# Exploring the Paradox of Amelino-Camelia's DSR

In summary, Amelino-Camelia's DSR (Doubly special relativity) is a theory that states that an object's length (in the direction of travel) will tend towards the Planck length, but never quite reaches it. It also invokes non-commutative geometry which I personally don't think is very well defined as a physical concept.
Is there not an inherent paradox in Amelino-Camelia’s DSR (Doubly special relativity)?

Every solid piece of matter is the sum of its parts. Put another way, all the building blocks of a meter stick must be less than a meter in length (e.g. the lines dividing centimeters, millimeters, the atoms and molecules that make up the meter stick etc.)

If my understanding of DSR is correct, an object traveling close to the speed of light will undergo length contraction that tends towards the Planck length, but never quite reaches it, just as its velocity can only approach c.

Now let’s say we had a meter stick that was traveling fast enough for it to appear to be just 10 times the Planck length to a stationary observer. What measurement would that stationary observer then give for the length of the centimeter marks on the meter stick, considering that there are 100 of these on the meter stick, they are all of equal length and the sum of their lengths cannot exceed 10x the Planck length.

Your question does not apply to DSR but to standard SR.

Firstly in either case there is no paradox as both DSR and SR are Classical theories and don't even know what "the Planck length" is. At worst they are inconsistent with quantum theory... but I don't think even that is a problem.

What your observation means I think is that if you found such a meter stick traveling at such a velocity relative to you then any attempt on your part to resolve the marks would create a gravitational singularity (black hole). Of course given the relative kinetic energy in such an assumed fast moving meter stick any interaction with a particle moving slowly in your frame would probably create a gravitational singularity.

both DSR and SR are Classical theories and don't even know what "the Planck length" is

I'm not sure I agree.

I quote this from wikipedia
Doubly-special relativity (DSR)— also called deformed special relativity or, by some, extra-special relativity — is a modified theory of special relativity in which there is not only an observer-independent maximum velocity (the speed of light), but an observer-independent minimum length (the Planck length).

I could be wrong here, but in my understanding a difference between SR and DSR is that in SR an objects length (in the direction of travel) tends towards 0 as v tends towards c.

In DSR, an objects length (in the direction of travel) tends towards the Planck length as v tends towards c, and nothing can be smaller than the Planck length.

Ahhh... My apologies. I was thinking of another version of DSR (in which there was an invariant upper limit on acceleration).

Well from the beginning the currently offered DSR invokes non-commutative geometry which I personally don't think is very well defined as a physical concept. They deform not only the particular group but deform the Lie group/Lie algebra structure to that of a so called "quantum groups" and "quantum algebras". (I don't know what it means to "quantize" a mathematical object. We can express quantization in a deformation setting BUT the relativity groups/Lie algebras stay the same, it is the lower-level logic algebra which is deformed. )

Looking at the introductory article:
http://arxiv.org/pdf/hep-th/0405273v1"

This author implies that the theory itself isn't complete yet. I personally think they've made a mistake in that the Poincare group has only two deformation "expansions" (reverse contractions). These are the deSitter and anti-deSitter groups. In both cases there is an invariant length but it is very large not very small. That is the radius of the spatial universe (equivalently the radius of curvature which relates to the cosmological constant) in the de-Sitter case and the negative radius of curvature in the anti-deSitter case.

I can't argue about a paradox in a not-yet-theory, but presumably the resolution to your "paradox" is supposed to rely on that "non-commutative geometry" business. I think it will have to imply that you can't get your ruler to act as a ruler as seen from such a relative speed.

With regard to "non-commutative geometry", I see spatial coordinates as non-physical i.e. parametric quantities like time especially since they mingle with time in SR. Thus making them "non-commutative" is not physically meaningful.

The generators of the Poincare group are (as canonical variables used in Poisson brackets) Linear momentum+energy, angular momentum, and the boost generators which are like space-time pseudo-angular momenta. The only deformation would be to change the commutation relations on the momenta which equates to changing partial derivatives to covariant derivatives. This is already done in GR (or rather the pre-GR extension of SR to curved space-time) where we then get
$$[P_\mu,P_\nu] = R_{\mu\nu} + B_\mu^\alpha P_\alpha$$
The R terms is the Riemann curvature tensor (with two indices suppressed ) and B is the Burger's vector (when you don't choose a coordinate basis).

Thus you get curved space-time which is locally deSitter or anti-deSitter depending on the curvature.

If they are deforming a larger set of generators then they are not deforming SR but rather ... I don't know, SR + Heisenberg's relations or SR+ the canonical algebra. This latter is an embedding algebra and not the algebra of physical symmetries.

Now having qualified why I don't like DSR I do think there are physical implications of a deSitter or anti-deSitter geometry which haven't been fully explored.

Finally, though I'm quite well versed on Lie algebra/group deformations (within the category of Lie algebras/Lie groups) I have not spent too much time on these more general mathematical deformations (thinking them not worth my time) so I am both biased and a bit ignorant. You should find someone to explain the meaning of these more general deformations in the context of a physical theory.

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## 1. What is the Paradox of Amelino-Camelia's DSR?

The Paradox of Amelino-Camelia's DSR (Doubly Special Relativity) refers to a theoretical framework proposed by physicist Giovanni Amelino-Camelia, which attempts to reconcile the principles of special relativity with the existence of a minimum length and a maximum energy in our universe.

## 2. How does DSR differ from traditional special relativity?

DSR introduces a new fundamental scale, known as the Planck scale, which sets a minimum length and maximum energy in our universe. This means that at extremely small scales, the laws of physics may be different from what we observe at larger scales. In traditional special relativity, there is no such fundamental scale.

## 3. What is the significance of the Planck scale in DSR?

The Planck scale is significant because it represents the smallest possible length and the largest possible energy that can exist in our universe. This means that at scales smaller than the Planck length, the laws of physics may be fundamentally different and cannot be described by traditional theories.

## 4. Are there any experimental evidence for DSR?

Currently, there is no direct experimental evidence for DSR. However, some phenomena, such as the Cosmic Microwave Background Radiation, may provide some indirect support for the existence of a fundamental scale in our universe.

## 5. What are the implications of DSR for our understanding of the universe?

DSR challenges our traditional understanding of space and time, which are considered absolute in special relativity. It also has implications for theories such as quantum gravity and the search for a unified theory of physics. However, more research and experimental evidence are needed to fully understand the implications of DSR for our understanding of the universe.

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