Relativistic Effects

[granthutchison]

No, tachyons have negative mass squared, i.e. imaginary mass!

Square-root of negative mass? Surely a quantity of mass that is the square of a negative number would merely be a positive amount of mass?

Grant

[Cham]

...
Since time stops at the speed of light i would imagine it would speed up again when its past by and object of negatiive mass (im supposing the tachyon is a particle of negative mass?).

...

No, tachyons have negative mass squared, i.e. imaginary mass! Negative mass is physically indistinguishable from positive mass, i.e. that sign is not measurable. It's taken positive by convention in particle physics.

Bye Fridger

Hmmm! Sorry t00fri, but this is an eresy !

Negative mass is really not physically indistinguishable from a positive one. For example, Newton's equation will say that a negative mass will accelerate in the opposite direction of the net force applied to it :

F = m a,

with m < 0. If you apply ONLY a gravitationnal force alone, then yes, the mass disappears from the equation, because

F_grav = G M m / r^2.

But if you apply a gravitationnal force AND some electromagnetic force at the same time, the mass cannot disappear from the Newton equation. Also, a negative mass can lead to violation of energy and can make the vacuum in trouble (unstable). The reason why nature doesn't let negative masses to exist is still a mystery. We can't solve this problem without a better understanding of inertia (and its source).

Also, about the tachyons, there is still some confusion around them, especially on the web. The imaginary mass is just a matter of convention. You can define a tachyon mass as a positive real number, and its 4-velocity "u" obey the equation

u * u = - c^2.

It's a space-like 4-vector (I'm using the particle convention on the metric signature : +1 -1 -1 -1).

The problem with tachyons is they may violate causality. So coherence of the universe cannot accept them IF they may interact with normal matter. This is related to a subtle convention in special relativity, which most people aren't aware, even in the physics community ! The metric signature and the kind of trajectories in space-time (time-like for all particles). Really, this is a fascinating subject.

[t00fri]

No, tachyons have negative mass squared, i.e. imaginary mass!

Square-root of negative mass? Surely a quantity of mass that is the square of a negative number would merely be a positive amount of mass?

Grant

Grant,


i=sqrt(-1); i^2 =-1

A tachyon's mass has the imaginary form:

M_tachyon = i*m; m=real

hence it's mass^2 is

M_tachyon ^2 = i^2 * m^2 = - m^2 <0

That's what I said.

Bye Fridger

[granthutchison]

That's what I said.

Ah, the simple matter of a hyphen. "Negative mass-squared" would have made it unambiguous, whereas what you wrote is more readily interpreted as meaning that the negative value of the mass is squared ... which is what I understood you to be saying. Sorry!
(Perhaps it's easier to be unambiguous in German, where you can pile all the related words into one big word?)

Grant

[t00fri]

Hmmm! Sorry t00fri, but this is an eresy !

Right you are Cham .

Sorry I forgot to clearly specialize to /particle physics/, which traditionally comprises all interactions /except/ gravity. Now things are rapidly changing and one has to be more careful...
I did implicitly specialize to particle physics above, but not in a sufficiently clearcut way.

In strong, electromagnetic and weak interactions among elementary particles the sign of the mass is indeed not measurable. This is a bit intricate and has to do with the mixing of the 3 generations of quarks and leptons.

If gravitation is considered in addition, the sign seems to be measurable at the classical level. But remember in electromagnetism, you may change the sign of the mass without measurable consquences. Did you take this into account in your above argument?

As to the effects of quantized gravity, the last word is still not said.

The problem with tachyons is they may violate causality. So coherence of the universe cannot accept them

Sure, I have also pointed this out in an earlier post of mine. As you also note, problems only arise, if tachyons interact with normal matter. As long as they are living an "isolated life" in a world with super-luminous speeds things are fine.


Bye Fridger

[t00fri]

That's what I said.

Ah, the simple matter of a hyphen. "Negative mass-squared" would have made it unambiguous, whereas what you wrote is more readily interpreted as meaning that the negative value of the mass is squared ... which is what I understood you to be saying. Sorry!
(Perhaps it's easier to be unambiguous in German, where you can pile all the related words into one big word?)

Grant

I am sorry....continuous trouble with these foreigners

But "imaginary" should have been ambiguity resolving.

Bye Fridger

[granthutchison]

But "imaginary" should have been ambiguity resolving.

That's "ambiguity-resolving", as I suspect you already know.
Yes, it should have been. But unfortunately I wasn't aware of an ambiguity that need to be resolved - I perceived only one possible meaning when I asked my questions, and only cottoned on to the ambiguity when I read your reply. (Sigh.)

Grant

[Cham]

In my opinion, that imaginary tachyon mass concept is an outrageous idea that should be banned forever from all the physics arena. It just doesn't make any sense.

In relativity theory (special and general), space-time is a real manifold, whose points (events) are represented by REAL coordinates. All vectors embeded in it MUST be made from REAL components. Even for tachyons.

By definition, the 4-velocity of any point moving in space-time is made up of four real components :

u == dx/ds

where x are the four coordinates parametrised by the scalar parameter "s". With this definition, the metric gives the scalar product (in space-time)

u * u = +/- c^2.

The particules associated with the plus sign are called time-like (normal matter), while the particles associated with the minus sign are called space-like (tachyons). The sign is an invariant under any change of coordinates, and under any change of reference frame. A time-like particle cannot turn into a space-like particle, along its trajectory, etc.

Now, the particle's momentum is defined by

p == m u,

where m is a real positive number, called the "proper mass" of the particle. Then, for a space-like particle, we can write

p * p = - m^2 c^2,

and this can also be written p * p = n^2 c^2, with n == i m, the "imaginary mass" of the tachyon. Well guys, this is useless and plain vanilla BS !

