Formation of Bound Systems, Stars & Galaxies in General Relativity

[Frank Castle]

In particular how does matter "clump" together to form stars and planets, and how do Galaxy/star systems form?

For the latter question is the answer simply that near massive enough bodies, the spacetime curvature is significant enough that the geodesics within its vicinity are closed curves?! If so, then how does one explain how matter passing by the vicinity of star, such as light, for example, is able to pass through the system (albeit perturbed from its original path whilst passing by the star)? Is the point here, that with an object with a sufficient amount of energy is able to deviate from the natural geodesic paths imposed by the presence of the star?

 

[Orodruin]

For the latter question is the answer simply that near massive enough bodies, the spacetime curvature is significant enough that the geodesics within its vicinity are closed curves?!

Certainly not, closed geodesics would imply coming back to an earlier time in your history.

The easiest example is probably a smaller object being caught in the gravitation of a much heavier one so that you can approximate the situation with the Schwarzschild space-time. A bound object is then an object which never escapes to spatial infinity, there is nothing saying that the projection of the orbit onto a spatial slice must be closed - not even in classical physics (although it is an implication of a Kepler potential). In fact, the perihelion precession of Mercury was one of the observations that did not match Newtonian gravity that was known before GR and was accurately described by it.

 

[Frank Castle]

Certainly not, closed geodesics would imply coming back to an earlier time in your history.

You're right, sorry I didn't think this one through properly. I was think something along the lines of them being spatially closed.

The easiest example is probably a smaller object being caught in the gravitation of a much heavier one so that you can approximate the situation with the Schwarzschild space-time. A bound object is then an object which never escapes to spatial infinity, there is nothing saying that the projection of the orbit onto a spatial slice must be closed - not even in classical physics (although it is an implication of a Kepler potential). In fact, the perihelion precession of Mercury was one of the observations that did not match Newtonian gravity that was known before GR and was accurately described by it.

Thinking in terms of the curvature of spacetime around a massive body, what causes a less massive object to become gravitationally bound to the more massive one? In the context of Newtonian mechanics I found this a little easier to conceptualise, since gravitationally bound objects are those that have insufficient energy to escape the potential well of the more massive object, but this is when one thinks of gravity as a real force as opposed to in GR where it is considered as a manifestation of the non-trivial curvature of spacetime.

 

[Nugatory]

Thinking in terms of the curvature of spacetime around a massive body, what causes a less massive object to become gravitationally bound to the more massive one?

Start with the Schwarzschild spacetime around a mass, as Orodruin suggests. Let's say it's the Earth, just for ease of visualization. You are standing on the surface of the earth. Select an event - any event - along your worldline. There are geodesics in every "direction" (scare-quotes because these are "directions" in spacetime not space) through that event, and you can set an object moving on one of those geodesics by throwing it in the appropriate spatial direction with the appropriate initial velocity and hence energy.

Some of these geodesics reach out to infinity. These are the trajectories of unbound objects, and in classical terms you threw the object with a speed greater than escape velocity. Other geodesics intersect the surface of the gravitating mass; if you just drop the object, or throw it weakly so that it falls back to earth, it's following one of these. Finally, there are geodesics that neither reach out to infinity nor intersect the surface of the earth; these are the orbital trajectories of gravitationally bound objects.

 

[Frank Castle]

Start with the Schwarzschild spacetime around a mass, as Orodruin suggests. Let's say it's the Earth, just for ease of visualization. You are standing on the surface of the earth. Select an event - any event - along your worldline. There are geodesics in every "direction" (scare-quotes because these are "directions" in spacetime not space) through that event, and you can set an object moving on one of those geodesics by throwing it in the appropriate spatial direction with the appropriate initial velocity and hence energy.

Some of these geodesics reach out to infinity. These are the trajectories of unbound objects, and in classical terms you threw the object with a speed greater than escape velocity. Other geodesics intersect the surface of the gravitating mass; if you just drop the object, or throw it weakly so that it falls back to earth, it's following one of these. Finally, there are geodesics that neither reach out to infinity nor intersect the surface of the earth; these are the orbital trajectories of gravitationally bound objects.

