E really does equal mc^2
- Thread starter K.I.L.E.R
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What is a triangle? Three points connected by three straight lines. So what I have described on the surface of a sphere is a triangle if you allow the definition of straight lines to vary (which you have to in order to describe curved space, and it turns out that there is a unique way to do it). And you don't even need extra dimensions to do it, either.Crazyace said:
What you've described, however, is not a triangle. It is no more than a projection of one, which is entirely different.
Well, yes. It can be. When that would be the 2D representation of a triangle on a surface curved in multiple directions that would make that shape the shortest distance to interconnect the dots.K.I.L.E.R said:
Which is essentially what they're talking about in relation to gravity, expressed as deforming the "fabric" of our space-time reality. Which is, so to say, the thing against which everything else is measured in the first place. In other words: when the geometry changes, everything inside gets shuffled around as well.
The point raised is, that that deforming force is instantaneous, as described in special relativity. Although it would be more correct to say, that the propagation of the changes is instantaneous, in that it seems to make the whole space-time continuum shift to the new state in the next possible measurement. It's the strongest reason to believe that time is quantified as well (happening in frames, so to say), and that the geometry is updated in between those.
We can only measure that through effects that respect the speed of light. Which is the main reason why there is no direct relation between special relativity and quantum mechanics.
Another thing mentioned is, that quantum mechanics stipulate that all interactions are quantified (ie. use integer math), while the effects have a probability that's represented by a fractal number, although the possible permutations are integer as well.
Like, you can state the possible actions (permutations) that a particle could perform, which are finite and clearly defined, but the probability of them occuring is fractal, although the thresholds at which they can happen are clearly defined as well.
Like: you could have a particle, that is bound in a certain layer. If you increase the energy to the amount needed for it to flip to the next layer (which is a fixed amount), the actual transition will happen at the moment the particle is in the right state to do so, which only has a probability of occuring. And there is no way for a particle to be somewhere else than in a strictly defined layer. It cannot be somewhere between them.
Which is what makes Planck units so interesting, as you can count all those effects in multiples of those, or in indirect units that use those as the base unit.
Yes, I agree.Chalnoth said:
Shouldn't it be better to talk about direction and speed in this? Because I have not the faintest idea how the geometry itself looks like. (Does anyone?) I only know that I can measure things that exist in the same location, and proclaim it "flat", as far as the measurements are concerned. As everything would exist within the same piece of geometry, and be subjected to the same forces.
Well, mathematically it's rather difficult to do so. You don't talk about forces any longer, for example, in the framework of General Relativity. In particular, you can't talk about light as undergoing a gravitational force (thinking just in special relativity, for instance, time does not move at all as far as the photon is concerned, as it is moving at the speed of light, and as such any force applied could never act over any finite amount of time and change its direction).DiGuru said:
So it turns out that it's a much more general statement to just talk about objects following geodesics (shortest space-time distance between two points). Photons, for example, follow null geodesics (where the space-time distance is zero: this is just saying that the time that the photon feels is zero over the path it follows).
You also have to realize that in the framework of General Relativity, it's typically desirable to perform operations in a coordinate-free representation. The reasoning behind this is that it is possible to have apparent anomalies come into your analysis that turn out to just be based upon the particular coordinate system you chose. One simple example is spherical coordinates: at theta=0 (the north pole, typically), the phi coordinate (your lattitude) has no meaning. In GR, this can appear to produce a singularity at the poles, one which is obviously is not really there.
Chalnoth said:
My point is that you have changed the definition... The definition of a triangle is built up on the definition of a straight line.. If you change the definition of a straight line then you are no longer talking about the same thing - My point is that the angle rules etc are based on the strict definition and are complete in that definition. When you play semantic tricks you haven't invalidated those rules, you've just incorrectly labelled something they shouldn't apply to.
Anyway, merry Xmas ( or other seasonal greeting
)But you have to change the definition of a straight line when dealing with curved space or curved spacetime. To put it simply, a cartesian coordinate system where all parallel lines never meet just doesn't work in curved space: at some point, your grid will break down. And what's worse, if you attempt to hold to this system of parallel lines not intersecting, you'll find that your definition of straight becomes completely arbitrary.Crazyace said:
I believe that parallel transport is the unique way of defining a straight line in curved space (It's the only way I know about, but I actually think it's been proven, though I'm more into the physics than the math). Parallel transport is the idea that a "straight" line is one that remains parallel to itself as you move along the line (the result of implementing parallel transports gives you the aforementioned geodesics).
