[Herra Tohtori]

Ah, yes, tachyons. Quite interesting things, you can do a whole lot of stuff with those, I just recently stumbled upon a device that emits tachyons to "help you balance you inner energy patterns" or some crap like that.
But let's take a look at real world physics. What are tachyons really? Basically, until proven otherwise, they are one solution of Einsteins special theory of relativity. But does that mean they exist? In my opinion: NO.
I'll illustrate my point by an example of Pythagoras (yes, the triangle guy). As you all know, his famous formula for triangles is a² + b² = c², c being the hypotenuse. Now, if you solve this formula, you get not one but two solutions for the hypotenuse, c and -c. As far as math is involved, both solutions are correct. But if you try to take this solutions to real world physics, the solution -c disappears. A triangle with a negative length for the hypotenuse simply doesn't exist. It's basically the same with tachyons: they are the negative hypotenuse of your average particle.

Ah, but what if the sides of the triangle are vectors instead of just lines with some length?

Vectors have two properties, length and direction. Scalar quantities only have one property, amount (or magnitude or whatever) and that can be anything depending on the quantity measured, though most scalar quantities do usually have positive values only.

However, one property of particles with negative (or imaginary, whichever interpretation you think better suits reality) mass is that they would always move at superluminal speeds, whereas particles with zero invariant mass would move exactly at c and particles with positive invariant mass would move at sub-light speeds. This means that the question of what a tachyon's rest mass (or invariant mass) actually would be is irrelevant since they can not be at rest any more than photons can. Which makes it much more interesting to just ignore the outlandishness of the concept of negative mass for a while and look at how such a thing would effect the properties of an object. Mainly in this case, momentum... or rather, four-vector, seeing how the origin of the .

When you consider the fact that the mass of moving objects always demonstrates as momentum more than anything else, negative mass becomes much less of a problem because momentum is a vector quantity - negative mass would just invert the vector (at least in the traditional sense of "momentum", I dunno how exactly the four-vectors would behave, but I suspect inversion of either time or the 3-vector would happen, both being completely in line with the tachyon hypothesis...).

Gravitational interactions between tachyons and bradyons would probably be a nightmare to define, though... 

Anyhow, inversion of momentum is exactly what you would expect from the interpretations of what tachyons would look like; After the tachyon hit you, you would essentially be able to see the tachyon gaining distance from yourself... in fact, tachyons cannot transmit information because their emission and absorption are indistinguishable from each other, which would mean that a tachyon detector would actually look like a tachyon emitter. Yes, they are hypothetical, and haven't been observed, but that doesn't make them any less possible. The black hole was completely hypothetical assumption based on some pretty exotic solutions of general relativity; similarly the neutrinos were predicted long before they were confirmed to exist. Right now the Higgs' boson and Higgs' field are under the scrutiny; quantum theories predict their existence, but they haven't yet been observed.

Tachyons are, obviously, a bit different; they aren't a direct prediction but rather a way of General Relativity to deal with all the possibilities - it just says that to exceed light speed, particles would need to have imaginary (or negative) mass; it doesn't really take any real opinion on their existence, for or against. But the black holes are a good example of how a mathematical strangeness can actually exist in real world... although I still don't think real singularities exist inside the event horizon. Singularities are even more worrisome than imaginary mass mathematically, since they imply infinities in a definite universe...

[Mika]

Singularities are even more worrisome than imaginary mass mathematically, since they imply infinities in a definite universe...

Don't worry. I have never observed a singularity that is defined in the mathematical sense.

Mika

[Herra Tohtori]

Don't worry. I have never observed a singularity that is defined in the mathematical sense.

Mika

 

Well, that's re-assuring to know.

[Aardwolf]

Complex mass >> Complex momentum >> Complex velocity >> Complex position >> BAD!