Great explanation! That makes a lot of sense.

Can one use this heuristic analysis to explain why objects passing by a massive object, such as the Earth, either "fall" into the Earth, become gravitationally bound on an orbit around it, or simply follow a slightly perturbed trajectory as they pass by, but otherwise carry on past the Earth?! Is it simply that in each case, the initial velocity of the smaller body (and the distance that it is from the Earth) dictates which of the possible geodesic it follows in the vicinity of the Earth?!

(By the way, is the Schwarzschild metric used to describe the geometry of the solar system? Is it also the case that the geodesic that the Earth follows in this geometry is a helical path?)

 

[Nugatory]

(By the way, is the Schwarzschild metric used to describe the geometry of the solar system? Is it also the case that the geodesic that the Earth follows in this geometry is a helical path?)

The Schwarzschild spacetime is the solution to the Einstein Field Equations for the vacuum outside a spherically symmetrical mass (with negligible electric charge and rotation). Thus you can use it for any problem in which the classical solution starts with point masses and Newton's ##F=Gm_1m_2/r^2##, including planetary motion and satellite orbits.

 

[pervect]

In particular how does matter "clump" together to form stars and planets, and how do Galaxy/star systems form?

For the latter question is the answer simply that near massive enough bodies, the spacetime curvature is significant enough that the geodesics within its vicinity are closed curves?! If so, then how does one explain how matter passing by the vicinity of star, such as light, for example, is able to pass through the system (albeit perturbed from its original path whilst passing by the star)? Is the point here, that with an object with a sufficient amount of energy is able to deviate from the natural geodesic paths imposed by the presence of the star?

An interesting question. I did a bit of reading to review what information was out there.
"How did astronomers discover how stars were formed?" http://www.astronomycafe.net/qadir/BackTo212.html talks a bit about observations of star formation, basically we see that protostars form first, dense areas in a a gas cloud or a nebulae, which then condense and become real stars that ignite and start to burn through fusion.

I would tend to view the general process as being similar to the phase change where a gas condenses into a liquid. One initially has a gas cloud that's hot and unbound - so hot that the atoms forming the cloud can escape the pull of gravity due to the cloud, making the cloud unbound. As the cloud cools, eventually the atoms don't have enough energy to escape, so this system can be considered to be bound. As the cloud cools more, the binding becomes even tighter.

There are probably several cooling mechanisms, the two that come immediately to mind are cosmological expansion, and the emission of electromagnetic radiation. I'd expect that the first mechanism may be the most relevant to the question of how the first stars formed. As the universe expanded, it cooled - when it cooled enough, the phase change process began, and galaxies and stars started to condense out of what had previously been the nearly homogenous interstellar media, the condensation process starting at areas which started out as being slightly more dense than average. I believe I recall reading something about sound waves being important in the cosmology of the early universe, said waves causing some of the density fluctuations, but the details are escaping me at the moment.

The second mechanism is probably important for later generation stars.

This answer is a bit speculative, due to the lack of sources :(. But it's my best attempt at the moment, and I'm hoping it will serve as a useful springboard for discussion.

 

[Frank Castle]

The Schwarzschild spacetime is the solution to the Einstein Field Equations for the vacuum outside a spherically symmetrical mass (with negligible electric charge and rotation). Thus you can use it for any problem in which the classical solution starts with point masses and Newton's ##F=Gm_1m_2/r^2##, including planetary motion and satellite orbits.

The Sun does rotate though. Is it simply that the rotation is slow enough that it can be considered negligible for practical purposes?!

Do the possible geodesics that an object can take depend on its velocity and distance from the massive body (I'm thinking heuristically in terms of Newtonian mechanics and a uniform gravitational field where the path of an object is ##\mathbf{x}(t)=\mathbf{x}_{0}+\mathbf{v}t+\frac{1}{2}\mathbf{g}t^{2}##)? If this is the case, then it makes sense that certain objects will follow spatially bounded trajectories, whereas others are unbounded.

