In fact, as we shall illustrate throughout this book, it is rarely worthwhile to introduce a basis and take components of a tensor at all, let alone to worry about how these components change under a change of basis.

This is like in Euclid’s Elements. You can do all of the proofs without any measuring om arcs or lengths. But if you want to see whether the earth is flat or a sphere, or doing surveying, you will have to measure the arcs and lengths of the sides of triangles.

Likewise, you can do most, if not all, of GR without actually filling in the numbers for the tensors. But if you want to actually see what happens, e.g., when calculating the slowing down of pulsars, spinning neutron stars, you will have to measure masses, times, and distances.

]]>General Relativity

Just a quick footnote, Robert M. Wald “General Relativity” U. Chicago Press (1984) develops the theory using modern intrinsic geometry, with a minimal use of coordinates.

Extract discussing multiple indices and coordinates versus intrinsic geometric formulation:

“ In other treatments, equation (2.3.8) often is used as the defining property of a tensor. The definition we have given here has the advantage that it generally is much easier to define a quantity as a tensor by displaying it as a multilinear map on vectors and dual vectors than it is to display it as a collection of numbers associated with a coordinate system which changes according to equation (2.3.8) when we change coordinate systems. In fact, as we shall illustrate throughout this book, it is rarely worthwhile to introduce a basis and take components of a tensor at all, let alone to worry about how these components change under a change of basis.”

]]>That’s only because the physics has been corrected, augmented, totally revised etc. to accommodate the accumulation of what we have looked at as we go along.

That’s called the scientific method. It gave us quantum mechanics, electricity and computers, and microbiology and antibiotics. Without it you and I would not be corresponding and probably would not even be alive.

today, M-theory or quantum loop?

Both are pure mathematics, ie, theoretical physics. Astronomers do not want to touch them. Neither theory has observations to back them. That is why they cannot decide which one to choose. There still is no credible successor of General Relativity. GR has boatloads of observational evidence to back it.

But to put the local patches together seems to at least implicitly require the intrinsic point of view.

General Relativity is doing that. If you formulate physics in Tensor mathematics, all of physics works in any manifold. Astronomers have looked how far out you can do this, and it looks you can do this as far out as you can look.

There are exiting developments around dark matter, but those behind it tell us it is too early to draw any conclusions.

]]>Wherever we have looked, physics does work flawlessly.

That’s only because the physics has been corrected, augmented, totally revised etc. to accommodate the accumulation of what we have looked at as we go along. It’s not flawless while new data and phenomena are in the process of being understood. E.g. today, M-theory or quantum loop? Or are they really the same ? Stay tuned for further episodes.

Without coordinates, you quickly end up in qualitative statements

Not so. It’s ok to start with coordinates, just as the definition of manifold starts with local patches mapped to Euclidean coordinate space. But to put the local patches together seems to at least implicitly require the intrinsic point of view.

]]>Not sure how this is relevant.

Your questions are always regarding places in the universe where we have not yet been able to look. Wherever we have looked, physics does work flawlessly. Speculating about possible deviations of current physics in places we are unable to look sounds a lot like those old arguments, or new ones in creationism, that God is hidden in the gaps of science.

My question there was whether there is a treatment of Physics, General Relativity say, that proceeds without using coordinates for everything.

Physics, and science in general, tries to predict/model measurements. Measurements are quantitative and hence require coordinates. Without coordinates, you quickly end up in qualitative statements which cannot distinguish between possible theories.

As the current method of science, which we can trace back to Euclid and Plato, is very successful, scientists feel little inclinations to go qualitative.

But then, scientists do not look for the *Answer to the Ultimate Question of Life, the Universe, and Everything*.

Just throwing up questions does not help much. Especially not if you do it without looking whether they have already been answered.

Sure it does or can. Not being an expert except in a small area, one asks questions (probably foolish ones) hoping for on return a flash of insight or guidance for intuition, i.e. a “teaching:learning” moment. It doesn’t help much to just dive into the literature.

Specifically regarding homogeneity and isotropy, this just sounds totally wrong to my mathematical gut. In the small there is homogeneity perhaps, but there is nothing by nature like this. It would say the universe is just a big cancer.

So far for today’s foolishness.

]]>A God of the gaps will shrink with the gaps

Not sure how this is relevant. My question there was whether there is a treatment of Physics, General Relativity say, that proceeds without using coordinates for everything. In mathematics, this intrinsic approach greatly clarified and made explicit what is going on, compared to the coordinates and indices based approach of say the 19th century (Levi-Civita).

]]>This is perhaps valid locally but what can be said globally ?

Astronomers have looked far in space and time (~10B lightyears) and have not found any evidence that the universe is not homogeneous and isotropic. If you want non-observational evidence, do not look at physics, or science.

This is achieved in mathematics but has this intrinsic expression been presented on Physics ?

The preeminent physical theory that depends on this is General Relativity, and that has so far weathered every test thrown at it, down to recording gravitational waves from merging black holes. Again observations do not find any evidence of a non-homogeneous or non-isotropic universe.

Just throwing up questions does not help much. Especially not if you do it without looking whether they have already been answered. A God of the gaps will shrink with the gaps.

]]>the universe is homogeneous and isotropic

This is perhaps valid locally but what can be said globally ? E.g. Eddington in The Mathematical Theory of Relativity discusses the possibility of the signature of space/time changing, e.g. two dimensional time.

Physics does look the same whatever basis or units

This is similar to the definition of a differentiable manifold as a (topological) space that is everywhere locally faithfully mappable to Euclidean space (of some dimension), i.e. coordinate patches; and for which the the transformation between local maps (where they overlap) is differentiable. Then the whole analysis is to find the intrinsic characteristics of the space expressed without reference to coordinates.

This is achieved in mathematics but has this intrinsic expression been presented on Physics ?

]]>So the laws of physics are expressed using arbitrary non-natural quantities.

But all our laws of physics are “observer independent”, that is, independent of the selection of vector base, scale, or units.

There are “natural units”, e.g., the Planck units [1]. But these are not very practical. Constants like the speed of light in vacuum are independent of the quantities chosen and the universe is homogeneous and isotropic, so there are no special positions or directions. These symmetries are deeply connected to preserved quantities, e.g., mass/energy and linear and angular momentum [2]. That was the point of Einstein’s Relativity Theory, both Special and General.

All together, this tells us that Physics does look the same whatever basis or units you chose to describe it. Or, from another viewpoint, there are no natural units.

[1] ‘https://en.wikipedia.org/wiki/Planck_units

[2] ‘https://en.wikipedia.org/wiki/Conservation_law

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