geometry vs. force

Factual Questions
1

I understand (as an idiot) general relativity. Space-time is curved by massive objects, and the world-lines of nearby objects are affected by this curvature. So, a planet orbiting the sun, or an apple falling from the tree doesn’t have any force acting on it, it is simply occupying the space it is supposed to. The curvature of that space causes it to move relative to other objects. Great.

So why is it necessary for gravity to be a fundamental force? Why are gravitons postulated? What would a graviton do?

Thx,

dgr

2

I’m not an expert on gravity, but the gist is something like this:

We can deal with all the other forces (besides gravity) really well using quantum field theory. If you want to treat gravity on that same footing, you have to introduce a particle that mediates the force of gravity (namely the graviton), and it has to have certain properties (like a spin of 2, but I won’t get into what that means right now). But then the math for your theory doesn’t end up working out anyway. (In technical terms, it’s nonrenormalizable).

So you’re left with one theory for describing gravity (general relativity), and another for describing everything else (the Standard Model of particle physics, based on quantum field theory). Both are very well tested. But the problem is, they aren’t really compatible, so you need a new theory (quantum gravity) to understand how gravity fits with everything else.

No one really knows for sure what this theory is, but maybe it looks something like string theory.

3

String theories are one option (albeit manifestly incomplete) to unify GR and QM, or integrate gravity into the Standard Model, but there are others (loop quantum gravity being the leading one); however, to date, none have made predictions that are falsifiable by any current or even conceptual means.

General Relativity postulates a plenum for spacetime described as the metric tensor in the Einstein field equations; that is to say, it describes the “shape” of spacetime in four dimensions just as a topographical map visually “describes” the contours and distances of a mountain range in three. This is all well and good until some wiseacre asks the question, “What is spacetime made of?”, the only answer to which is, “Shut up and do your problem set.” Unlike the batholith of the Sierra Mountains, we can’t see, touch, or otherwise directly measure spacetime except by its influence on other objects. If this sounds like some kind of con game where someone promises to turn your $10,000 investment into a million bucks in three months time, well, one couldn’t be blamed for feeling a little lead on. On the other hand, the calculations work insofar as we’re able to test them physically, which has been done to high precision in all but the most extreme conditions, so even if General Relativity is ultimately misleading about the underlying mechanic, it still adequately models physical behavior of mass in motion.

This does have the problem, however, of not really working with the other forces (electromagnetic, strong nuclear interactions, weak nuclear interactions), which are described and experimentally verified as quantized exchanges; that is to say, these forces are mediated by the exchange of particles of discrete energies. The electromagnetic force, for instance, is described by the exchange of real or virtual gauge bosons called photons as force carriers, and we know photons really exist because of the photoelectric effect and Compton scattering. The other forces are mediated by other gauge bosons (gluons, W and Z bosons). These exchanges allow interactions between leptons (like electrons) or quarks, but not between the two. There are theories that allow these particle exchanges to be described as fields (quantum electrodynamics or QED for electromagnetic and by extension electroweak interactions, and quantum chromodynamics or QCD for strong nuclear interactions) and thus be described in terms that work with Special Relativity.

We’re pretty confident that the tenets of QED are right, or at least on the right track, because the predictions made by the theory have been verified to a very, very high degree of precision. (Because of this, it is often referred to as the “crown jewel of modern physics.”) QCD, insofar as it is currently developed, has also provided very extensive and exacting predictions which have been experimentally verified. So by doing a little mathematical fudging we can describe these interactions as smeared out fields that distort an underlying plenum (again, of a type that we can’t directly experience) that reacts to electric or color charge in analogue to how spacetime reacts to mass. However, but no one has yet quite figured out how to describe these theories in concert with gravity for General Relativity, or indeed, with each other. (Unification of electroweak and strong interactions would be a Grand Unified Theory, or GUT.)

If we assume that these forces all work in the same general scheme, i.e. the distortions in their fields are due to quantized interactions between mediating particles, then it follows that distortions in spacetime from mass that “cause” gravity must be due to some kind of exchange of a different type of gauge boson, dubbed the “graviton”. Like other gauge bosons, gravitons don’t directly interact with bosons of other forces, at least not under normal conditions, and because gravity is a very weak, long range force, it is extremely difficult to find any direct evidence of them; in fact, it is impossible, by any means conceived, to detect the action of a single graviton. (On the other hand, your eyes can easily detect the absorption of a single photon.) And attempts to describe gravity as exchanges of quantized particles (quantum gravity) have run into substantial problems; as time314 says, the theory is nonrenormalizable, and so you end up with infinities that don’t cancel out, giving nonsensical results.

