Check my equation of I x F = E

[Sepoquro]

I x F = E

 

I: infinity (unlimited stretch of space)

F: forever (unlimited stretch of time)

E: everything

[John Cuthber]

It's wrong.

http://en.wikipedia.org/wiki/Dimensional_analysis

[ajb]

I x F = E

 

I: infinity (unlimited stretch of space)

F: forever (unlimited stretch of time)

E: everything

 

Forgetting the units for a moment, I do not see how this is different to [math]\infty \times \infty = \infty[/math] which is ok (lets not worry about different kinds of infinity for now) What non-trivial information is supposed to be in this equation?

[Daedalus]

Forgetting the units for a moment, I do not see how this is different to [math]\infty \times \infty = \infty[/math] which is ok (lets not worry about different kinds of infinity for now) What non-trivial information is supposed to be in this equation?

 

 

A more accurate and correct statement would be:

 

An infinite length divided into an infinite number of segments would produce a segment length of one. This is in relation to the set of integers, Z. If we take each segment and sub-divide them into an infinite number of parts, then each part will be equal to the infinitesimal 1 / Infinity. This is in relation to the set of real numbers, R. Some will argue that this length is zero. However, a decimal with an infinite number of zeroes followed by a one in not equal to an infinite number of zeroes followed by a zero. The proof lies in the fact that two points will always have some length seperating them. If two consecutive real numbers did not have a length seperating them, then we would get:

 

Sigma {1 -> Inf, l} = 0 where l = 0, is the length seperating two points.

 

However, if l = 1 / Infinity we get:

 

Sigma {1 -> Inf, l} = 1

 

Now for any finite measurable length L, that we apply the same process, we get l = L / Inf. Therefore the infinite sum of the infinitesimal is:

 

Sigma {1 -> Inf, l} = L

 

 

It is because of the mathematics shown above, that no two points will ever be able to come together due to scaling. This is one of the problems I have with the Lorentz factor because it is a scaling effect that states that relative length will be zero if an object traveled at the speed of light. Since photons travel at the speed of light, am I to assert that they have no relative length in the direction of forward motion? I will for now concede to Einstein, however I believe that photons have some infinitesimal length in the direction of forward motion. But I cannot prove this even through I have a highly speculative modification to the Lorentz factor that accounts for this.

 

Accepted length contraction equation

 

L' = L Sqrt(1 - v^2 / c^2)

 

L' = 0 when v = c

 

However a photon should have some size, maybe it dont. But the equation for this would be:

 

l = L Sqrt(1 - (c ^ 2 / c^2) * F)

 

where l is an unknown universal constant that represents the relative length of a photon

and F is an unknown factor that is applied to the Lorentz contraction. Solving for F we get:

 

F = 1 - l^2

 

plugging this back into the modified equation we get:

 

L' = L Sqrt(1 - (v^2 / c^2)(1 - l^2)

 

when v = 0 we still get L' = L. But, when v = c we would get:

 

L' = l

 

If l is extremely small, then we would not observe the effects of l as both, accepted and modified, equations produce near identical results except when dealing with relative length at the speed of light. If l is truly zero, we get the original equation back. Factoring the modified equation we arrive at:

 

L' = L Sqrt(1 - (v+vl)(v-vl) / c^2)

 

Enjoy!

Edited July 3, 2011 by Daedalus

[ajb]

...infinitesimal 1 / Infinity.

 

Well, usually dividing by infinity is not well defined. Infinity is not a number so it is problematic to define division. You have to approach this as a limit, and then you get zero.

 

What you want to do is extend the real numbers to include infinitesimals. You will probably be interested in non-standard analysis.

 

 

 

Since photons travel at the speed of light, am I to assert that they have no relative length in the direction of forward motion?

 

I am not really sure what you mean by this. Classically photons are point like anyway.

 

I expect your thinking here maybe to do with picking an inertial frame for a photon and thus a proper size?

 

Anyway, you should start a new thread on your ideas of generalising the Lorentz group rather than hijack this one.

Edited July 3, 2011 by ajb

[Daedalus]

Well, usually dividing by infinity is not well defined. Infinity is not a number so it is problematic to define division. You have to approach this as a limit, and then you get zero.

 

What you want to do is extend the real numbers to include infinitesimals. You will probably be interested in non-standard analysis.

 

 

 

 

 

I am not really sure what you mean by this. Classically photons are point like anyway.

 

I expect your thinking here maybe to do with picking an inertial frame for a photon and thus a proper size?

 

Anyway, you should start a new thread on your ideas of generalising the Lorentz group rather than hijack this one.

 

 

Sorry, I'm new here. Didn't mean to hijack this thread. Just giving an opinion and I do understand what you are saying. I actually completed finite calculus and numerical analysis. I was just trying to give an explanation and opinion that the OP may relate too. Your statement to deriving infinitesimals is a relative matter. You can either start from the small and work towards the large or vice-versa. However, the approach you have stated is considered proper compared to the one I demonstrated. On the topic of the Lorentz contraction, I am new to this area in physics. I am currently going back to college to work on a degree in Physics and look forward to exploring all of these areas.

Edited July 3, 2011 by Daedalus

[Yoseph]

The problem I see lies in the units... Let's say the unit for time is seconds and the unit for space is in meters... How can "Everything" be defined in Meter Seconds? You need to define your "Everything" better. Is there a use for this equation your suggesting?

[md65536]

The problem I see lies in the units... Let's say the unit for time is seconds and the unit for space is in meters... How can "Everything" be defined in Meter Seconds? You need to define your "Everything" better. Is there a use for this equation your suggesting?

Perhaps Infinity I is multidimensional. Perhaps Everything is Meters³ Seconds. Then it makes sense.

 

However, a decimal with an infinite number of zeroes followed by a one in not equal to an infinite number of zeroes followed by a zero.

