A What is the formal definition of spacetime in physics?

[Enrico]

If you mean, how do we know which metric to use for whatever particular spacetime we are modeling, we do that by solving the Einstein Field Equation.

First you have to derive the equation then. How do you do that if you still have to define what spacetime is?

 

[Ibix]

I have manually created a link with a specific visible text. Is that the only way?

Depends. The button should give you a URL. If you are in the plain text editor (buttons in the grey bar at the top of the edit box are red) you then have to create the link manually as [url=https://whatever]post #24[/url]. If you are in the WYSIWYG editor (buttons are black) there's an "add link" button that looks like two circles forming two links in a chain. That gives you a dialog with a place to paste the link and a place to enter the text you want to display. You can switch between the two editor modes by pressing the button that looks like a pair of square brackets - if that's not visible click the rightmost $\vdots$ and it will pop up.

Your final option is to quote some of the post you want to refer to. Highlighting a bit of it should give you a popup with Quote and Reply. Clicking Reply inserts the selected text into your edit box, and the "<User> says" text at the top of the quote paragraph is a link to the post you quoted.

 

[Enrico]

At this point let me ask a question. Just out of curiosity. What is everybody's favorite definition of a tensor?

 

[weirdoguy]

A function $T:V^p\times (V^*)^q\rightarrow \mathbb{R}$ linear in every ergument. $V$ is of course some vector space.

 

[Enrico]

If that is too far, then it would be good to declare your starting point.

Obviously I'm not willing to start explicitly from there, but actually when I think of fundamentals the proof that $1+1=2$ is what comes to my mind (I had seen that from the Peano axioms. Thanks for quoting the PM, I'll put that into the infinite pipeline of things I  could read in my life).

 

[Enrico]

A function $T:V^p\times (V^*)^q\rightarrow \mathbb{R}$ linear in every ergument. $V$ is of course some vector space.

Actually this is a multilinear application. This definition is as saying that a matrix is a linear application. Just as a matrix represents a linear application, I'd say that a tensor represents a multilinear application.

 

[vanhees71]

No, a matrix can be used to arrange the components of a 2nd-rank tensor in a handy scheme. The tensor is an invariant object, independent of any basis, and the definition given in #54 is the established definition in math and physics.

 

[weirdoguy]

It was.

Then show us explicity where.

 

[robphy]

Was this a disputation between physicists? Because if you look at the actual physics literature, there is no issue at all about the derivation or validity of the Lorentz transformations, and hasn't been for decades. So whatever level of rigor physicists have used for that seems to work just fine.

It was. With more than one. In particular, I had a paper about this very subject. Submitted to three different reviews. Two of them stated "it's correct but not suitable for us". The third stated "it's suitable for us but wrong". And the reason why they thought it was wrong was impossible to dispute exactly because it was about the relation between frames of reference and the actual spacetime (and the consequent unique relation between two different frames of reference). Similar issues I had when discussing with  some physicists directly, and it was impossibile to get an agreement exactly because of the lack of a precise definition of the object under consideration (i.e. an algebraic model for spacetime).

For an algebraic model, the only starting points I know of
are discussed in approaches to non-commutative geometry
(which I am aware of, but know practically nothing about).

From https://alainconnes.org/wp-content/uploads/noncommutative_differential_geometry.pdf
(bolding mine)

What such spaces have in common is to be, in general, badly behaved as point sets, so that the usual tools of measure theory, topology and differential geometry lose their pertinence.
These spaces are much better understood by means of a canonically associated algebra which is the group convolution algebra in case b).
When the space V is an ordinary manifold, the associated algebra is commutative.
It is an algebra of complex-valued functions on V, endowed with the pointwise operations of sum and product.

A smooth manifold V can be considered from different points of view such as
α) Measure theory (i.e. V appears as a measure space with a f‌ixed measure class),
β) Topology (i.e. V appears as a locally compact space),
γ) Differential geometry (i.e. V appears as a smooth manifold).

Each of these structures on V is fully specif‌ied by the corresponding algebra of functions, namely:
α) The commutative von Neumann algebra $L^\infty(V)$ of classes of essentially bounded measurable functions on V,
β) The $C^*$-algebra $C_0(V)$ of continuous functions on V which vanish at inf‌inity,
γ) The algebra $C^\infty_{c}(V)$ of smooth functions with compact support.

If that is too far, then it would be good to declare your starting point.

Obviously I'm not willing to start explicitly from there, but actually when I think of fundamentals the proof that $1+1=2$ is what comes to my mind (I had seen that from the Peano axioms. Thanks for quoting the PM, I'll put that into the infinite pipeline of things I  could read in my life).

I feel rather frustrated that we are all taking stabs in the dark to pin down where you want to start.
It's as if nothing we suggest is a good enough starting point for you.
Please explicitly declare your starting point.

 

[Dale]

I feel rather frustrated that we are all taking stabs in the dark to pin down where you want to start.

I also feel that it is foolish to demand to start in a specific place instead of starting in the usual place, and further that it is self-defeating to demand that you must proceed from that place only by construction.

Frankly, the problem in this thread is not theoretical physics, but the unreasonable demands of the OP:

1) standard starting point is not acceptable
2) acceptable starting point is secret
3) complete mathematical rigor is required
4) mathematical axioms are forbidden

Such an approach is wholly inconsistent with the professional scientific literature (because it is a horrible approach), and IMO does not belong here. I think that these demands are guaranteed to produce a failed outcome.

 

[Dale]

@Enrico you have not provided a good reason to reject an axiomatic formulation. In fact, you accepted it for vectors but rejected it for spacetime. Your justification was to capitalize the word THE. That is not a valid justification. Nor is bold or italics or underlining.

Since the professional scientific literature uses axioms then those axioms are acceptable here on PF. Material has been provided using those acceptable axioms as a starting point and reasoning from there. As usual, there are multiple such equivalent starting points and approaches that have been provided.

If you find that unsuitable for your preferences then it is up to you to show that your preferences are also suitable by finding a professional scientific reference that embodies your preferred starting point and construction. If your preferences are incompatible with the literature then your preferences should change.

When you have found such a reference please feel free to open a new thread on the topic. Alternatively, while reading the existing material that has been provided please feel free to open a new thread on any point in any of those references that you find confusing.

As this thread has become unproductive and inconsistent with the professional scientific literature it is now closed. The question in the OP has been answered, and the restrictions that you wish to apply are inappropriate, unjustified, and unclear.

 

[PeterDonis]

I don't feel comfortable by pointing to the real physical object. If we want to do maths

Adding one additional note: we're not doing math here. We're doing physics. Relativity is physics, not math. To be doing physics and yet not feel comfortable pointing to real physical objects does not make sense.