How to deal with this? The Standard Model predicts that there is a more fundamental gauge boson, called the Higgs boson, which mediates reactions between other gauge bosons (including, oddly, itself), and an invariant field called the Higgs field that gives the property of inertial mass to fundamental particles. So gravitons may just be an exchange between areas of this field where distortions that create mass exist. This would unify all forces under a single mechanic, albeit one that is “broken” into seperate, non-interacting forces except at very high energy levels.

So, to directly address the questions posed by the o.p., gravity is a fundamental force because it involves interactions between fundamental particles, i.e. all particles with mass. Gravitons are postulated so that the force works in the same manner as the (mostly) verified quantum theories of the other forces do, and thus gives an ability to relate all fundamental forces together into one unified mechanic. And gravitatons “exchange” the force, resulting in attraction between massy objects. This can be alternatively (and more easily) represented as distortions in the underlying spacetime, but we don’t really know what that is other than something that gets distorted in the presence of mass, which is a bit like answering a question by asking the same question in response, which is the sort of thing that so annoys critics of playwrite David Mamet. So, while that really answers any of your questions in a definitive way, it is about the best that the present state of physics can manage.

Stranger

4

Beautiful. A piece of work. <applauds>

5

Thanks for your thoughtful and interesting reply.

I never appreciated that GR was so incomplete. Attributing everything to the terms of distortions of spacetime without defining spacetime is like saying “God made it.” I still don’t “get” how exchanging quantified particles causes attraction (other than in a similar fashion as with the other forces, I imagine), but at least I understand that GR really speaks to how gravity behaves and what its effects are rather than it’s underlying causes.

Thank you again, Stranger!

DGR

6

I wouldn’t call General Relativity incomplete; indeed, given the expansiveness of the theory and the accuracy to which predictions made from applying the theory it should be very successful as a theory that unifies Special Relativity and the effects of mass-energy (i.e. gravitation). However, GR doesn’t attempt to define what “spacetime” is; it is content to merely describe the topology of it. This goes one better than the Newtonian model of gravity which merely assumes some mysterious force arising between two masses without any conceivable linkage. (Not to bash Newton, who himself was uncomfortable with the notion of “action at a distance”; prior to his conception of gravity, nobody even really noticed or questioned that any attractive force was necessary to explain ballistic motion, which is a bit like walking into your bedroom and not noticing that an elephant is laying in your bed.)

Einstein himself recognized that the whole business of the metric tensor was an abstraction, and that furthermore the (then known) fundamental forces of electromagnetic and nuclear interactions should all somehow cozy together into one unified theory. Analogues between Maxwell’s equations and gravity in GR suggested that a symmetry existed which would actually define them as different aspects of the same fundamental interactions. However, Einstein focused on geometric solutions to this problem, as he was uncomfortable with the tenets of quantum mechanics (despite having received the Nobel Prize in Physics for having described the photoelectric effect, a basic demonstration of QM) and so missed out on much of the theorizing that ultimately lead to the Standard Model and the electroweak force, a relativistic unified field theory of electromagnetism and the weak nuclear interaction.

Today, we know as much about the composition of “spacetime” as Einstein did, which is to say not at all. Since we can directly observe or measure it, we can only attempt to describe the action and reaction of it in response to energy, but fundamentally, it is something so different from anything we experience in normal life that an attempt to describe it in relation to a concept we do understand is ultimately prone to failure. It is not a bedsheet (although that analogy is descriptively useful), nor a whirlpool (although it can behave like that), nor a lump of expanding bread dough, or the skin of a balloon, or any other familiar concept. Similarly, the business of forces being the result of exchanged particles is but a very rough analogy, albeit one that works mathematically.

So we don’t know anything about what spacetime is, and we will probably never know in the sense that we know what the price of milk is, or where Portugal is located, but we can spin beautiful theories which, if they predict results that we then confirm, at least let us cope with the effects better than predicting tax revenues via the Laffer Curve. Similarly, we may never know what fundamental particles really are (although we might think of them as vibrating strings, or twisted vacuum, or little pan-dimensional hypermonkeys) but if we can describe the results of their exchanges we can at least take comfort in knowing that they work, if not how they work. It’s not so much faith as an acknowledgement of the limitation of understanding. The alternative is to ramble on about the implicate order of the universe, which is a fun pastime for metaphysical theoriests but doesn’t really get us any closer to free energy and flying cars.