[perihelion]

Herra, I still don't get your posit that the ideas of negative and imaginary mass are interchangeable.  Unless the formulae you are using can be factored to m² where ever mass is mentioned, there is a whooping big difference between -m and m*i.  Negative mass would essentially curve space-time in the opposite fashion compared to normal mass.  I can wrap my mind around that.  I have doubts that such particles actually exist, but I can handle that conceptually.  You start talking about imaginary mass and you suddenly start implying that mass has a phase component.  It is no longer a scalar quantity.  That is not compatible with the Standard Model.

[Herra Tohtori]

Herra, I still don't get your posit that the ideas of negative and imaginary mass are interchangeable.

Not really interchangeable, I'm just trying to assign some kind of sensible (lol, I know) physical equivalent to imaginary mass that appears in the equations. Since negative mass offers a simpler way for physical interpretations of tachyons, one might as well use it.

Mathemathics is all fine and dandy, but in physics you need to have some kind of physical equivalent to what you're describing in the equations, otherwise it isn't physics. Imaginary mass is just as non-scalar quantity as negative mass, due to the non-scalable nature of imaginary numbers themselves. It is a very probably possibility that I'm wrong about it, but by Occam's razor I do find actual imaginary mass a lot less probable than interpreting it as negative mass.

Unless the formulae you are using can be factored to m² where ever mass is mentioned, there is a whooping big difference between -m and m*i.

Yes. I know that. But the fact of the matter is that the imaginary mass still needs some kind of physical interpretation aside from mathematical means to take square roots of negative numbers.

Also, if you go by the numbers and calculate the time dilatation for objects traveling at velocities greater than c, using imaginary numbers to get some result from the equations, you would probably notice that the time for tachyons is flowing backwards... or the other interpretation is that their perception of direction of time is interchangeable; the end is the beginning of tachyon's journey from absorption to emission. This backwardness would also essentially invert the perceived vectors such as velocity and, indeed, momentum. And I can't really postulate any other interpretations for inverted momentum than negative mass.

Negative mass would essentially curve space-time in the opposite fashion compared to normal mass.  I can wrap my mind around that.

You're assuming that tachyons have simialr qualities as bradyons as far as their interactions with space-time continuum are concerned. They might or they might not. And if the earlier wild hypothesis of tachyons being responsible of gravitational interactions would in fact be accurate, then they would actually be responsible for making up the curvature of space-time... or effects of it. After all, in General Relativity Einstein just gives us an energy tensor and says that mass and energy affect the local space-time continuum; explanations for it are less than vague. It works but we don't really know how or why mass/invariant energy curves space-time...

I have doubts that such particles actually exist, but I can handle that conceptually.  You start talking about imaginary mass and you suddenly start implying that mass has a phase component.  It is no longer a scalar quantity.  That is not compatible with the Standard Model.

Then it isn't. It should be quite obvious by now that I can't actually provide any experimental data to support my hypotheses, and this is mostly just fun and games for me without the exactness of experimental science.

Also, imaginary mass automatically kinda adds a phase component to the concept of mass due to the fact that phase is an inherent part of imaginary numbers... or even better it makes mass a vector quantity on imaginary plane. What that amounts to in physical reality is, in my physical intuition, negative mass. Effectively, if not mathematically. You could go on calling it exotic mass or whatever, but I would still guess that it would behave as if the mass was negative.

Besides, for all we know, mass could actually have a phase component, it's just that we only observe the mass that has zero phase angle as "normal" mass. Perhaps we would perceive mass with 180 degree phase angle as negative mass. And what's in between would be something... else.  However I find this literary interpretation of imaginary mass somewhat more complex than simply saying that the imaginary mass in the equations corresponds to negative mass of particles. Because accurate mathemathics doesn't mean that all the results are physically sensible. It's obvious to use that triangles don't have sides with negative length.

[CP5670]

However I find this literary interpretation of imaginary mass somewhat more complex than simply saying that the imaginary mass in the equations corresponds to negative mass of particles.

That makes even less sense than just talking about complex mass though. The equations simply aren't physically meaningful with complex quantities, so there isn't any point in trying to assign an interpretation to them.

