How do we calculate galactic gravitation?

[PhDnotForMe]

<Moderator's note: Several threads merged and moved to cosmology.>

So I’m wondering about gravity. I was under the impression that gravity was created due to the curvature of 4d spacetime and whatnot. Can someone explain to me what gravitons are in relation to that? Are the gravitons making spacetime curve or are they the curvature themselves or what? Please explain:)

 

[PeroK]

So I’m wondering about gravity. I was under the impression that gravity was created due to the curvature of 4d spacetime and whatnot. Can someone explain to me what gravitons are in relation to that? Are the gravitons making spacetime curve or are they the curvature themselves or what? Please explain:)

Gravity is the curvature of (4D) spacetime. The curvature is caused by mass, energy and, in general, the stress-energy tensor. The curvature is related to the stress-energy tensor through Einstein's field equations.

That is GR in a nutshell!

But, an outstanding question for physics is how gravity arises at the elementary particle level. And, in fact, more generally, how GR is reconciled with Quantum Mechanics. The graviton is a hypothetical particle in theories of quantum gravity, which are very much work in progress.

There must be lots online about gravitons!

 

[haushofer]

So I’m wondering about gravity. I was under the impression that gravity was created due to the curvature of 4d spacetime and whatnot. Can someone explain to me what gravitons are in relation to that? Are the gravitons making spacetime curve or are they the curvature themselves or what? Please explain:)

Well, in QM we describe interactions between particles via "particle exchange" (do not take this too literally) in a flat spacetime. If we apply this mechanism to electromagnetic fields, the excitations of these fields become "massless spin-1 particles" called photons. So we can do the same with the gravitational field, which gives us "massless spin-2 particles" called gravitons. The nice thing is, that if we describe this "graviton exchange" between particles, we get a theory which looks very similar to GR! The spacetime curvature "emerges" from the graviton-exchange. However, this approach has several problems:

1) The resulting theory is not renormalizable: in these interactions, you sum over all possible graviton-energies, which leads to infinite answers. So you have to "cut-off" the energy scale, i.e. restrict the region of validity of the theory.
2) You have to choose a background (Minkowski) to start with, but only after a lot of hard work you'll find out that this background is permitted in the first place, i.e. that it is a solution to the field equations (string theory has the very same issue). This is the problem of "background dependence", and some people make it more of a problem than others. If you want to do perturbative calculations, you have to start out with a vacuum anyway. Also, conceptually it is weird to assume a smooth background while you're quantizing gravity, because Einstein taught us that gravity is spacetime curvature.

The theory I just described is called "Fierz-Pauli theory". So, short answer: gravitons are the hypothetical "messenger particles of gravity" if you apply a naive quantization scheme to GR, just like photons bring about electromagnetic forces. But from technical reasons we expect this quantization to be too naive, so we don't know whether gravitons exist in the first place.

 

[ohwilleke]

A quantum gravity theory based upon a massless spin-2 graviton should, in the classical limit, reproduce general relativity (GR) (I have yet to see any really rigorous proof of this piece of folk wisdom, and many accounts inaccurately exclude the limitation that this is true only in the classical limit). But, such a theory isn't and can't be, completely identical to GR (although devising an experimental test of whether it is one or the other is a question that has stumped physicists so far).

There are some pretty generic qualitative differences between classical GR and any theory of gravity based upon graviton exchange. Here are fifteen of them. In a quantum gravity theory:

1. Gravitational energy is localized (also https://www.researchgate.net/publication/262635777_Quantum_gravity_evaluation_of_stimulated_graviton_emission_in_superconductors)- to the extent the Heisenberg uncertainty principle permits (this is not true in GR).
2. Gravitational energy is perfectly conserved (this is not true in most interpretations of GR).
3. Graviton self-interactions and graviton interactions with other particles would look the same mathematically, while in GR gravitational field self-interactions do have an impact on space-time curvature, but while all other kinds of mass-energy inputs make their way into Einstein's equations via the stress-energy tensor, gravitational field self-interactions are treated differently mathematically.
4. Gravitons deliver gravity in tiny lumps, while space-time curvature does so continuously; i.e. sometimes graviton should act like particles instead of waves, while GR has only wave-like gravitational behavior.
5. Gravitons ought to be able to exhibit tunneling behavior that doesn't exist in classical GR.
6. A graviton based theory is stochastic; GR is deterministic (except in hypothetical systems that don't actually exist in our particular universe).
7. It is much less "natural" to include the cosmological constant in a graviton theory than in GR where it is an integration constant. See also here. In a quantum gravity theory there is a tendency to decouple dark energy from other gravitational phenomena. (Admittedly, this is something of an aesthetic judgment and is a matter of opinion to some extent).
8. In a quantum gravity theory, gravitons couple to everything so a creation operator from a pair of high energy gravitons could give rise to almost anything (in contrast, photoproduction can give rise only to pairs of charged particles that couple to photons); likewise any two particles with opposite quantum numbers could annihilate into gravitons instead of, for example, photons. Neither creation nor annihilation operations exist in GR in quite the same way, although seemingly massive systems can be converted into high energy gravitational waves. Some of these interactions (but not all of the possible ones) are discussed here and here and http://preprints.ihes.fr/2014/P/P-14-32.pdf. See also this powerpoint presentation.
9. In some graviton based theories, properties of a graviton must be renormalized with energy scale like all of the SM physical constants; in others there is a cancellation or symmetry of some kind (probably a unique one) that prevents this from happening (illustrating this possibility with the cosmological constant). One or the other possibility is true but we don't know which one. Classical GR doesn't renormalize and doesn't need to, because the singularities it gives rise to have physical meaning rather than being merely mathematical side effects of the way quantities are usually calculated in a theory that cancels out in the end and doesn't have a physical analog.
10. In graviton based theories lots of practical calculations require approximating infinite series that we don't know how to manage mathematically; in GR, in contrast, infinite series expressions are very uncommon and the calculations are merely wickedly difficult rather than basically impossible.
11. In GR singularities like black holes can be absolute; in a quantum gravity theory they can be only nearly "perfect" but will always leak a little, because they are discontinuous and stochastic.
12. In quantum gravity it ought to be possible to have gravitons that are entangled with each other (also here) while in GR this doesn't happen.
13. In quantum gravity with gravitons, the paradigmatic approach is to look at the propagators of point particles (also https://www.researchgate.net/publication/262635777_Quantum_gravity_evaluation_of_stimulated_graviton_emission_in_superconductors); GR is conventionally formulated in a hydrodynamic form that encompasses a vast number of individual particles (although it is possible to formulate GR differently while retaining its classical character).
14. In quantum gravity, calculations for almost every other interaction of every kind need to be tweaked by considering graviton loops; in GR the gravitational sector and the fundamental particles of the Standard Model operate in separate domains. For example, even if Newton's constant does not run with energy scale due to some symmetry in a quantum gravity theory, the running for the strong force coupling constant with energy scale would be slightly different than in the SM without gravitons.
15. Adding a graviton to the mix of particles in a TOE qualitatively changes what groups can include all fundamental particles that exist and none that do not; while in GR where gravity is not fundamental particle based, it does not. (See also here.)

 

[mitchell porter]

[In] any theory of gravity based upon graviton exchange...

1. Gravitational energy is localized (this is not true in GR).
2. Gravitational energy is perfectly conserved (this is not true in most interpretations of GR).

Where are you getting this from? Is there some argument that this is true in perturbative quantum gravity? (which is where gravitons arise). Because that would mean that the delocalized, nonconservative aspects of GR have a nonperturbative origin in the quantum context, which would be interesting.