Stranger

7

Well, I wouldn’t go that far. “Explaining” things with “God did it” isn’t predictive, whereas General Relativity makes testable predictions (including ones which have in fact been tested and confirmed).

To put it another way, “spacetime” is as defined as it needs to be in order to make testable predictions, and that’s not a trivial thing. That said, I’m not sure how to answer questions like “What is spacetime made of?” (nor am I sure precisely what Stranger means in posing such a question – Does it have to be made of anything?) Perhaps instead of saying “spacetime is curved” it would seem less vague to say “The paths that objects follow through space and time are changed from what they would be in the absence of gravity.” Newton’s 1st law tells us that objects travel in straight lines unless acted on by a force. So when gravity changes the path an object follows, you can either say “There’s a force acting on the object”, or you can say “The definition of a ‘straight line’ has changed.” General Relativity essentially takes the second approach.

Anyway, there’s a school of thought that says the purpose of physics is to predict what will happen when you do experiments, and curved spacetimes and exchanges of virtual particles are all just convenient calculational tools for generating these predictions. That take on things is a little extreme for my tastes, but there it is.

EDIT: I typed this up before seeing Stranger’s most recent post (but procrastinated on hitting submit), so it might cover some of the same ground.

8

Well, we don’t expect it to be made of stuff, but from an ontological perspective we’d like it to be formed from something more substantial than angels’ wings and fairy breath. The reality is that it is unlike anything our primitive monkey brains cope with on a regular basis, and so we resort to crude analogies and sophisticated but ultimately unrevealing mathematical models to describe it. If you are a very noteworthy and revered physicist working in quantum mechanics you can wax lyrically about “wholeness and the implicate order” which sounds real good but doesn’t really get you anywhere closer to groking the underlying nature of the fabric of reality.

I believe the quote you are looking for is from Niels Bohr: “It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.” He also gave us such gems as “There are some things so serious you have to laugh at them,” and, “Your theory is crazy, but it’s not crazy enough to be true.”

Stranger

9

To give a more concrete reason for why quantum gravity is necessary: Using our current, separate, understandings of general relativity and quantum mechanics, we can predict that black holes evaporate. This prediction can be made in a number of different ways, and is about as strong as a purely theoretical prediction can be. Unless there are serious flaws in many different aspects of our understanding of relativity, quantum mechanics, and thermodynamics, black hole evaporation must occur.

Well, all of our current models for black hole evaporation predict that the energy of the particles emitted by a black hole depend on the mass of the hole. Specifically, the typical energies of the particles should be inversely proportional to the hole’s mass. This works just fine, for stellar-mass black holes, since the energies involved end up being completely miniscule.

But what happens when we extrapolate this? If the energies of the particles emitted are inversely proportional to the hole’s mass, then there must be some black hole mass small enough (specifically, in the vicinity of the Planck mass) that, were the same relationship to hold, the hole would be emitting particles with more energy than the hole itself. This is, of course, absurd. Like so many other extrapolations, this one clearly breaks down when you take it too far.

But why does it break down? There must be some physical phenomenon which occurs with relation to black holes whose mass approaches the Planck mass, which changes the behaviour of the Hawking radiation produced. Since this phenomenon involves the Planck mass, it must necessarily inherently involve both gravitation and quantum mechanics. The explanation for such a phenomenon, whatever it is, must therefore be a theory of quantum gravity.

10

This might seem a foolish question to folks like tim, stranger, and yourself, but I have to ask anyway: If a black hole emited more particles than it’s energy equivelent, why wouldn’t it “disappear?” What would be left?

11

Not a silly question. Chronos isn’t saying that evaporation ceases when you reach a sufficiently small mass. He’s saying that our non-quantum theories for how this evaporation occurs clearly break down at that point. Particle theorists come at the problem the other way and ask things like “What particle decay modes might a teeny tiny black hole have?” and “What would be its lifetime?” General relativity doesn’t offer us anything useful at this scale, though. There’s a wide theoretical gap to be bridged.

12

Well, presumably that is what happens – at some point the black hole radiates itself out of existence. But I think what Chronos is saying is that General Relativity doesn’t tell us how this happens. That is, if GR worked all the way down to the Planck scale, you could have a black hole that was radiating more energy than it contains. This makes no sense, so in the real world GR must be superceded by a theory of quantum gravity which describes how black holes cease to exist.

[simulpost with Pasta]

13

Let me take a breath.

