that we must consider to evaluate the curvature of spacetime produced by Earth, but the parabola in dimension 2 + 1, where the time axis enters with a factor c, which makes the curve much more like a straight line. This shows that the curvature of space-time generated by the Earth is very small.

 

[A.T.]

Hi, A.T., I am sure that I am not referring to Cartesian vc polar coordinates.

Then you should show a picture of what you mean.

since the OP proposes a discussion about the evolution from a initial system of geodesics to a final one, I guess the time dimension is being considered here as well.

If you want to show an evolution of space-time, then you need time as a coordinate in your grids.

 

[Dale]

Looking up the initial chapter of the book Gravitation from Misner, Thorne and Wheeler, I found it considerably difficult to understand the basic properties of the metric matrix and the machinery as a whole.

Sean Carroll's online lecture notes are probably better to start with; MTW is a classic, but it's not necessarily a good introductory textbook, it's more of an advanced comprehensive resource.

There is also a brief introduction from the same author that is quite good:

https://preposterousuniverse.com/wp-content/uploads/2015/08/grtinypdf.pdf

 

[PeterDonis]

If you want to show an evolution of space-time, then you need time as a coordinate in your grids.

More precisely, "space-time" includes "time", so any valid coordinate grid on spacetime will include time. Or, to put it in coordinate-free terms, "spacetime" is a 4-dimensional geometry that already includes all "evolution". It doesn't change; it just is.

 

[DaTario]

The inclusion of time as a third dimension (2 + 1) in situations like the one shown below lead to the clearification of the notion of curvature of spacetime. I have ploted the time axis using c = 100 just to give an idea.

Regarding the graphics of cartesian and polar coordinates, with good humor but also very respectfully, I must say that is an invitation to a trap by A.T. . I mean something that could well be represented by the schemes of these two coordinate systems. But the important difference is in the fact that in each system, the principal lines are to be understood as the trajectories a massive body would follow as a manifestation of Newton's first law.

 

[PeterDonis]

The inclusion of time as a third dimension (2 + 1) in situations like the one shown below lead to the clearification of the notion of curvature of spacetime.

Chapter 1 of MTW has a similar discussion (see in particular Box 1.6).

 

[A.T.]

But the important difference is in the fact that in each system, the principal lines are to be understood as the trajectories a massive body would follow as a manifestation of Newton's first law.

So instead of a grid that shows the geometry of space-time, you want to show trajectories or world-lines of free falling objects? Note that those lines can get very complex. Not sure if this would make the illustration more clear.

 

[DaTario]

So instead of a grid that shows the geometry of space-time, you want to show trajectories or world-lines of free falling objects? Note that those lines can get very complex. Not sure if this would make the illustration more clear.

I am also not sure. But when you refer to world-lines of free falling objects, aren't these lines geodesics?

 

[Dale]

I am also not sure. But when you refer to world-lines of free falling objects, aren't these lines geodesics?

Yes, but remember that these are lines in spacetime. An orbiting satellite’s worldline is a helix, not a circle.

 

[DaTario]

Yes, Dale, I am aware of this fact. It is like the extended parabola I presented in #25 (not the 2D one).

 

[Dale]

Yes, Dale, I am aware of this fact. It is like the extended parabola I presented in #25 (not the 2D one).

Ok, I just thought that would have been one possible reason for believing that a linear geodesic could turn into a circular one.

 

[DaTario]

Ok, I just thought that would have been one possible reason for believing that a linear geodesic could turn into a circular one.

Thank you, Dale. This comment of yours was enlightening. So, it seems that we can say that there are no circular geodesics, at least as long as no time travel is considered.
If this is the case, then it could have been a nice answer to my question.