Don't worry. I have never observed a singularity that is defined in the mathematical sense.

A singularity can be just about anything with locally non-smooth or non-analytic behavior. There are many situations where some basic physical quantity remains bounded but its derivative blows up, for example. Phase transition in materials or turbulence in fluids are types of singularities in some sense.

[Aardwolf]

I had an idea just now, and although I have nothing to back it up, it sounds sort of interesting so I thought I'd bounce it off you guys.

I was thinking, since gravity is effectively equivalent to a curvature of space, but there has yet to be a generalization of this law to the other forces...

What happens if you look at charge as the imaginary part of a complex mass value? I haven't done the math, but it seems like it might explain how gravity always attracts, and electrons and protons attract or repel depending on what they're interacting with. Furthermore, both gravity and electric interactions have a falloff rate of 1/r².

I have no idea how this and the tachyons thing would fit together though.

What do you think?

Edit: I was trying to do some maths on this, assuming that the force (times d² of course) should be proportional to the product of the two "complex masses." However, this gave complex results in many situations. So I don't really know what the correct solution would be.

[Bobboau]

what if there exists a complex axis to the universe? what if that is why gravity is so weak, cause it's bleeding over into the complex end of the universe. of the complex interactions we only see the real component. the three spacial dimensions are not scalar but in fact planar. this is frik'n nuts **** to be thinking about.

[Scuddie]

what if there exists a complex axis to the universe? what if that is why gravity is so weak, cause it's bleeding over into the complex end of the universe. of the complex interactions we only see the real component. the three spacial dimensions are not scalar but in fact planar. this is frik'n nuts **** to be thinking about.

It is?  Good thing to know.  Gives me another reason not to waste my time thinking about it.

* Scuddie throws black holes into the equation, just to piss people off.

[Mika]

A singularity can be just about anything with locally non-smooth or non-analytic behavior. There are many situations where some basic physical quantity remains bounded but its derivative blows up, for example. Phase transition in materials or turbulence in fluids are types of singularities in some sense.

Then the derivative is incorrect and is neglected. As long as you can have two discrete measurement points, you can always approximate between them, and construct the real derivative from the measurement data. Those listed examples I consider as discrete valued functions which result from the fact your measurement setup is not quick / accurate enough.

Locally non-smooth functions have never been that much of a problem. Approximate the steps with spline and voila, you have continuous functions.

Yeah, I know these tricks are not mathematically valid, but unfortunately work all too well in reality.

Mika

[CP5670]

Then the derivative is incorrect and is neglected. As long as you can have two discrete measurement points, you can always approximate between them, and construct the real derivative from the measurement data. Those listed examples I consider as discrete valued functions which result from the fact your measurement setup is not quick / accurate enough.

I am talking about the underlying continuous-domain processes (or one of their derivatives) having jumps, not just a sequence of measurements of them. A derivative doesn't even make sense for discrete functions.

Locally non-smooth functions have never been that much of a problem. Approximate the steps with spline and voila, you have continuous functions.

That won't ever make them analytic, so you still have a singularity in one sense of the term.

[Mika]

I am talking about the underlying continuous-domain processes (or one of their derivatives) having jumps, not just a sequence of measurements of them. A derivative doesn't even make sense for discrete functions.

And yet somehow I have still managed to optimize a discrete function with a gradient based method... and back then the computed gradient made perfect sense. And if I'm walking down the stairs, I tend to think that the direction of my trajectory is perfectly predictable, even though there are discrete steps.

Regarding the underlying processes, it could be argued that physically you can't even know if the processes are really discrete or continuous. Why? Because you can't measure them continously (shortest physically achievable sampling time at the moment is of order 10^-18 seconds), and theoretically shortest possible time interval is Planck time!

That won't ever make them analytic, so you still have a singularity in one sense of the term.