So, at a Palanck scale, under GR mathematics a tiny black hole would contain enough gravitation to consume half of a Hawking’s radiation partical pair - causing it’s twin to radiate, and that twin would have greater energy than the black hole itself.

If I got this right, I understand why it’s impossible - but it also seems impossible for an anti-matter particle to be consumed by a black hole smaller than itself. So, probably I don’t have this right.

14

Forget the last question. I misread somthing on Wikipedia. The Planck scale black hole is not smaller than the partical. Thanks for all of your help.

15

There is an analogy here from the beginning of the 20th Century with classical electodynamics and quantum mechanics in the ultraviolet catastrophe, to wit that electromagnetic energy in a reflecting cavity should have an infinite number of modes, and by the (classical) Rayleigh-Jeans law, each frequency should have power proportional to frequency squared; summing all that up would give you an infinite amount of energy in even the smallest cavity, which is of course, impossible. Albert Einstein resolved this with his proposed explanation to the photoelectric effect using a relationship developed by Max Planck in 1900 which was intended to explain the divergence of classical theory at high frequencies (those at ultraviolet and higher) by limiting the vibrational modes radiating at discrete energy levels proportional to frequency, such that at high frequencies the number of energy quanta radiated is reduced, and the sum of all radiated energy is finite.

The irony here is that Planck considered his energy quanta to be naught but a mathematical formalism, and while Einstein accepted that light came in discrete packets (now called photons) and won the Nobel Prize for his theory that became the basis for quantum mechanics, he was forever uncomfortable with the probabilistic nature of reality implicit in quantum mechanics and insisted that the theory was seriously incomplete. One of his arguments against it, known as the EPR paradox, was invalidated by Bell’s Inequality, demonstrating that in fact that a theory using local hidden variables can’t exist, and therefore either reality is probabilistic in some fashion or non-local causality (which seriously violates precips of Special and General Relativity) underlie quantum phenomena.

Anyway, an analogous theory may exist to join General Relativity and quantum mechanics by quantizing GR in the same way that Planck’s Law quantized EM. And as with electromagnetic theory, the classical theory (which is still taught today in engineering heat transfer classes) is suitable for the vast range of everyday phenomena, while the quantum theory is only required when one gets to very small energy scales, those which approach the inviolate energy quanta. Certainly relativistic phenomena involving Special Relativity and electromagnetic force interactions (quantum electrodynamics or QED), and SR and strong nuclear force interactions (quantum chromodynamics or QCD) has provided a multitude of predictions that have been verified to extremely high precision. A similar quantized field theory for gravity would permit solution to the kinds of apparant paradoxes mentioned by Chonos. But thus far this has proven elusive to the best minds.

But hey, we’ve had fire for about 8000 years, Newton for about 300 years, quantum mechanics for just over 100 years, and the iPod for about 6. So there is still plenty of time before the theory becomes to worn out to be of use.

Stranger

16

It’s interesting to note that despite what’s frequently claimed in physics textbooks, Planck wasn’t really concerned with the “ultraviolet catastrophe”. Planck wanted to provide a theoretical explanation for the empirical data on blackbody radiation (which followed Wien’s law at high frequencies and deviated from it at low frequencies).

The Rayleigh-Jeans law (which in addition to containing unphysical divergences was contradicted by the experimental data) was derived from the equipartition theorem, which neither Planck nor apparently Rayleigh and Jeans themselves believed was universally valid. It was only later pointed out that the quantum theory resolves the question of why Rayleigh and Jeans law does not hold. (In fact, the name “ultraviolet catastrophe” was coined by Paul Ehrenfest in 1911 – more than a decade after Planck derived his law of blackbody radiation.)

In many modern textbooks the story is condensed to “Planck wanted to show why the Rayleigh-Jeans law was wrong” rather than “Planck wanted to explain the empirical data, and in the process revealed why the Rayleigh-Jeans law was wrong.”

My source for all this is this interesting article.

It’s also interesting to note that it was Planck, and not Boltzmann, who (in his work on blackbody radiation) was the first to write what is now known as Boltzmann’s entropy equation in its well known form S = k log W, and it was Planck who first worked out the value of “Boltzmann’s constant” k. Planck commented on this unusual circumstance in his Nobel Prize acceptance speech.

17

It’s actually a somewhat common phenomenon in physics that the second person to develop an idea ends up being the one who gets credit for it. Hawking was the second person to discover Hawking radiation, and Penrose was the second person to develop Penrose diagrams.