So I suppose that a better way of describing this isn't that the definition of a straight line has been changed to deal with curved spacetime, but rather that the definition has been generalized to included curved spacetime. This is a very important concept for dealing with cosmology, for example.
Chalnoth said:
Good points.
How about this (disclaimer: I never formally studied these things, so this might be really stupid):
According to quantum mechanics, all possible "slots" that can hold the propagating waveform of any particle, do so. (Like, how we divided the electromagnetic spectrum into channels, so they can each carry a transmission). Albeit ruled by probability, so there might be some temporal gaps and overlap, while transitions are about to take place. Or they might not (see the following).
So, we would have a multi-dimensional object, that is filled with all those slots, all of them taking up a certain "space" in that object. With no empty room in between where other things could exist.
When the gravity forces a change on that object (the whole universe), that will change the probability of those slots being in a possible location. So they change their location relative to one another, although the dimensions and the boundaries of the slots themselves don't change. And the propagation of the changes seems immediate, because everything just fills the smalles (lowest energy state) "volume" it can at all times.
So, while there is no immediate change in the waveforms themselves, the location of the slots in relation to the ones around them shifts around a bit. Which has a small impact on the possible interactions of the different waveforms, in the sense that it changes the requirements for collapsing and interacting with one another slightly.
This is (AFAIK) completely different than string theory (in as far as I know anything about that), in that there is no direct interaction in between the different stands for the geometry (no gravitons and such), but that they together just occupy the smalles possible "volume".
Does that makes any sense?
Edit: after thinking about it some more, I think the relative positions of the slots in relation to the ones around them won't shift as such, but rather the overlap and "height". A bit like (in 3D) you might assemble an onion again and again, by slapping flat pieces on top and next to one another, while the size and boundaries of the pieces stays the same, but the geometry changes a bit every time, due to how you stack them. And flatten them into the space available of the underlying layers. And the change in the geometry would be the effect of gravity.
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I'm sorry, but not to me. I'm just not sure what it is you're attempting to get across.DiGuru said:
I will say, however, that early in my grad school career I had an idea that I thought sounded really good and was different from string theory. Then I studied a little bit string theory, and found that my idea basically was string theory, though string theory has been taken much, much further than a simple idea, of course.
Anyway, string theory really is the most mathematically-beautiful possible description of our world there is. Basically, you take the following premise: assume that the fundamental objects have one spatial dimension (they're strings). Then demand that the numbers one uses to describe the string have no effect on how the string behaves. Now add specific ways in which the string can oscillate (there are a couple of simple ways of doing this). From this all of string theory follows, though it's not easy to see.
It took something like thirty years, for example, to discover that string theory also predicts that you will get fundamental objects of all possible numbers dimensions, such that string theory isn't just limited to strings. The main problem today is discovering whether or not it's possible to make any actual predictions based upon string theory.
No. Because this is highly related to the question: how do we know that we live in flat space? You really have to describe how one would perform measurements to tell whether or not you live in flat space without exiting the three spatial dimensions in which we live.Himself said:Yeah, but isn't that really just changing the concept of a triangle as a flat geometrical construct? The shape you get on a sphere should be called something else.
This is akin to testing whether or not the Earth is flat. Without leaving the Earth and looking down on it to say, "Oh, yeah, that's round allright," you would have to perform an experiment like drawing a large triangle and looking at the angles. A similar experiment, and possibly one that's easier to perform on the Earth, is to look at parallel lines. In flat space, parallel lines never intersect. On the Earth they do. You can never have, for example, two straight roads on the Earth remain the same distance apart for long.
In California, for instance, there is a large area that is covered in farmland, and is criscrossed by a relatively regular grid of roads. If you look at this grid on a small scale, it looks like all the roads are straight and parallel. But if you zoom out a bit, you'll notice that every once in a while two roads join to become one: this is because due to the curvature of the Earth, these parallel roads are always getting closer together.
On a cosmic scale, the question is equally important: how do we know we live in flat space? We find a way to "draw" a huge triangle, and measure its angles.
Ah, ok. Yes, I agree, it's probably the basic idea of something established, or just something that has been dropped long ago as not making any sense. I don't really think I can come up with something better, but I lack the background to know. Then again, because of that I might come up with a different way to look at it. Which might be interesting, or just stupid.
Anyway, I'll try to explain it a bit better, and I would really appreciate your feedback.
If we look at the basics of quantum mechanics, it says something like, that anything in existence is basically an energy packet (particle), that normally should be described as an expanding wavefront, except when interacting with another particle.
