E really does equal mc^2

[KimB]

Gravitational wave, like the wave front of the lightspeed information? Or actually something else, measured? If so, what exactly? I didn't know things like gravitons or such were actualy proven to exists.

No, just plain old classical GR predicts gravitational waves. These would just be ripples in spacetime that propagate at c.

 

[Frank]

Ok, I think I'm going beyond my understanding on this, but here goes.

I think you're missing the underlying problem here. Consider Newtonian gravitation (ommitting the constants):

F = -m1*m2/r^2

We we consider that mass is energy, we can write:

F = -E1*E2/r^2

Now, I'm going to look at a gravitational field by just one of these objects:

a_g = -E2/r^2 (with the appropriate vector direction ommitted for all of these equations)

Ok, you're taking about the gravitational field exhibited as a force from a single mass while disregarding the interaction.

Now we must recognize that this is a quantum-mechanical object. It doesn't have a well-defined energy in our chosen scenario: its energy is in a superposition of two states. We can write:

|particle2> = 1/sqrt(2)|1> - 1/sqrt(2)|2>

In the language of quantum mechanics, this says that particle 2 is in a superposition of the first and second excited energy states. If we made a measurement of energy, the particle would be forced into one state or the other, with 50% probability (due to the factors in front that I chose).

Yes, it has gotten excess momentum that could flip it to two different shells with equal probability. I'm with you so far.

So, the question is: for this quantum mechanical particle, what is the gravitational force on particle 1?

Hm, while we scrapped/simplified the interaction? Ok, continuing with my view, I would say that the effect of the gravity would be a slight compression and the slight enlargement of the surface area that could interact. In other words: the increase in mass distribution would flatten the volume occupied slightly, and make it more likely that the two waveforms would interact due to the larger surface area shared between them.

It would be pretty hard to say what would happen, depending on the spin and such, but if the universe really strives to the lowest possible energy state, a reduction of the volume would be better.

So I would think, that the particle would want to get rid of the excess energy and try to occupy the smallest possible volume.

There are a few ways you can attempt to resolve this problem. The first is just stating that the interaction between these two particles consitutes a measurement of energy, so the particle picks one state.

Sounds reasonable.

But what if there was no particle one? Space itself is reacting to particle 2's gravitational field, so that alone could consitute a measurement of energy. But this then becomes a statement that no particle could ever be in a superposition of energy states, because every particle is always interacting with the gravitational field. This is clearly nonsense.

Hm, I don't know about that. While I follow your logic, it depends on the particle interacting with space itself or other particles, as far as gravity (packing it) is concerned. While I stipulate that that never happens.

The second method one might use could be to state that the gravitational field is set by the expectation value of the energy. In this case, E2 = <E> = (1/2) <1|E|1> + (1/2) <2|E|2> (the expectation value of energy would be 1/2 the energy of the first excited state plus 1/2 the energy of the second excited state, again just due to the particular numbers chosen). This has been investigated, and it turns out that you get a problem that you end up with a self-interaction effect that leads to runaway acceleration, so this idea is pretty much ruled out as well.

I don't know why it would lead to runaway acceleration, but would that be in the mathematics, or an observed effect? If the first, it might be a problem with the math. But don't ask me to fix it!

A third method would be to attempt to quantize space, to somehow allow spacetime to be in a superposition of the two chosen states.

Ok, to be able to determine the interaction, you need to have a distinct frame of reference. Otherwise, you cannot equalize the math.

Then again, if it's only a stacking problem, with the superposition being expressed by the increased size of the wavefront and increased probabilities, that would be no problem.

The first problem that this has is that most obvious ways of quantizing space result in a breaking of Lorentz invariance (i.e. special relativity, which has been tested extensively, no longer holds...this is probably going to be an issue with your idea).

This is a difficult point for me. Because I don't know enough about what makes things covariant or invariant. But, it holds a loophole as well: it states clearly that it doesn't hold for gravity. And as far as I can see, all the other effects would go for both particles equally.

The second problem is that if one attempts to create a particle description of gravity (the graviton), and looks at what properties it must have in quantum mechanics based upon its behavior on large scales, one finds that it is impossible to do any calculations: the theory is divergent.

Yes, agreed. There is no single solution to that.

To date, the only successful theory of quantum gravity of which I am aware is string theory. It does have the nice feature that gravity is not added to string theory: it is produced automatically. But string theory does have other very significant problems.

I believe you. And I'm not saying mine is better, it most likely isn't, but I'm prepared to see if it holds up.

Er, curvature is not a constant in general. It's actually a fourth-rank tensor (i.e. a tensor with four indices. It can be visualized as a stack of four 4x4 cubes of numbers at every point in space). Note that there aren't 4^4 degrees of freedom due to the symmetry of the tensor as well as gauge and coordinate freedom, but there's a whole lot more than one.