 

I think it's equal.

It is because of the mathematics shown above, that no two points will ever be able to come together due to scaling.

Perhaps I missed something but how is it that scaling by a factor of 0 is disregarded?

 

 

 

[Bignose]

Perhaps Infinity I is multidimensional. Perhaps Everything is Meters³ Seconds. Then it makes sense.

 

No, plenty of things (that I would assume would fall under the umbrella of 'everything') can't be described by units of m³s. A unit of energy, for example?

[Daedalus]

Perhaps Infinity I is multidimensional. Perhaps Everything is Meters³ Seconds. Then it makes sense.

 

 

 

I think it's equal.

 

Perhaps I missed something but how is it that scaling by a factor of 0 is disregarded?

 

 

 

 

I'm sorry. I wasn't clear as to the type of scaling I was referring to. I was talking about inverse proportional scaling of the form 1 / n. The Lorentz contraction exhibits similiar behavior. I fully understand that the mathematics support the observed length decreasing to zero at the speed of light. I'm just making a statement about infinitesimals. Given my above explanation about the existence of some distance in between any two infinitesimal points, and that the Lorentz contraction is not a subtractive operation, that a body in motion being observed scales along the direction of motion such that we still see the object as we should, albeit a distorted view, and not some fractional portion of it such as the front when we should see both the front and the side of the body in motion. So the faster you go relative to an observer in a different frame of reference, the smaller you appear to that observer and vice-versa. If we place everything we know about Physics aside except for the fact that a body in motion, such as a spaceship, will always reflect or emit light along each and every point that it traverses along its trajectory. Then if said body could somehow move at the speed of light relative to an observer, the observer should be able to record the information of the photons emanating from the body at each and every point along the body's observed path. However, the Lorentz contraction states that the observer would not be able to see the object's length along the direction of motion because it has been contracted to zero and this seems to be a contradiction to the logic stated previously. So this is where I have the problem and suggest that a photon must have some observable "length" for this special theoretical case. Now I understand that relativity solves this by only allowing particles with zero mass to obtain this speed. These particles are said to have relativistic mass, but this mass does not directly equate to the type of mass we have being comprised of atoms. Therefore bodies such as space ships will never be fully contracted as they can never reach the speed of light.

 

But back to the original point I was trying to make. Intuition tells me, although it may be wrong, that we would still observe a single strip of photons along the direction of motion if a body could move at the speed of light. Therefore, the length is not zero but that of an infinitesimal value that is expressed as a physical constant. I am also aware that the dimensional analysis of this hypothetical "l" universal constant reveals that it is unitless. The fact that it is multiplied by L to arrive at L' preserves the unit of measurement. This would allow for the body to maintain some detectable length along the direction of motion, and all of the information contained within the photons along this length would be conserved, even though they would all be compressed into a single strip of photons equal to the object's length perpendicular to the direction of forward motion. This suggests that infinitesimals are inherent in the structure of space-time. The confusion comes from the fact that our mathematics is simplified to treat all particles as point-particles that exhibit fields that have no definite length, and as in the case of gravity propagates through the entirety of space. I'm suggesting that infinitesimals play thier role in the structure of space-time to preserve the information of the photons emanating from a body in motion relative to an observer because all of the photons that have their positions transformed due to the Lorentz contraction would still exist even if the equations predict that they would be compressed to a zero length. Thus, this hypothetical universal constant "l" is an information conservation mechanism and as shown in the mathematics from my original post, preserves the original Lorentz contraction equation and therefore affects the underlying Lorentz factor. This is because if "l" is truly non-existent, zero, we get back the original equation. If "l" does exist, then we no longer obtain infinite density due to finite mass being concentrated in a dimensionless point because the fields exhibited by point particles can never trully collapse to a zero length due to relativistic velocities. There is support for the existence of this universal infinitesimal constant, "l", because the Lorentz factor can be expressed using infinite series operators. This means that the Lorentz factor can be viewed as a recursive process that is expressed as the number of recursions approach infinity. It is important to understand that infinitesimal numbers exist from -infinity to +infinity. This means that they can approach the limits of transcendentals, reals, integers, etc... The fact that we can derive huerisic shortcuts in mathematics for recursive processes, finite or infinite, is just a simplification of the underlying nature of the processes for which we are deriving relationships. When dealing with processes that have limits that approach infinity, we must accept the consecquence that the result we get is a limit and not an actual value that represent the true nature of the result. This is because we can never reach infinity. Therefore there will always be an infinitesimal that is present that accounts for uncertainty in measurements and provides a mechanism for chaos to emerge from an ordered system and vice-versa. I believe that infinitesimals will be at the heart of the Grand Unification Theory and it will most likely be based on equations that describe quantum gravity as an infinitely recursive fractal pattern similar to how the recusive Fibonacci sequence seems to govern the pattern that life on Earth uses to build its structures. But it may be the case that I do not full understand what it means to have a perceived length of zero.

 

I am only a software engineer that develops class 2 electronic gaming devices using C++ and C# on Windows and Linux (Gentoo Rules). C# allows me to quickly create utility apps. So I will admit that I could be completely wrong as I only took the necessary Physics courses as required by my chosen field. But I am heading back to college this fall after eight - nine years in the field to work towards a degree in Physics. I'm particularly interested in ring singularities and theories on time. I can't wait to apply my knowledge of computer science to model complex quantum processes.

 

P.S.

 

Aspergers and Dyslexia can be blessing and a pain in the ass. The means that you will never get a short answer from me and that there is a chance that I will reverse words, concepts, and numbers without realizing it. I am also moving this thread as AJB suggested to the Lorentz group to see how far we can take this topic. So you can follow me there if you like.

Edited July 4, 2011 by Daedalus