Doubly special relativity
violent green
forward by Amelino-Camelia that physics (more specifically SR) might need to
be modified so as to include a second observer independent scale (a length
or mass scale, most likely identified with the Planck scale).
I was just wondering if anyone here has been paying attention, and if so,
what they think of the idea.
Cheers,
Violentgreen
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Eric A. Forgy
"violent green" <violen...@hotmail.com> wrote in message
> I was just wondering if anyone here has been paying attention, and if so,
> what they think of the idea.
I see that Lee Smolin has just coauthored a paper on this subject:
http://www.arxiv.org/abs/gr-qc/0207085
Generalized Lorentz invariance with an invariant energy scale
Authors: Joao Magueijo, Lee Smolin
It seems interesting. I'll try once again to fish for comments on what
people think :)
I don't know if this is at all related, but something about it seems
to resonate in my mind. I recently read this interesting paper:
http://www.math.iupui.edu/~zshen/Finsler/history/chern.html
Notices of the AMS September 1996
Finsler Geometry Is Just Riemannian Geometry without the Quadratic
Restriction
Shiing-Shen Chern
>>From what I understand, even Riemann's original work considered more
general inner products than the quadratic one that we use in
"Riemanniann geometry."
Again, I don't know if the two are at all related, but when I looked
at Magueijo and Smolin's paper and saw his expression:
E^2 = p^2 + m^2 + lambda*E^3 + ...
the first thing I thought of is, "Hey, that is just an alteration of
the usual inner product of Minkowski spacetime."
Then, that resonated with Chern's paper on Finsler geometry.
Could the two ideas be at all related?
Could "doubly special relativity" be the consequence of a slight
modification of the inner product from the simple quadratic form?
Thanks,
Eric
Jeremie Vinet
Eric A. Forgy wrote:
>I see that Lee Smolin has just coauthored a paper on this subject:
>
>http://www.arxiv.org/abs/gr-qc/0207085
>Generalized Lorentz invariance with an invariant energy scale
>Authors: Joao Magueijo, Lee Smolin
>
>It seems interesting. I'll try once again to fish for comments on what
>people think :)
Thanks for the help! That first post of mine hasn't exactly stirred up
passions here :)
>Again, I don't know if the two are at all related, but when I looked
>at Magueijo and Smolin's paper and saw his expression:
>
>E^2 = p^2 + m^2 + lambda*E^3 + ...
>
>the first thing I thought of is, "Hey, that is just an alteration of
>the usual inner product of Minkowski spacetime."
The actual relation in Magueijo and Smolin's version of doubly special
relativity is
(E^2 - p^2)/(1-lambda*E)^2 = m^2
where clearly, the invariant "m" is not equal to the rest energy
anymore. (This brings up an interesting question: which of the rest
energy and the invariant "m" plays the role of inertial mass? And of
gravitational charge?)
>Then, that resonated with Chern's paper on Finsler geometry.
>
>Could the two ideas be at all related?
>
>Could "doubly special relativity" be the consequence of a slight
>modification of the inner product from the simple quadratic form?
That would certainly be an interesting way of looking at it, and one
which to my knowledge hasn't received much (any?) attention. However,
I don't believe it's at all clear that the inner product on momentum
space behaves in the same way as it does in spacetime in doubly
special relativity. In fact, finding a spacetime which is compatible
with DSR appears to be a highly non trivial task.
Since this subject doesn't seem to have stirred up much interest on
this ng so far, I'll try to summarize what I understand about it in a
few lines.
The basic motivation behind DSR comes from the observation that if the
Planck length/energy is truly a fundamental quantity, then different
observers looking at, say, a photon with Planck energy, should all
agree that it has E=E_p. This is clearly not the case according to
Special Relativity, where the only observer independent scale is the
speed of light. So the aim of DSR is to modify SR so as to accomodate
a second observer independent scale, hence the name "doubly" special
relativity. (See hep-th/0012238, gr-qc/0012051).
This is achieved by modifying the generators of boosts on momentum
space. More specifically, boosts are coupled to dilatations in such a
way that the commutation relations between the modified boosts and
(unmodified) spatial rotations still satisfy the ordinary lorentz
algebra.
There's more than one way to do this, so there is more than one
version of DSR, leading to profoundly different physical predictions
(hep-th/0201245).For example, in Magueijo and Smolin's proposed
realisation (hep-th/0112090), both the energy and the momentum become
bounded by a maximum value, in such a way that there still exists a
maximum value for dE/dp which corresponds to the speed of light. In
the original version of DSR proposed by Amelino-Camelia, only the
momentum is bounded, which leads to dE/dp diverging at large energies.