Those wavefronts (and their limits) are defined by different properties, like the orientation and magnitude (spin) and amplitude (energy density). And they are all bound in a certain shell, around or in relation to other particles. All those are determined by probability, and defined by thresholds. Unless they interact and collapse, of course.
Starting with a single atom, we can distinguish between different particles that all occupy a kind of volume inside or around each other. The boundaries are clearly defined, and they cannot exist outside those. Unless they interact and the combined particle has to occupy a different volume.
But it doesn't stop there. The layers/shells around atoms go on infinitely, although after a certain "distance", they stop being part of the atom and are just "random" particles. Especially when you have a lot of particles close together (any kind of matter), and the other particles start occupying volumes around the combined mass. And occupying a larger and larger volume.
The boundaries of a particle are determined by where the probability of the wave propagation which determines where it can be at that location and time are below the threshold of that speciific shell. So, there can be many in or around the center of atoms, even existing within the same location (although still having different probabilities and boundaries), while there can only be a single one, occupying a large volume in "empty" space.
Or, in other words: all possible shells (volumes) that can hold a particle do so, and the more other particles there are around (the more massive something is), the smaller the volume they occupy.
But, a particle isn't a point in space. It's a volume, holding a wavefront. Even so much so, that multiple particles in very close proximity (like the core of an atom), seem to be a massive particle all together, due to interference (did I say that right?).
Back to gravity.
The main problem with gravity is, that it seems to happen instantiously. String theory goes around that by making all strings one-dimensional. But a different explanation might be, that there is NO direct interaction between the individual particles at all. Rather, they all keep on occupying the volume they do, in relation to the masses around.
Picture all those probable volumes, that all hold their particle(s), all clearly defined. And put them all together by stacking them all on top of each other. So far, we're just keeping it within quantum mechanics.
Now, rather than stipulating that there is an energy packet that transfers gravity, and adjusting the dimensions of the particles to something that makes it possible for that graviton to interact with them, whatever their perceived "real" location (one dimensional strings), we just say that all those volumes are stacked all the time as close together as possible. Deforming them if needed, as long as the volume and other properties are unaffected. We could see that as the universe always trying to be in the state that requires the least energy. Which can be the thing that defines the thresholds of the boundaries of the individual wave propagation shells in the first place.
That way, the instant change of the geometry due to gravity can just stay what it is. And the only effects we would notice is the slightly changed interactions between the different waveforms, due to them occupying a slightly different location every time.
And that would also be a good explanation of how gravity could have an effect on photons. Even while the properties of the photon itself aren't changed in any way. Only the location and the probability of it interacting with other particles.
Did that make more sense?
Anyway, I'll try to explain it a bit better, and I would really appreciate your feedback.
If we look at the basics of quantum mechanics, it says something like, that anything in existence is basically an energy packet (particle), that normally should be described as an expanding wavefront, except when interacting with another particle.
Those wavefronts (and their limits) are defined by different properties, like the orientation and magnitude (spin) and amplitude (energy density). And they are all bound in a certain shell, around or in relation to other particles. All those are determined by probability, and defined by thresholds. Unless they interact and collapse, of course.
Starting with a single atom, we can distinguish between different particles that all occupy a kind of volume inside or around each other. The boundaries are clearly defined, and they cannot exist outside those. Unless they interact and the combined particle has to occupy a different volume.
But it doesn't stop there. The layers/shells around atoms go on infinitely, although after a certain "distance", they stop being part of the atom and are just "random" particles. Especially when you have a lot of particles close together (any kind of matter), and the other particles start occupying volumes around the combined mass. And occupying a larger and larger volume.
The boundaries of a particle are determined by where the probability of the wave propagation which determines where it can be at that location and time are below the threshold of that speciific shell. So, there can be many in or around the center of atoms, even existing within the same location (although still having different probabilities and boundaries), while there can only be a single one, occupying a large volume in "empty" space.
Or, in other words: all possible shells (volumes) that can hold a particle do so, and the more other particles there are around (the more massive something is), the smaller the volume they occupy.
But, a particle isn't a point in space. It's a volume, holding a wavefront. Even so much so, that multiple particles in very close proximity (like the core of an atom), seem to be a massive particle all together, due to interference (did I say that right?).
Back to gravity.
The main problem with gravity is, that it seems to happen instantiously. String theory goes around that by making all strings one-dimensional. But a different explanation might be, that there is NO direct interaction between the individual particles at all. Rather, they all keep on occupying the volume they do, in relation to the masses around.
Picture all those probable volumes, that all hold their particle(s), all clearly defined. And put them all together by stacking them all on top of each other. So far, we're just keeping it within quantum mechanics.