Now, it appears that a space-like particle is always moving for ANY time-like observer. You cannot see a standing tachyon (or else it would be time-like), so its proper mass "m" cannot be measured. BUT, if some force can be applied to the tachyon, let say an electromagnetic force alone (no gravity) if the tachyon is bringing an electric charge with him, then we could write the Lorentz force :

du/ds = (q/m) F * u.

This is consistent for a space-like particle. Notice here that the "s" parameter isn't the "proper time" of the particle. It's just an affine parameter that parametrise the particle's trajectory in space-time. Because of this, the coefficient "q/m" doesn't have any clear physical meaning.

[t00fri]

Cham,

I am confused about what you wanted to say.

I particle physics & quantum field theory in general, we frequently encounter particles
with p*p <>m^2 (units c=hbar=1). Notably particles that have spacelike momenta p*p <0, but m^2>0. These are as we say not on their mass shell. This happens as a consequence of quantum physics and is nothing special. Certainly such particles are NOT tachyons and did not appear at all in your above discussion. Therefore I have trouble following it. It seems much too restricted.

Tachyons cannot be discussed consistently in a purely classical framework.

Quantities that are measurable classically, often cease to be measurable quantum mechanically or in quantum field theory.

I can only assure you that in quantum field theory the only consistent definition of tachyons refers to m^2<0.
Things are really nontrivial on the quantum level since you have to diagonalize the respective operators before you know what you are doing. This again involves a detailed knowledge of the (quantum) dynamics.

Bye Fridger

[Cham]

Cham,

I am confused about what you wanted to say.

I particle physics & quantum field theory in general, we frequently encounter particles
with p*p <>m^2 (units c=hbar=1). Notably particles that have spacelike momenta p*p <0, but m^2>0. These are as we say not on their mass shell. This happens as a consequence of quantum physics and is nothing special. Certainly such particles are NOT tachyons and did not appear at all in your above discussion. Therefore I have trouble following it. It seems much too restricted.

Tachyons cannot be discussed consistently in a purely classical framework.

Quantities that are measurable classically, often cease to be measurable quantum mechanically or in quantum field theory.

Bye Fridger

Well, "m" is always real and positive. Even for tachyons. In the classical field theory, we always have m^2 > 0. The squared "norm" of the momentum (or the velocity) is negative in the case of tachyons. Don't confuse the mass and the momentum, these are very distinct concepts. Yes, tachyons can be discussed in a classical theory, but their causality violation isn't interesting, that's why we forget about them.

In quantum field theory, yes, we can have a particle "off-shell" : p * p < 0. This is because of the uncertainty principle. And this can lead to a violation of causality at the microscopic level. However, because of the statistical nature of quantum theory, it never lead to a problem in the macroscopic world.

[Cham]

t00fri, I just realised that the word "tachyons" may have two very different meanings. This may be the source of the confusion here. In some of my quantum fields books, tachyons refer to particles for which m^2 < 0. In some other books, "tachyons" are associated with p * p < 0. These are two completely distinct things.

In the classical regime, m^2 < 0 doesn't have any sense at all. Those particles just doesn't exist on a classical theoretical sense. However, the classical tachyons are the particles for which p * p < 0. Those particles (space-like) may exist, in a classical theory.

In quantum theory, the m^2 < 0 idea contains a lot of confusion. I think it isn't related to mass. The "tachyon" mass "m" (for which m^2 < 0) isn't really a mass, it's just a parameter in the lagrangian. We should write m^2 == - lambda, instead.

Is it what you were talking about ?

[t00fri]

t00fri, I just realised that the word "tachyons" may have two very different meanings. This may be the source of the confusion here. In some of my quantum fields books, tachyons refer to particles for which m^2 < 0. In some other books, "tachyons" are associated with p * p < 0. These are two completely distinct things.

In the classical regime, m^2 < 0 doesn't have any sense at all. Those particles just doesn't exist on a classical theoretical sense. However, the classical tachyons are the particles for which p * p < 0. Those particles (space-like) may exist, in a classical theory.

In quantum theory, the m^2 < 0 idea contains a lot of confusion. I think it isn't related to mass. The "tachyon" mass "m" (for which m^2 < 0) isn't really a mass, it's just a parameter in the lagrangian. We should write m^2 == - lambda, instead.

Is it what you were talking about ?

Cham,

correct. In quantum theory, p*p and m^2 are fundamentally disjunct/independent quantities while classically, particles are always "on-shell", i.e. they satisfy p*p=m^2. In quantum theory, only non-interacting particles are always on mass-shell, but that happens only in asymptotic limits.

In quantum theory, it is plain wrong to associate tachyons with particles having p*p<0. There we define masses in terms of zeros of inverse propagators, but things are really subtle. So-called "bare" masses (defined in absence of interactions) are meaningless and typically infinite. Only the so-called "renormalized" masses have physical meaning and thus are finite and well defined. In electrodynamics for example, a renormalized electrical charge would correspond to the charge in the presence of vacuum polarization that screens the original (bare) charge....

These examples may just illustrate that if we want to discuss sensibly about tachyons, we should definitely "switch gears"

In other words, in your arguments you have merely considered tachyons from a more or less "kinematical" point of view which is far from satisfactory. The existence and properties of tachyons can only be discussed sensibly in a truly dynamical microscopic framework like gauge theories.

Bye Fridger

[Cham]

Like what I've said, in the classical theory, tachyons are only space-like particles (p * p = - m^2 < 0, but m positive real number).

In quantum theory, we use the same word for another thing (which is very unfortunate, because of all the confusion). Well then, how do you define "tachyons" in a quantum theory ? I guess it's very technical and I'm not a quantum field expert anymore, since I specialised in the classical general relativistic field.