If we are to identify dE/dp with the physical velocity of particles,
then particles can have velocities exceeding c (!). However, just
what quantity corresponds to the velocity of particles is ambiguous,
due to the following fact. In DSR, the value of dE/dp is not equal to
"v", the relative velocity between the rest frame and the boosted
frame. In fact, it is even worse: dE/dp depends on the rest mass of
the particle, so that on the face of it, two particles of different
mass which are at rest in a given frame will be seen to be moving
toward or away from each other in a boosted frame (hep-th/0207031).
Going back to the way in which an observer independent scale of
energy/length is introduced, i.e. by modifying the generator of boosts
on momentum space, it turns out that this corresponds to something
called kappa-poincare algebra. Seen from this point of view,
spacetime is to be identified with the coalgebra sector of the kappa-
poincare algebra. When this is done, one finds that space and time do
not commute anymore. Rather, [x_0,x_i] ~ lambda x_i.
(hep-th/0107054,hep-th/0203040, hep-th/0203065,hep-th/0204245).
However, in Magueijo and Smolin's latest paper (gr-qc/0207085), they
propose that the spacetime coordinates should be identified with the
generators of translations in momentum space. They then show that in
this case, spacetime is still commutative, at least in their proposed
realisation of DSR.
I think I'll stop here for now, hoping that this will have been of at
least some interest to some of the good folk here!
Jérémie (the physicist formerly known as Violentgreen)
Urs Schreiber
[...]
> The basic motivation behind DSR comes from the observation that if the
> Planck length/energy is truly a fundamental quantity, then different
> observers looking at, say, a photon with Planck energy, should all
> agree that it has E=E_p. This is clearly not the case according to
> Special Relativity, where the only observer independent scale is the
> speed of light. So the aim of DSR is to modify SR so as to accomodate
> a second observer independent scale, hence the name "doubly" special
> relativity. (See hep-th/0012238, gr-qc/0012051).
Is this, explicitly or implicitly, related to the old idea by
L. Nottale dubbed "scale relativity" (e.g. [1])? Nottale
proposed to replace the "Galilean" sum c of two "scale
transformation" (i.e. magnifications) with factors a,b,
c = a*b <=> ln c = ln a + ln b
by the "relativistic" sum
ln c = (ln a + ln b) / (1 + ln a * ln b/ ln^2 k ) ,
where k is the fundamental scale (e.g. the Planck scale). Like
the analogous sum for relativistic velocities implies (or
expresses) the fact that the sum of two boosts can never yield
a velocity greater than the fundamental velocity c, the above
sum would imply that no scale transformation can resolve a
distance smaller than the fundamental scale k.
As far as I know this has remained "just an idea", but what
you wrote about DSR sounds very similar, bearing in mind that
length scales correspond to energy scales.
[1]
p. 231 of
L. Nottale, Fractal Space-Time and Microphysics, World
Scientific (1992)
Danny Ross Lunsford
> Is this, explicitly or implicitly, related to the old idea by
> L. Nottale dubbed "scale relativity" (e.g. [1])? Nottale
> proposed to replace the "Galilean" sum c of two "scale
> transformation" (i.e. magnifications) with factors a,b,
>
> c = a*b <=> ln c = ln a + ln b
>
> by the "relativistic" sum
>
> ln c = (ln a + ln b) / (1 + ln a * ln b/ ln^2 k ) ,
>
> where k is the fundamental scale (e.g. the Planck scale). Like
This doesn't sound right. Say for example instead of an extra space
dimensions, there is an extra time one. One would expect the space and
time 3-spaces to be related by a single c-like parameter (same
analysis that worked for Minkowski space) and so one would expect a
linear relation among the actual parameters and not their logs,
because they are components of a unit time vector.
The thing with the logs however reminds me of the addition of angles
under a projective metric, in the sense of Klein. It works like this -
the distance between two points can be invariantly specified if one
first specifies a quadratic form. Any line will have two intersections
with the associated quadric surface. Produce the line connecting the
given points and determine the two intersection points. Take the
cross-ratio (projective invariant) of these 4 points. The distance is
then
Dab = k log (Xa - Y1)(Xb - Y2)/(Xa - Y2)(Xb - Y1)
where Y1 and Y2 are the intersection points. k is a dimensional
parameter (cosmological constant? :)
To do angles, we just go over to plane coordinates. Then one is
calculating the cross ratio of 4 planes of a pencil. Two of the planes
are tangent to the associated quadric. Two planes are perpendicular
with respect to this quadric when their cross ratio is -1 ("harmonic"
position).
-drl