Worldlines & Curves in Spacetime: Exploring Possibilities

[SaintRodriguez]

Can there be worldlines that are neither timelike, nor null, nor spacelike? They can
Are there curves in spacetime that are neither timelike, nor null, nor spacelike? Why?

 

[malawi_glenn]

Can there be integers that are neither zero, positive nor negative?

 

[Ibix]

You can have curves that change character along their length - so are spacelike, null, and timelike at different events. But that's all - see the previous post for why (although note that tangent vector inner products with themselves can be any real number, not just integers).

 

[PeroK]

Can there be worldlines that are neither timelike, nor null, nor spacelike? They can
Are there curves in spacetime that are neither timelike, nor null, nor spacelike? Why?

It depends what you allow as a "curve". In real analysis, there is the Weierstrass function, which is continuous everywhere, but differentiable nowhere.

https://en.wikipedia.org/wiki/Weierstrass_function

I suspect a mathematician could construct a continuous curve that was neither timelike, spacelike or null on any finite interval.

 

[PeterDonis]

Can there be worldlines that are neither timelike, nor null, nor spacelike?

Mathematically, one might be able to construct something like that (@PeroK describes an example).

Physically, no, we don't observe any actual realizations of curves that are neither timelike, nor null, nor spacelike.

 

[Mark44]

Can there be integers that are neither zero, positive nor negative?

Gaussian integers; e.g., 1 + 2i. Reference - https://en.wikipedia.org/wiki/Gaussian_integer

 

[PeterDonis]

Gaussian integers

The squared norm of a 4-vector in spacetime can't be a Gaussian integer. It can only be an ordinary real number.

 

[Mark44]

The squared norm of a 4-vector in spacetime can't be a Gaussian integer. It can only be an ordinary real number.

My comment was specifically a response to this statement, and nothing more:

Can there be integers that are neither zero, positive nor negative?

 

[PeterDonis]

My comment was specifically a response to this statement, and nothing more

But that statement was not made in a vacuum. It was made in response to the OP of this thread. In that context, it seems evident to me that the statement was intended as a reference to the fact about squared norms that I stated.

 

[Ibix]

But that statement was not made in a vacuum.

It was if he was talking about Minkowski spacetime!

(...I'll get my coat)

 

[Mark44]

Can there be worldlines that are neither timelike, nor null, nor spacelike?

Can there be integers that are neither zero, positive nor negative?

But that statement was not made in a vacuum. It was made in response to the OP of this thread.

malawi_glenn's question seemed to me to be a rhetorical question about a scenario with presumably exactly three possibilities, but mathematically, there is one that wasn't listed.

I'm a very literal person...

 

[Ibix]

malawi_glenn's question seemed to me to be a rhetorical question about a scenario with presumably exactly three possibilities, but mathematically, there is one that wasn't listed.

The additional point necessary, then, is that the squared norms of tangent vectors to curves are always real. That excludes your case of Gaussian integers and leaves the three possibilities to which @malawi_glenn referred.

I must admit I wasn't aware of the term "Gaussian integer", so between that and the Weierstrass function it's been a learning day for me.

 

[Mark44]

That excludes your case of Gaussian integers and leaves the three possibilities to which @malawi_glenn referred.

I would buy this if malawi_glenn's post had included some context about 4D coordinates. However, it mentioned only integers, with no other context. My purpose was to enlighten readers that there are integers that are neither negative, zero, or positive. Nothing more.

 

[malawi_glenn]

My purpose was to enlighten readers that there are integers that are neither negative, zero, or positive

I had no idea that Gaussian integers were elements of $\mathbb{Z}$

Perhaps catfish are mammals too?

If you read your wiki article you can see for yourself that it is written "Gaussian integers share many properties with integers"
and one does therefore conclude that they are not integers.

Anyway, back to the topic.
The concept of wordline has no afaik universal definition. Some say it is a curve parametrized by the particles proper time, and such curves are either null, lightlike or timelike. However, some call all curves in Minkowski space for wordlines.

 

[strangerep]

The concept of wordline has no afaik universal definition. Some say it is a curve parametrized by the particles proper time, and such curves are either null, lightlike or timelike.

Umm,... are you sure you don't want to edit that slightly?

 

[malawi_glenn]

Umm,... are you sure you don't want to edit that slightly?

Yes I will :) I have a fever now so I can't think or write proper :D

 

[malawi_glenn]

Ok here it is

Worldlines are sometimes restricted for curves that are time-like, those curves can be parametrized by the particles proper time.

 

[martinbn]

I would buy this if malawi_glenn's post had included some context about 4D coordinates. However, it mentioned only integers, with no other context. My purpose was to enlighten readers that there are integers that are neither negative, zero, or positive. Nothing more.

It wasn't said explicitly, but it was understood, that he meant rational integers. Because this is what is meant by "integers" if there are no additional qualifiers such as "Gaussian".

 

[martinbn]

I had no idea that Gaussian integers were elements of $\mathbb{Z}$

Perhaps catfish are mammals too?

If you read your wiki article you can see for yourself that it is written "Gaussian integers share many properties with integers"
and one does therefore conclude that they are not integers.

Back to off topic: they are integers in the sense of algebraic integers. Just like complex numbers are numbers although they are not a subset of $\mathbb R$.