## Friday, July 13, 2018

### Infinitesimal analysis of geometry and locality

The geometric axiom would appear to be in conflict with the locality axiom. The geometries are defined by symmetry transformations that relate points to distant points. Under locality, points only relate to nearby points, not distant points.

The resolution of this conflict is that the geometric operations actually act infinitesmally. The Lorentz transformations do not actually act on spacetime, but on the tangent space at a given point.

Here “infinitesimal” is a mathematical shorthand for the limits used in computing derivatives and tangents. Locality allows a point to be related to an infinitesimally close point in its light cone, and that is really a statement about the derivatives at the point.

In general relativity, matter curves spacetime, and spacetime does not necessarily have rotational or other symmetries. But locally, to first order in infinitesimals, a point looks like the Minkowski space described earlier. That is, the tangent space is linear with the metric −dx2−dy2−dz2+c2dt2, in suitable coordinates.

A causal path in a curved spacetime is one that is within the light cones at every point. As the light cones are in the tangent space, this is a statement about the tangents to the curve. In other words, the velocity along the path cannot exceed the speed of light.

There is a mathematical theory for curved spaces. A manifold has a tangent space at each point, and if also has a metric on that, then one can take gradients of functions to get vector fields, find the shortest distance between points, compare vectors along curves, and calculate curvature. All of these things can be defined independently of coordinates on the manifold.

Spacetime (Riemann) curvature decomposes as the Ricci plus Weyl tensors. Technically, the Ricci tensor splits into the trace and trace-free parts, but that subtlety can be ignored for now. The Ricci tensor is a geometric measure of the presence of matter, and is zero in empty space.

There are whole textbooks explaining the mathematics of Riemann curvature, so I am not going to detail it here. It suffices to say that if you want a space that looks locally like Minkowski space, then it is locally described by the metric and Riemann curvature tensor.

The equations of general relativity are that the Ricci tensor is interpreted as mass density. More precisely, it is a tensor involving density, pressure, and stress. In particular, solving the dynamics of the solar system means using the equation Ricci = 0, as the sun and planets can be considered point masses in empty space. Einstein's calculation of the precession of Mercury's orbit was based studying spacetime with Ricci = 0.

Next we look at electric charges.

1. You write, "Under locality, points only relate to nearby points, not distant points."

So when Euclid related two points by stating that a line is defined by two points, how close must the two points be for Euclid's definition to hold true, in your view?

2. Euclid's line is imaginary, an abstract concept. Try it with a wooden stick, 2 points do not define it, only a collection of directly ajacent points make it a real stick, in this case atoms.
The idea for space then becomes the concept of 'no ajacent stuff, no causality', and since we have a causality, there must be ajacent stuff. The point is to demistify and no longer think of abstract spacetime, but of ajacent real entities making up space, and time emerging from it.

3. You write, "The idea for space then becomes the concept of 'no ajacent stuff, no causality',"

So when light travels from the sun to the earth, can the light pass through space and cause the earth to warm up? Or, are you saying that this is impossible because the sun is not adjacent to the earth, and thus there is no causality?

4. You write, "The point is to demistify and no longer think of abstract spacetime, but of ajacent real entities making up space, and time emerging from it."

Suppose an electron passes a meter stick at .999999999999999999999999999c. From the electron's reference frame, are the ends of the meter stick "adjacent" in your theory, or are they non-adjacent?

At what electron velocity will the electron see the ends of the meter stick as adjacent?

5. Points do not and never have had mass.
There are no point masses in actual physical existence, anywhere. Points are just as imaginary a construct as lines. They can carry no mass, and have no ability to be physically affecting other things, carrying forces, or be used other than as imagined place holders diagrammatically, as they are abstract ideas, not actual things. They have ZERO geometric extension, thus they have no volume (which is required in a calculation to determine density in order to determine mass), and because a point has no actual shape (zero sized volume has no measurable extension, it's a reference to location, not a shape), it can not spin or rotate, zero sized entities also can't have physical movement, as there is nothing there to move or be observed rotating, much less interact with anything, as a zero sized entity can not interact with a another zero sized entity or a non zero sized entity.

Black hole lunacy relating to getting infinite masses or singularities all revolve around misusing a point to convey mass in a purely mathematical space with Ricci = 0, meaning there is no energy or mass in the spacetime, and yet mass is surreptitiously (by saying so or magic) inserted back into the spacetime without any attributable volume of mass there to assign it to, which is essentially hogwash.

1. Just to clarify...

If you start with a quantity defined over a finite volume and then subject this definition to a limiting process towards vanishing volume, you _can_ get point-quantities that look contradictory to what you are saying above.

For instance, if there is a velocity field in a fluid, it means that there exist an infinity of points (even within a finite region) where velocity and density are defined, and so, obviously, you can say that something is moving at each point. By going from Eulerian to Lagrangian description, you can say that the point is moving.

You can also have more complicated and locally (``at point'') defined quantities such as stresses, strains, conductivity tensors, etc., none of which can be defined for a literally zero-sized entity, but which, though first defined over finite volumes or areas, by subjecting them to a limiting process, can be taken as point-quantities.

Personally speaking, I took an inordinately long time to understand this point: viz., that there can be infinitesimals within an infinitesimal, and even further, that there can even be an _infinity_ of infinitesimals within an _infinitesimal_. It all comes about because of the continuum hypothesis.