A bit like bezier curves? But what do they actually represent?

No, just plain old classical GR predicts gravitational waves. These would just be ripples in spacetime that propagate at c.

Yes, but what do they effect? The geometry will remain consistent, and propagate changes all at once. Only the (in my words) interactions between the wave packet volumes take some time to settle down. Not the initial adjustments of the volumes and location, only the propagation of the changed probabilities of interaction that caused.

 

[KimB]

Hm, I don't know about that. While I follow your logic, it depends on the particle interacting with space itself or other particles, as far as gravity (packing it) is concerned. While I stipulate that that never happens.

GR tells us that gravity does both. Gravity must interact with other particles because we see gravitationally-bound systems. It must interact with space if gravitational waves exist (since gravitational waves carry off energy).

I don't know why it would lead to runaway acceleration, but would that be in the mathematics, or an observed effect? If the first, it might be a problem with the math. But don't ask me to fix it!

It's in the math, and I'd be rather surprised if it were avoidable.

Ok, to be able to determine the interaction, you need to have a distinct frame of reference. Otherwise, you cannot equalize the math.

From what we know of the way the universe works, you should be able to calculate an interaction from any frame of reference you choose and still get the correct result. You can't expect, of course, that the interaction will look the same in different frames, but they should be related to one another through simple transformations.

Then again, if it's only a stacking problem, with the superposition being expressed by the increased size of the wavefront and increased probabilities, that would be no problem.

I guess I'm still not understanding what the effect of this "size" is supposed to be. And these are shells around what? And what determines the shape of the shells?

A bit like bezier curves? But what do they actually represent?

Not really. One way to think about it is that General relativity replaces the acceleration field one can make use of in Newtonian gravity with curved spacetime. So if the curvature tensor R determines how spacetime is curved, then it must also contain directionality information as well as just the amount of curvature.

Yes, but what do they effect? The geometry will remain consistent, and propagate changes all at once.

Well, this is where gravitational radiation comes in. Large changes in the gravitational field don't propagate all at once: they ripple outward in a gravitational wave, which propagates at c. The detectors in question are built to measure distances to incredible accuracies. As a gravitational wave passes, the distance between the objects in question is going to change ever so slightly (returning to the previous value when the wave has passed, of course).

 

[Xmas]

Because any quick change in an object's position and velocity won't propagate immediately: the new information will be sent out via a gravitational wave. This is why large objects under large amounts of acceleration are the current best candidates for detecting gravitational waves.

I thought you were arguing about the "instantaneous" effect of gravity? If gravity propagates faster than c, then c shouldn't be the limit for information propagation.

 

[KimB]

I thought you were arguing about the "instantaneous" effect of gravity? If gravity propagates faster than c, then c shouldn't be the limit for information propagation.

No, I was arguing against it. I should probably talk to the resident gravitation expert at UCD before attempting to explain why there is some apparent infinite-speed propagation, but I can say with certainty that you still won't be able to get any new information faster than the speed of light: any apparent "infinite-speed" effects you see won't give you new information: they'll just tell you what you already knew about the current position which you could infer from an optical measurement.

 

[Xmas]

No, I was arguing against it. I should probably talk to the resident gravitation expert at UCD before attempting to explain why there is some apparent infinite-speed propagation, but I can say with certainty that you still won't be able to get any new information faster than the speed of light: any apparent "infinite-speed" effects you see won't give you new information: they'll just tell you what you already knew about the current position which you could infer from an optical measurement.

How so? I can see that "old" optical information can help "predict" the true current position of an object, however that is indeed just a prediction assuming nothing unforeseen happen(s/ed) to alter the course of the observed object.

 

[KimB]

How so? I can see that "old" optical information can help "predict" the true current position of an object, however that is indeed just a prediction assuming nothing unforeseen happen(s/ed) to alter the course of the observed object.

And my point is that if something unforseen does happen, then that information won't be carried instantaneously: it'll be carried off in gravitational waves.

 

[Frank]

There are some problems with gravity and gravity waves.

For starters, it's extremely difficult to detect it/them. Gravity by itself is easy. Just jump. But measuring the effect of radical changes is hard. There aren't many good candidates for that, and they're all very far away. It just requires an extremely huge force to generate any change that might be measurable, a short distance away. We cannot do it. Which might be for the best, because anything clearly measurable would probably rip the Earth apart.

Then again, gravity is ruled by special relativity. Which says nothing about gravity as a force, or the propagation. It only stipulates that it bends the geometry of space. And at least that is proven pretty well, mostly by observing that photons do indeed change their vector slightly when they travel close to a very heavy body, like a star. And, as they have no mass at all (only spin and momentum) and no time in which any force can react on them, it has to be due to space being deformed.