An interesting question. I did a bit of reading to review what information was out there.
"How did astronomers discover how stars were formed?" http://www.astronomycafe.net/qadir/BackTo212.html talks a bit about observations of star formation, basically we see that protostars form first, dense areas in a a gas cloud or a nebulae, which then condense and become real stars that ignite and start to burn through fusion.

I would tend to view the general process as being similar to the phase change where a gas condenses into a liquid. One initially has a gas cloud that's hot and unbound - so hot that the atoms forming the cloud can escape the pull of gravity due to the cloud, making the cloud unbound. As the cloud cools, eventually the atoms don't have enough energy to escape, so this system can be considered to be bound. As the cloud cools more, the binding becomes even tighter.

There are probably several cooling mechanisms, the two that come immediately to mind are cosmological expansion, and the emission of electromagnetic radiation. I'd expect that the first mechanism may be the most relevant to the question of how the first stars formed. As the universe expanded, it cooled - when it cooled enough, the phase change process began, and galaxies and stars started to condense out of what had previously been the nearly homogenous interstellar media, the condensation process starting at areas which started out as being slightly more dense than average. I believe I recall reading something about sound waves being important in the cosmology of the early universe, said waves causing some of the density fluctuations, but the details are escaping me at the moment.

The second mechanism is probably important for later generation stars.

This answer is a bit speculative, due to the lack of sources :(. But it's my best attempt at the moment, and I'm hoping it will serve as a useful springboard for discussion.

Thanks for your response. I'm hoping to understand how this structure formation occurs in the context of general relativity. From what we discussed in this thread so far it seems to me that structure formed due to density perturbations in the early universe. Particles propagating in the vicinity of these density perturbations were forced to follow geodesics imposed by the warping of the local geometry of spacetime (due to these density perturbations), and some of these geodesics intersected with these density perturbations, resulting in some of the particles "falling into" the density perturbation. Over time, this caused such perturbations to accumulate mass and eventually cause structure to form.

 

[Nugatory]

The Sun does rotate though. Is it simply that the rotation is slow enough that it can be considered negligible for practical purposes?!

Yes. (When the rotation is not negligible, we use the Kerr solution instead of the Schwarzschild solution.)

Do the possible geodesics that an object can take depend on its velocity and distance from the massive body

The "shape" of the geodesics (scare-quotes because these are lines in spacetime not space) through a given event varies with the spatial distance of the event from the massive body, because the curvature also varies with distance.

The spatial velocity of an object in free fall passing through that point tells us which of these geodesics the object is following. (If the object is not in free fall, then its worldline is not a geodesic).

 

[m4r35n357]

In the context of Newtonian mechanics I found this a little easier to conceptualise, since gravitationally bound objects are those that have insufficient energy to escape the potential well of the more massive object, but this is when one thinks of gravity as a real force as opposed to in GR where it is considered as a manifestation of the non-trivial curvature of spacetime.

You can look at the GR situation is a similar way. Look at equations 7.47 and 7.48 in this (advanced) article, where the GR case is treated as addition of a ##1 / r^3## term (in other words "extra" gravitation close to the source) in the effective potential.

 

[Frank Castle]

The spatial velocity of an object in free fall passing through that point tells us which of these geodesics the object is following. (If the object is not in free fall, then its worldline is not a geodesic).

Won't the spatial velocity of an object (in free fall) dictate which geodesic it will follow as it passes by the vicinity of a more massive body?! Is the point that in GR it is only meaningful to consider the velocity of an object relative to an observer at the point it passes that observer (since there is in general no unique path along which one can parallel transport tangent vectots between two points and so the velocities of objects spatially separated from an observer are not well defined)?!

Also, is any of what I wrote at the end of post #8 about the formation of structure in the universe correct at all?