Since the hypothesis is that powerful, you can always start with any quantity defined for a finite volume and then make a point quantity out of it. Yes, inasmuch as a quantitative measure can be specified for shape, you can say, after the limiting process, that the resulting points have zero sizes but they also have shapes. The contradiction resolves when you realize that Euclid's definition of a point itself does not make sense. The only way a point can be defined is via methods of calculus. So, a point is not zero-sized; it is: infinitesimally small-sized.

Best,

--Ajit

6. You can divide up space any way you like, but there are physical limits where science is concerned, as well as the actual calculation itself. When you have something without size, no, you aren't really doing anything computationally with it, unless you wish to claim you can divide zero. I don't think you want to claim that.

Yes, a zero is of no size, (read up on Cauchy and the size of a point) it is a place on a number line, zero is place holder. To get to some size you have to have distance between one thing and another. There is no distance between a number and itself on a number line, just as there is not a distance between a point and itself on a graph. Calculus is actually a linguistic/logical sleight of hand, it plays with defined definitions and then changes those definitions elastically to get whatever it wants by conflating small (or something not zero) with zero. What should be used is finite calculus, as there are no measurable infinitesimals, that's more like rubber numbers or an elastic ruler. In reality, there is only change over measurable distances or over a measurable period of time...with margins or error. There is no change over no period (no time) or an instant, or over some small distance you can't actually measure or that is swamped by your margin or error. Saying otherwise is to not understand what is going on, but to get lost in the process rules of math. NO, math is not reality, it is a abstract construct we overlay on top of reality which can sometimes be useful, like a map. Math is not constrained by reality, math is constrained by created logical rules designed to be self consistent. Math in a sense is as non physical as Euclidian geometry in that it places itself outside of the actual reality that contains it (it is a subset of reality not vice versa), it is depicted as independent of time and actual physical process while claiming to do process outside of time. As a person who has studied what calculation actually is, and how it is represented or performed in various logical, mechanical, and electronic formats, it actually is always a FINITE process in some amount of time, even if you pretend to not acknowledge it. No matter the system, there is always some plain old finite bean counting going on in a finite amount of time, there are no infinitesimals, just a finite number of decimal places, which really is no different than really big numbers. No matter how big your number, No, you don't get any closer to infinity, you haven't got forever, and your memory and computational architectural are always finite. Likewise, in terms of going small, you never really approach zero either unless you skip by some fixed increment, and face it, there are an infinite number of points between any two different numbers, just add another decimal place. An infinity of points all added together is nothing, largely because two infinity of points is no larger than one infinity of points, or a google of infinities, they have no fixed size, so stacking them or attempting to add them doesn't mean anything, as infinity isn't actually a number, just a direction on a number line.
Once again, a point has no size, and you can't just make up a size because then everyone can do that which makes it meaningless for purposes of calculation or comparison. A mile is only a mile if we agree on how to determine a fixed length. There is no infinitesimally small, that is meaningless except as a convenient pretense for hand waving. You also can't conflate zero with not zero, they are not equivalent.

1. 0. You write many points that seem interesting, but they all come embedded in such a fine fabric of arguments that to isolate the points of differences and agreements would be too daunting a task. Certainly not meant for a comment-by-comment exchange on a blog post!

1. But, yes, in talking to you, I made one error towards the end, and you picked it up right.

I said in my above comment:

``The contradiction resolves when you realize that Euclid's definition of a point itself does not make sense. The only way a point can be defined is via methods of calculus. So, a point is not zero-sized; it is: infinitesimally small-sized.''

This is wrong, I soon later realized, but only after clicking the ``Publish'' button.

Yes, through analytic geometry, a point can be defined as the geometric counterpart of a number---which means, the definition of a point isn't via infinitesimals, but at the more basic level of the identity that is a number. Yes, since size refers to a difference between two numbers, a point can be taken as an entity without a size.

2. However, no, I am not as against the ideas of calculus as you seem to be.

No, one does not have to commit the mistake of intrincisim, of thinking that numbers exist in the metaphysical reality out there as apart from the mind that grasps them.

But this position does not mean that numbers are completely free-floating abstractions unconnected from reality either.

3. Since I regard infinitesimals as just a short-hand that captures the _definite_ _trends_ in the numbers belonging to infinite sequences, I am quite OK with them, as also with the methods of calculus. I do think that calculus is a towering achievement, and refers to _de-finite_ things, that it's not a sleight of hand. The position depends on an appreciation of the idea of limits, esp. of the idea that a limit of a sequence need not belong in the values that make up that sequence, and yet, does capture something which is given in the definite succession of definite numbers that is the sequence.

4. If interested, please see a couple of posts of mine. Major Caveat: Both are far too long, and are very boringly written; they need a second and a third editorial pass and considerable shortening and focusing. But even as rough notings, they do succeed in conveying a certain definite sense regarding my positions. The posts are:

More generally, see (if interested):

and

I am not ready for a very prolonged or detailed exchange even on those write-ups, though. Currently, am interested in other things (as indicated by the currently sticky post at my blog).

Bye for now, and best,

--Ajit

7. @CFT

Exactly.

@GNR

I illustrated a philosophical approach concerning infinities, the infinitesimally small, points,..:

On pages 9-12 :

https://www.scribd.com/doc/343283628

And a summary for a discrete gravitational field incorporating the propagation of light, accompanied by a causal principal for gravity in the following link.
It comes with model, interpretation (a.o. a different interpretation of so called cosmological redshift), prediction, and concept for an experimental test with clocks, with true capability of falsifying instead of 10^500 possibilities:

'A New Breeding Ground to Uncover the Discontinuum'

https://www.scribd.com/doc/361404451