So far, so good. But, special relativity states that everything is subjected to the speed of light, and particles cannot do anything else but hold to the inverse square law of distance. Which is quite a bummer, as it turns out that all mass in the whole universe has to influence everything else through gravity.

There is no possible way how particles (gravitons) can be able to distribute and propagate the force (if gravity is one) onto anything else, unless you remove location from the equation as well. Which is basically what string theory does. And stipulate that there is about a whole extra universe (multiple extra dimensions) interwoven within the observable one, that only holds gravitons. Which seems to be a bit extreme to me.

Then again, nobody has been able to prove (or even hint experimentally) that gravitons might actually exist at all. Which might re-enforce the notion that they are in some kind of parallel universe to ours. Which is also what is used to show how other effects could behave like they do when they shouldn't.

I don't know about that. Sounds too much like: "God did it, stupid!", to me. Then again, it's almost certain that our universe is just some hypersphere in a much larger whole, so who knows.



Ok, back to gravity, and especially gravity waves.

We seem to have two problems with (all) the current theories:

1. There is no way whatsoever how gravity could propagate through particles when they would have to be part of our universe.

2. Although the gravity caused by any mass clearly seems to have an effect on every particle in the universe, that would require an instant force at any distance.

Therefore, the idea that gravity acts on (or essentially is) the geometry of space itself, directly, is very hard to circumvent. But, in that respect, it is clearly a violation of special relativity (c) as well as quantum mechanics (particles). The two most tested and reliable theories in the field of physics, that are the foundation of things like the highly respected and tested to death Standard Model. Or even mundane things like IC's.

To circumvent the most glaring problems, science postulates gravity waves. What those mean is: although the propagation of changes caused by gravity happen instantly on any scale and distance, the observable effects happen through ripples in the fabric of the universe, that respect the speed of light and are quantified. Neat solution, isn't it?

Then again, just like gravitons and the violation of locality, nobody has yet been able to prove the existence of gravity waves, or even conducted a neat experiment that shows (or even might hint at) their actual existence.


In short: gravity is an enigma, and gravity waves are just a highly doubtful proposition to make things fit.

 

[KimB]

1. There is no way whatsoever how gravity could propagate through particles when they would have to be part of our universe.

Not a problem in the least. Particles pass through one another all the time, and gravitons are exceptionally weakly-interacting.

2. Although the gravity caused by any mass clearly seems to have an effect on every particle in the universe, that would require an instant force at any distance.

Nah. I'm pretty certain it's just the position and velocity terms that together make the force appear instantaneous for small accelerations (I believe this comes out of gravitation requiring lorentz invariance to hold, but I haven't done the math). So for most objects, the gravitational position will appear to be the "real" position, but for objects undergoing extremely large accelerations, we'll see sort of a "delayed image," just as with visible light.

To circumvent the most glaring problems, science postulates gravity waves.

No. Gravitational waves are a natural result of General Relativity. They're not put in.

 

[Frank]

Not a problem in the least. Particles pass through one another all the time, and gravitons are exceptionally weakly-interacting.

They might be, as nobody knows. They're just speculation. But even so: how could they possibly do what they should? There cannot be enough of them, even if they're much more numerous and less able to interact than neutrinos. Then again, they do have to interact with their supposed target, don't they?

And then you would need a mechanism that releases a whole cloud of new gravitons on each hit (where would the energy come from?), as they're subject to the inverse square law of distance: they spread out.

Nah. I'm pretty certain it's just the position and velocity terms that together make the force appear instantaneous for small accelerations (I believe this comes out of gravitation requiring lorentz invariance to hold, but I haven't done the math). So for most objects, the gravitational position will appear to be the "real" position, but for objects undergoing extremely large accelerations, we'll see sort of a "delayed image," just as with visible light.

Special relativity and Lorentz both demand it, otherwise the math wouldn't work. Math is pretty single-state, after all.

No. Gravitational waves are a natural result of General Relativity. They're not put in.

Show me a single source that gives a plausible experiment, or even a probable mathematical explanation of them.


I would be very interested in the view of your gravity guru on this.


Edit: how would gravitons home in on their intended target? And how would they be able to impact just the right amount of negative momentum?

 

[KimB]

They might be, as nobody knows. They're just speculation. But even so: how could they possibly do what they should?

This is just a function of the strength of the gravitational force, which is exceptionally weak. And it's not that they're not interacting: they are. It's just that the interactions, on an atomic or subatomic level, are dwarfed by all of the other forces. Gravity is just so weak compared to these that it's exceptionally hard to notice the effect of a single graviton.

Special relativity and Lorentz both demand it, otherwise the math wouldn't work. Math is pretty single-state, after all.

Special relativity = Lorentz Invariance.