7. It is much less "natural" to include the cosmological constant in a graviton theory than in GR where it is an integration constant. In a quantum gravity theory there is a tendency to decouple dark energy from other gravitational phenomena.

"Dark energy" is a phenomenological concept. But the cosmological constant does naturally arise in quantum gravity, e.g. if constructed according to principles of effective field theory.

 

[PhDnotForMe]

The stars on the outer edge of the galaxy revolve faster than our current theories predict. I know dark matter is used to explain this. My question is how exactly do we go about executing our current theories to predict that the stars should be revolving slower? I'm unsure of the exact process. I'm assuming we use Newtonian physics, g = GM/r^2. Do we treat all the accountable mass as a point mass in the center of the galaxy, or do we calculate the gravitational pull on each individual star by finding each value of the gravitational pull between that particular star and every other star and then adding all of the inward directional components together??

Are those two values the same? As in can we treat the mass of our galaxy as a point mass.

Let me know if you need clarification on my question, my phrasing might not be the best but I am very curious about this. Thanks.

 

[PhDnotForMe]

The stars on the outer edge of the galaxy revolve faster than our current theories predict. I know dark matter is used to explain this. My question is how exactly do we go about executing our current theories to predict that the stars should be revolving slower? I'm unsure of the exact process. I'm assuming we use Newtonian physics, g = GM/r^2. Do we treat all the accountable mass as a point mass in the center of the galaxy, or do we calculate the gravitational pull on each individual star by finding each value of the gravitational pull between that particular star and every other star and then adding all of the inward directional components together??

Are those two values the same? As in can we treat the mass of our galaxy as a point mass.

Let me know if you need clarification on my question, my phrasing might not be the best but I am very curious about this. Thanks.

 

[phyzguy]

Perhaps reading about the Shell Theorem will help answer your question.

 

[PhDnotForMe]

Perhaps reading about the Shell Theorem will help answer your question.

Shell's theorem applies when the mass is uniform, or relatively uniform, and spherically symmetric as in the case for planets and stars. Shell's theorem allows us to treat large massive objects as point masses. However, a galaxy is not a large massive object. It's a collection of massive objects. Using Shell's Theorem, we can model the galaxy as a collection of point masses. Shell's Theorem says the mass can be treated as point mass if it is spherically symmetric and uniform. A collection of point masses modeling the galaxy is neither spherically symmetric nor uniform. Therefore, I would argue we cannot treat the mass within the galaxy as a singular point mass. Is this correct?

 

[PhDnotForMe]

We calculate what we think should be the velocity of stars on the outer edge of the galaxy and we find they are moving faster than expected. How do we calculate this? Where can I learn the exact process of how we go about it calculating this? I know we can't calculate the correct value, but that's all I can find. I cannot find the exact process. Do we account for outer stars pulling on the inner stars, lessening the inner stars inward acceleration? I just want to find a detailed explanation of why the current matter in our galaxy isn't enough, and mathematical evidence.

 

[Bandersnatch]

Try this:
http://www.astro.utu.fi/~cflynn/galdyn/

 

[Bandersnatch]

No, the galaxy is not treated as a point mass, nor as a uniform symmetric sphere. The process involves identifying structures in the galaxy (central bulge, thin and thick disc, halo), and calculating gravitational potentials for each structure. The shell theorem is only used for the structures that it applies to.
The potentials are then added together, and velocities in such potentials are compared with those observed.

In your other thread I linked to the course materials that go over this in detail. Here it is for redundancy, in case the two threads end up merged:
http://www.astro.utu.fi/~cflynn/galdyn/
(I requested admins to merge them, since it's essentially the same question).

Oh, btw, it's 'shell theorem', not 'Shell's'. Name comes from how the results are obtained (by dividing a sphere into thin shells), not from the surname of its originator - that'd be Newton.

 

[Drakkith]

Therefore, I would argue we cannot treat the mass within the galaxy as a singular point mass. Is this correct?