Analytic? I really had to check what that meant from Wiki. Seemed like stuff I normally ignore, so no problem with that in the functions either. Hasn't yet caused any kind of trouble. Sounds like those Mathematical proofs that it is possible to reconstruct a drum from the sound it is making. While Mathematically proven, most of the Physicists are probably going to say "not gonna happen" and the extremists would be like "cannot be done".

With the singularity I mean something like 1/r where r approaches zero. There function value approaches infinity as does it derivative.  If that is the case, then even I believe that practical Math is useless there, so one has to perform experiments to find the value.

This actually reminded me of one electronics course I took four years ago. Mathematically speaking, the amplification response of the circuit had amplification approaching infinity at some point, but still we could only get a bounded number from that place with measurements that was well within the measurement range of the instrument - no matter how hard we tried.
Mathematically speaking, it was a true singularity, but at that point reality came back and bit mathematics in the arse. As it usually tends to do in these cases. Of course, sound quality was horrible in that region, but that was not the point. Hence my earlier comment of never encountering a singularity.

Mika

[CP5670]

And yet somehow I have still managed to optimize a discrete function with a gradient based method... and back then the computed gradient made perfect sense. And if I'm walking down the stairs, I tend to think that the direction of my trajectory is perfectly predictable, even though there are discrete steps.

Sure, but you haven't dealt with derivatives anywhere in doing that.

Regarding the underlying processes, it could be argued that physically you can't even know if the processes are really discrete or continuous. Why? Because you can't measure them continously (shortest physically achievable sampling time at the moment is of order 10^-18 seconds), and theoretically shortest possible time interval is Planck time!

True, but that only applies to functions of time.

With the singularity I mean something like 1/r where r approaches zero. There function value approaches infinity as does it derivative.  If that is the case, then even I believe that practical Math is useless there, so one has to perform experiments to find the value.

That is one type of singularity, but you said "defined in the mathematical sense," which includes a lot more than just that. And of course, any non-smooth function can be reduced to this case if you just look at derivatives, which are often physically interesting quantities in themselves.

Density type functions can have even worse local behavior, and in fact distributions were originally created to deal with that situation.

This actually reminded me of one electronics course I took four years ago. Mathematically speaking, the amplification response of the circuit had amplification approaching infinity at some point, but still we could only get a bounded number from that place with measurements that was well within the measurement range of the instrument - no matter how hard we tried.
Mathematically speaking, it was a true singularity, but at that point reality came back and bit mathematics in the arse. As it usually tends to do in these cases. Of course, sound quality was horrible in that region, but that was not the point. Hence my earlier comment of never encountering a singularity.

I think you're confusing math with physical theory. If the result does not correspond to something in reality, that is because the model is incomplete, not because there is anything wrong with the math.

[Ghostavo]

Regarding the underlying processes, it could be argued that physically you can't even know if the processes are really discrete or continuous. Why? Because you can't measure them continously (shortest physically achievable sampling time at the moment is of order 10^-18 seconds), and theoretically shortest possible time interval is Planck time!

True, but that only applies to functions of time.

Out of curiosity, is there anything that disproves discrete space?

[perihelion]

Out of curiosity, is there anything that disproves discrete space?

No.  That is why theories of quantum gravitation like "Loop Quantum Gravity" and "Causal Dynamical Triangulation" are so interesting.  The latter is especially interesting to me because its approach makes minimal assumptions beyond the Standard Model, and yet it is able to predict space-time "evolving" into pretty much the way we have observed it thus far.

[Mika]

Sure, but you haven't dealt with derivatives anywhere in doing that.

Actually, it is a derivative, further proven by the fact it actually can decrease the error function. It is mostly a question how big deviations one is ready to take from the lim x->a definition of derivates.

And it is not only time of which it cannot be said if there is really a continuous process or discrete steps. The same applies to the position by Planck's length. Every law of Physics can only be measured in discrete fashion. Basically every measurement set is discrete.

The problems arise when both the function values and the derivatives blow out to infinity. Then there is pretty much nothing that can be said by theoretical work alone.