Show me a single source that gives a plausible experiment, or even a probable mathematical explanation of them.

Gravitational radiation? Well, the mathematical explaination is simple, really. If you remember Einstein's equation that I described earlier:
G = T

...where G and T are both second-rank tensors (two indices: a second rank tensor is sort of a special kind of matrix...sort of...). If we set the right hand side of the equation to zero (which means no matter content), then we have the equation:

G = 0

As it turns out, one can write the above as a second-order differential equation that takes on a form similar to:
(d/dt)^2(h) - (d/dt)^2(h) = 0

...where h is the deviation of the metric g from minkowski space. As you can clearly see, this is a wave equation, and the solutions to this equation will be valid whether or not there is matter content in the universe. I ripped this out of my graduate GR text.

Edit: how would gravitons home in on their intended target? And how would they be able to impact just the right amount of negative momentum?

Er, think about it as a cloud of gravitons around every object, through which objects pass. It's the same way with all forces.

Edit:
By the way, the only way in which one could possibly expect to see a graviton would be in a particle physics experiment. But at current energy scales, gravity is exceptionally weak compared to other forces, so even if you wrote down the new reactions that you would be able to see in the collider if some theory of quantum gravity was considered, you'd find that the cross sections for those reactions would be so tiny that they'd just never occur.

That said, the hope is that as we reach higher and higher energy scales, the force of gravity will actually increase to a level nearer that of the other forces. Unfortunately, naiive analyses as to what energy scale is required to have gravity equal to the other forces put that energy at the Planck scale (10^28eV: current experiments are probing about 10^12eV). So it is unlikely that we will ever see the effects of gravitons directly.

 

[K.I.L.E.R]

I believe I started a thread a while back about tachyons and I think Chalnoth said they don't exist.

Btw, from this perspective, I would see a particle that turns into a tachyon (having more momentum than the local value of c allows), as a particle where the boundaries of the volume it occupies become smaller, and so has too much energy. Which it emits in the form of cherenkov radiation. Although it would increase the probability of it jumping to a higher energy shell as well.

Which would be in line with the reason why it became a tachyon in the first place: too much other mass around.

 

[Serenity Painted Death]

Planck units are useful for equations.

They are the most annoying things imaginable once you try to practically apply them.

The Planck energy, for example, the energy necessary to probe the Planck length, and test M-theory... is, more or less, about a quadrillion times larger than what we can current produce. It is fantastic beyond imagination. The length itself is 100 billion billion times less than that of a proton. They make equations work pretty, sure enough, but there are few things more frustrating. For the forseeable future, we are utterly helpless to directly test so many important theories.

 

[K.I.L.E.R]

Why can't we get rid of planck units?

 

[Serenity Painted Death]

Because they are necessary to unlock the rest of the dimensions which exist outside of our perceptual ability, yet are the keys to the elegant simplification and unification of all known physical principles?

Superstring Theory, M-theory, or what have you... is essentially a theory and set of equations that describe creation itself. I think that obviates the need, without getting into technical details.

 

[KimB]

I believe I started a thread a while back about tachyons and I think Chalnoth said they don't exist.

More to the point, they can't exist. Any theory which includes tachyons is unstable (you get runaway accelerations).

 

[KimB]

Because they are necessary to unlock the rest of the dimensions which exist outside of our perceptual ability, yet are the keys to the elegant simplification and unification of all known physical principles?

Meh, Planck units have nothing to do with all of this. They're just fantastically easy to use when it comes to many calculations (like relativistic kinematics, for instance).

 

[Frank]

More to the point, they can't exist. Any theory which includes tachyons is unstable (you get runaway accelerations).

How do you call the state that particles get in when they exceed the current speed of c, and bleed of their extra momentum through Cerenkov radiation? I thought particles in that state were called tachyons.

 

[StefanS]

How do you call the state that particles get in when they exceed the current speed of c, and bleed of their extra momentum through Cerenkov radiation? I thought particles in that state were called tachyons.

Nope, tachyons are particles with imaginary mass [sqrt(-m) ] that exceed c_{vacuum}. Particles that emit Cerenkov radiation exceed c_{medium}

 

[KimB]

How do you call the state that particles get in when they exceed the current speed of c, and bleed of their extra momentum through Cerenkov radiation? I thought particles in that state were called tachyons.

The problem is that as they bleed off their "extra" momentum, they increase velocity (as the momentum of a tachyon approaches infinity, its speed approaches c, just like a normal particle: but as it approaches zero, its speed approaches infinity). That's the problem with their instability, and why they're basically ruled out.

Edit: Oh, misread you. No, a charged particle that is still travelling under the speed of light in a vacuum (as all particles must), but is travelling over the speed of light in the medium through which it is travelling is what it's called.

Tachyons are particles that always travel over the speed of light by definition.