No, not if you want accurate predictions for gravitational effects inside/near the galaxy. Once you're several galactic radii away the point mass approximation starts to work better.

We calculate what we think should be the velocity of stars on the outer edge of the galaxy and we find they are moving faster than expected. How do we calculate this?

Estimate the mass of the galaxy as a whole, and then star looking at how matter is distributed within the galaxy. Find where all the gas clouds are, where the globular clusters are, the density of stars in various regions of the galaxy, etc. When we do this with visible matter we run into a problem; our predictions aren't correct. The stars on the outer edges of the galaxy are rotating faster than our predictions. MUCH faster. However it turns out that if we include another type of matter into our models, a type which we can't see with light, our predictions become accurate again.

Interestingly, we already have particles that are observed to not interact via electromagnetism. Neutrinos and their brethren all interact solely through the weak nuclear force and gravitation. Hence they are already a type of dark matter candidate. It's not a large stretch in my opinion to imagine that another type of particle exists that interacts solely through gravitation.

 

[PeterDonis]

In a quantum gravity theory:

Please give a specific reference for the quantum gravity theory you are talking about that has all of these properties you are claiming.

 

[ohwilleke]

Please give a specific reference for the quantum gravity theory you are talking about that has all of these properties you are claiming.

I am not aware of any specific paper that discusses all of these issues at once in one place. I have added 35 link in that post and also five in this post, so that there is at least one link to each of the properties that demonstrate these propositions.

Generally speaking, I've included the first paper that I came across discussing a property, rather than looking for the "best" one to encapsulate it. These links are all to quantum gravity theories that quantize the gravitational field into a carrier boson, rather than to quantum gravity theories like loop quantum gravity that quantize space-time. There are three to six different main approaches to doing this depending on the extent to which you are a lumper or a splitter in your classifications, but the differences between these subtypes are immaterial to the statements made in my list.

Basically items 1, 2, 4, 5, 6, 9, 12, 13, 14 and 15 on my list are common features of what we mean when we say that something is quantum mechanical. Not only do they apply to any graviton based quantum gravity theory, they apply to any quantum mechanical theory, in general.

For example, points 6 and 13, taken together, simply say that all graviton based quantum gravity theories, like all quantum mechanical theories of forces mediated by carrier bosons generally, have a propagator that describes the probability amplitude for a graviton (or other force mediating boson) to go from point A to point B, in a close analogy to the photon propagator of QED, even though the details of how it is formulated differ in some subtle ways from one quantum gravity theory to another, and from one non-gravitational quantum mechanical theory to another. These are just a restatement of the very general path integral formulation of quantum mechanics. If a gravity theory mediated by a carrier boson didn't do those things, we wouldn't call it a quantum gravity theory.

In contrast items 3, 7, 10 and 11 on my list involve matters that are particular to hypothetical gravitons and aren't just general principles that apply to everything that is quantum mechanical and more or less define what we mean when we say something is quantum mechanical. But, they pretty much have to be true in general of all quantum gravity theories because they flow from basic principles of gravity in general.

For example, the point about differences between how quantum gravity and classical GR are formulated, made in point 3, can't be generalized to QED since photons don't couple to each other, and also can't be applied to QCD or to the Standard Model formulation of the weak force interaction (which doesn't have a snazzy three letter acronym), because QCD and weak force interactions don't have classical counterparts that are analogous to the quantum gravity-general relativity, or to the QED-Maxwell's equations relationships.

Point 7 is specific to quantum gravity because there is really nothing closely analogous to the cosmological constant in the SM. Arguably vacuum energy is similar, but one of the unsolved questions of physics is why the naive expectations for vacuum energy and the cosmological constant are vast numbers of orders of magnitude different from each other.