I think you're confusing math with physical theory. If the result does not correspond to something in reality, that is because the model is incomplete, not because there is anything wrong with the math.

Yes, I know. Unfortunately, you cannot make the model perfect. Or maybe it is just Physicists way of thinking that you really cannot trust everything that you write on the paper since there are always some factors you couldn't know.

Mika

[CP5670]

Out of curiosity, is there anything that disproves discrete space?

I don't think it is fully understood whether this is true or not, but I'm not really a physicist.

Actually, it is a derivative, further proven by the fact it actually can decrease the error function. It is mostly a question how big deviations one is ready to take from the lim x->a definition of derivates.

There isn't anything special going on there. Chances are that the discrete function you're optimizing is an approximation of a continuous-domain function and so the finite differences are approximations to derivatives.

It is probably possible to prove completely discrete-domain versions of most of these methods, if you put bounds on the modulus of continuity of the function (which basically does what you're saying) without requiring actual continuity.

And it is not only time of which it cannot be said if there is really a continuous process or discrete steps. The same applies to the position by Planck's length. Every law of Physics can only be measured in discrete fashion. Basically every measurement set is discrete.

What about probability distributions? As far as I know, those are continuous in many contexts according to current understanding. It all depends on what you consider to be physically significant.

[Mika]

My comment about Planck's length was related to the question of discrete space. According to current understanding, it is impossible to measure distances shorter than Planck's length. This means you cannot really know if it the space is continuous or discrete. But as a side note, even if you knew the answer to this question, it would really not change your life in any way. [Comments like these get Physicists most likely ridiculed in the future. But I'm willing to take the risk.]

There isn't anything special going on there. Chances are that the discrete function you're optimizing is an approximation of a continuous-domain function and so the finite differences are approximations to derivatives.

It is probably possible to prove completely discrete-domain versions of most of these methods, if you put bounds on the modulus of continuity of the function (which basically does what you're saying) without requiring actual continuity.

Yes. This is most likely case in the large amount of physical situations that I have stumbled across - actually that was one of the first lessons in Physical Mathematics : "In Physics almost all functions are smooth enough" [And you can quote that to ridicule Physicists. Unfortunately, it appears to be true, so far in my cases at least.] It is here where physical insight comes to play. Is the phenomenom under research reasonably well sampled with respect to what I want to measure about it? This is the question what Physicists must answer. [And carry to grave if they made a mistake about it.]

For this reason I said that for example, phase transition is simply undersampled linear phenomenom. If you sample it accurately enough, there is no sudden phase transition.

What about probability distributions? As far as I know, those are continuous in many contexts according to current understanding. It all depends on what you consider to be physically significant.

Probability distributions with respect to what?

Mika

[CP5670]

My comment about Planck's length was related to the question of discrete space. According to current understanding, it is impossible to measure distances shorter than Planck's length. This means you cannot really know if it the space is continuous or discrete.

I checked the wikipedia article on it and it looks like different models interpret this in different ways, especially its connection to gravity (for which the currently accepted models are fully continuous).

Yes. This is most likely case in the large amount of physical situations that I have stumbled across - actually that was one of the first lessons in Physical Mathematics : "In Physics almost all functions are smooth enough" [And you can quote that to ridicule Physicists. Unfortunately, it appears to be true, so far in my cases at least.

Well, how much is almost all and how much smoothness is enough?

As I said earlier, in some contexts a failure of analyticity is considered a singularity, and analyticity often fails somewhere with many physical situations (even say like, walking around a room).

For this reason I said that for example, phase transition is simply undersampled linear phenomenom. If you sample it accurately enough, there is no sudden phase transition.

I'm not sure this is correct, but in any case it's certainly not an analytic phenomenon and can be viewed as bad behavior in that sense.

Probability distributions with respect to what?

I was thinking along the lines of uncertainty principles, for which the discrete cases look a little different, but I guess that ties into the issue of discrete spaces.