Exploring the Validity of Lorentz Transform and Discrete Space-Time Models

[mhill]

If we impose that under no force or gravitational field Lorentz transform must hold, my question is if the validity of Lorentz transform means that space time must be continous, or on the other hand if there may be a discrete model of space-time that preserves Lorentz transform.

For example if we can only measure a discrete time and space interval, could we find a lorentz transform (group transform) that leave the (discrete) element

$$(\Delta x )^{2}-c^{2}(\Delta t)^{2}$$

invariant ? , for example if there could be a discrete space-time admitting some 'modified' Lorentz transform or approximate Lorentz transform.

 

[yuiop]

Would I be correct in assuming you are talking about the Planck length and time interval?

I also assume you might be reasoning along these lines. IF we assume the Planck length is a minimum possible length in nature, then if some hypothetical object had a rest length of 1 Planck length then it would not be able to length contract as observed in a frame moving relative to the particle. That means we would have to modify the time dilation in some bizarre way to compensate for the lack of length contraction.

I think the formal answer is that it is not clear that the Planck length is a minimum (non zero) length and that Planck units only make sense in terms or ratios. The ratio of the Planck length to the Planck time interval is the speed of light for example. It is also clear that the Planck mass is NOT the minimum possible (non zero) mass. The mass of Hydrogen atom is something like 19 orders of magnitude smaller than the Planck mass.

Peter Lynd analysed the Zeno paradoxes and came to the conclusion that there is no such thing as an instant of time so it is meaningless to ask or state where a moving object is at any given "instant" and this is his solution (that not everyone agrees with) to the Zeno paradoxes.

It is not my intention to confuse anyone so this last paragraph is purely a personal view. Say we conjecture that time is given a special status and is uniquely discrete and that length is not quantisized (i.e the Planck length unit is not a natural minimum). Now if in the rest frame an event takes a time interval of one Planck time interval (or some other minimum discrete time interval) then the same event measured in any other inertial reference frame would never be less than than the minimum time interval due to the way time intervals transform. This would allow us to preserve a minimum time interval. However the same constraint would not apply to the spatial interval. This solves a number of problems (and probably introduces some new ones :P) This way of looking at things gives an intuitive and simple solution to the Zeno paradoxes but I should point out that the formal modern view is that the Zeno paradoxes are "non-paradoxes" that do no not require solving. This does not directly answer the question (I think) you are asking. (Bear in mind that the uncertainty principle greatly complicates things at this scale). The final solution would involve the marriage of Relativity and Quantum Mechanics and a working Grand Unification Theory. Many great minds have been working towards that goal for many years and as far as I know they are still working on it.

P.S That is a fairly informal answer by way of discussion. I am sure there are many here that will give you a more formal answer.

 

[yuiop]

Hello?