Similarly, item 10 is particular to quantum gravity, because all of the other forces are renormalizable and hence can be solved with existing mathematical tools (albeit not always easily), and item 11 is particular to quantum gravity because none of the three Standard Model forces give rise to singularities or infinities that correspond to real physical phenomena. Indeed, items 10 and 11 are to some extent flip sides of the same coin because one of the reasons that you can't use renormalization to just banish the infinities from quantum gravity is because a quantum gravity theory needs to give rise to infinities in the classical limit (in particular, to black holes and the Big Bang) because they exist in GR.

Finally, Point 8 is specific to quantum gravity because no other fundamental boson couples to all other fundamental particles. But, it is also the case that the basic idea that in any quantum mechanical theory, any valid Feynman diagram can be rotated in the space-time plane and remain valid, and that Feynman diagrams can be created for every possible coupling of a fundamental particle to another kind of fundamental particle, is really the foundation of this point. So, this is between and betwixt the observations that are generally true for anything quantum mechanical, and those that are particular to quantum gravity while not applying to other quantum mechanical phenomena.

 

[PeterDonis]

I have added 35 link in that post and also five in this post, so that there is at least one link to each of the properties that demonstrate these propositions.

I'll take a more in depth look when I have time, but just on an initial reading, most of these papers (a) do not appear to be published in peer-reviewed journals, (b) refer to highly speculative quantum gravity theories that don't fall within either of the two main lines of research in quantum gravity (string theory and loop quantum gravity), and (c) don't seem to be quantum field theories of massless spin-2 gravitons (they either don't have gravitons or the "gravitons" aren't the massless spin-2 gravitons that appear in the "obvious" QFT whose classical limit is GR--further comments in a follow-up post).

 

[PeterDonis]

A quantum gravity theory based upon a massless spin-2 graviton should, in the classical limit, reproduce general relativity (GR) (I have yet to see any really rigorous proof of this piece of folk wisdom, and many accounts inaccurately exclude the limitation that this is true only in the classical limit).

The rigorous proof appears in papers published by various researchers in the 1960s and early 1970s; the best final treatment appears in a 1970 paper by Deser, which unfortunately does not appear to be available online. This line of research is described in the book Feynman Lectures on Gravitation, which I believe is out of print but used copies are available online. The key results of this research were:

(1) If you construct the "obvious" quantum field theory of a massless spin-2 field, by analogy with the known QFT of a massless spin-1 field (which is part of quantum electrodynamics and was well understood by the time this research was being done), the field equation for this theory, correct to all orders in perturbation theory, is the Einstein Field Equation. (A number of ingenious methods were invented for obtaining results correct to all orders in perturbation theory in the course of proving this.) Therefore, the classical limit of this QFT is classical GR, for the same reason that the classical limit of the QFT of the massless spin-1 field is Maxwell's Equations.

(2) The QFT constructed as above is not renormalizable: it contains a coupling constant which has units of inverse mass squared in "natural" QFT units (the value of the coupling constant is the inverse Planck mass squared), and therefore leads to an infinite number of counterterms. This is different from the QFT of the massless spin-1 field, which is renormalizable (its coupling constant is the fine structure constant, which is dimensionless).

Note that #1 above is not just "true in the classical limit" as far as the field equation is concerned: the Einstein Field Equation is the exact field equation of the QFT, not just the classical limit of some other field equation that appears in the full QFT.

 

[PeterDonis]

The resulting theory is not renormalizable: in these interactions, you sum over all possible graviton-energies, which leads to infinite answers. So you have to "cut-off" the energy scale, i.e. restrict the region of validity of the theory.

Getting infinite answers out of the "bare" theory without a cutoff in the energy scale happens even if the theory is renormalizable--for example, it happens in QED. What makes a theory non-renormalizable is that it is not possible to remove the infinities with a finite number of counterterms.

The theory I just described is called "Fierz-Pauli theory".

As I understand it, Fierz-Pauli theory has a massive graviton, not a massless one. The QFT that has GR as its classical limit is a QFT of a massless spin-2 field (graviton).