Now, rather than stipulating that there is an energy packet that transfers gravity, and adjusting the dimensions of the particles to something that makes it possible for that graviton to interact with them, whatever their perceived "real" location (one dimensional strings), we just say that all those volumes are stacked all the time as close together as possible. Deforming them if needed, as long as the volume and other properties are unaffected. We could see that as the universe always trying to be in the state that requires the least energy. Which can be the thing that defines the thresholds of the boundaries of the individual wave propagation shells in the first place.
That way, the instant change of the geometry due to gravity can just stay what it is. And the only effects we would notice is the slightly changed interactions between the different waveforms, due to them occupying a slightly different location every time.
And that would also be a good explanation of how gravity could have an effect on photons. Even while the properties of the photon itself aren't changed in any way. Only the location and the probability of it interacting with other particles.
Did that make more sense?
Actually, the main problem with gravity is in quantizing it. One simple way of seeing the problem is the source of gravity is momentum-energy of particles, so what happens when a particle is in a superposition of energy states? There is no mechanism within General Relativity to, for example, have space-time be in a superposition of two slightly differently-curved states.
A little bit more mathematically, the main equation that describes how gravity behaves is given by the Einstein equation. On the left side of the equation, we have the tensor (G) describing the curvature of spacetime. On the right side we have the tensor (T) that is given by the matter content. The right hand side of the equation is inherently quantum mechanical. The left hand side is not. Attempts to reconcile this difference so far have not yielded satisfactory results, for one reason or another.
The apparent "action at a distance" really isn't a problem at all. You're not actually going to ever be able to use gravity to obtain information faster than the speed of light. That is to say, if you are looking at Jupiter, even if it may appear to be in a slightly different position gravitatioanlly than visually, you could infer the correct position anyway from optical measurements.
A little bit more mathematically, the main equation that describes how gravity behaves is given by the Einstein equation. On the left side of the equation, we have the tensor (G) describing the curvature of spacetime. On the right side we have the tensor (T) that is given by the matter content. The right hand side of the equation is inherently quantum mechanical. The left hand side is not. Attempts to reconcile this difference so far have not yielded satisfactory results, for one reason or another.
The apparent "action at a distance" really isn't a problem at all. You're not actually going to ever be able to use gravity to obtain information faster than the speed of light. That is to say, if you are looking at Jupiter, even if it may appear to be in a slightly different position gravitatioanlly than visually, you could infer the correct position anyway from optical measurements.
In my interpretation of it, that would have two effects: it would increase the volume occupied by the particle (and thus the probability of it interacting with other particles), and increase the probability of the particle to jump to a more energetic shell.Chalnoth said:
The increase in volume can be unnoticeable, as the particle would still occupy the same shell. And thus it would inhibit the effects that go with that, as long as it doesn't interact and collapse. While it would keep the local matter content constant. But it would have a higher chance of interacting with other particles, as the size of the boundary expanded.
Yes, but there would never have to be one, as everything is just stacked as tightly as possible at all times. The size of the volume might very well not be measurable before collapse in the first place. And the superposition of the energy state would just be so much potential energy (albeit increasing the volume / amplitude). There don't have to be any special rules this way, as I see it.
Well, if you look at the curvature of the "surface" as occupied by the underlying volumes of the shells of the particles (in relation to the elevation of the shell of the particle you use to do the measurement), it might work out. In that case, G uses the constant that determines that curvature.
Agreed.
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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.
Which would be in line with the reason why it became a tachyon in the first place: too much other mass around.
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Because all things (particles) you can use to measure the gravity shift would respect the speed of light. If you want to measure that, they would have to be influenced by it. Like, travelling around the large masses you shift around.Xmas said:
But you could theoretically do something at a large distance, by using a planet closer by to inhibit the effect. Then again, that would require an even larger shifting of huge masses, with a really low bitrate.
So I don't think it would be practical at all. But it might be doable.
Edit: that is, if we can shift suns around at will, of course. And by that time, we might have come up with something better.
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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.Xmas said:
I think you're missing the underlying problem here. Consider Newtonian gravitation (ommitting the constants):DiGuru said:
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)
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).
So, the question is: for this quantum mechanical particle, what is the gravitational force on particle 1? 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. 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.
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.
A third method would be to attempt to quantize space, to somehow allow spacetime to be in a superposition of the two chosen states. 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). 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.
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.
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.Well, if you look at the curvature of the "surface" as occupied by the underlying volumes of the shells of the particles (in relation to the elevation of the shell of the particle you use to do the measurement), it might work out. In that case, G uses the constant that determines that curvature.
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