February 24

I'm looking for a totally mathematical description of Minkowski space, and I'm unable to find one. The wikipedia page says that its a 4-dimensional vector space equipped with a bilinear norm. However, I want a mathematical definition of what that vector space is. Is it R^4? Taabibtaza (talk) 08:11, 24 February 2024 (UTC)[reply]

No. Minkowksi space is a vector space, period. It is defined by the vectors that live in it; to start with, these are abstract objects. They can be related to ${\displaystyle \mathbb {R} ^{4}}$ through an isomorphism, i.e. a mapping from those abstract vectors and operations to, I'll say, matrices (${\displaystyle 4\times 1}$ or ${\displaystyle 1\times 4}$) in ${\displaystyle \mathbb {R} ^{4}}$. In fact there are (infinitely) many such isomorphisms distinguished by the choice of a vector basis; physically that corresponds to the choice of reference system. This distinction is in my view essential for a proper understanding of relativity: there is an absolute space-time, mathematically represented by Minkowski space, in which physical quanities, e.g. four-momentum, are given as vectors, geometrical objects, that are what they are independent of any observer. When we measure these things we measure vector components in a specific basis, our reference system, given by the experimental setup. These components (e.g. energy or one of the three spatial momentum components) are scalar products of the vector with one the basis vectors and they are relative in the sense that they depend on the choice of the basis vector, i.e. the reference system. --Wrongfilter (talk) 08:24, 24 February 2024 (UTC)[reply]

@Wrongfilter

I'm just starting to learn Minkowski space, so I'm assuming we have already chosen a coordinate system for Minkowski space such that it's origin corresponds to an inertial observer who has three axes attached to him such that the 3 spatial coordinates of any event correspond to the coordinates he will measure for that event using the axes that are attched to him, and such that the time at any event in spacetime corresponds to the time on our inertial observer's clock (whose clock was set to 0 at an arbitrary time) when that event happens. This is how the wikipedia introduces minkowski space (I'll learn how to shift to a different perspective later). Also, changing the basis while not changing the vector space won't won't allow us to change to an arbitrary frame of reference since our origin is still the same. Each vector space has a UNIQUE origin by the very definition of a vector space. The origin is the vector you get by multiplying any vector by the scalar "0". Anyway, now that we are done with the physical interpretation, think of Minkowski space mathematically.

Is that vector space R^4? If not, you have to give a definition of that vector space and its elements, you can't just say that vectors in that vector space are abstract objects. That's not how math works, even abstract objects have definitions. For example the elements of R^4 are defined as ordered quadruples of real numbers. Taabibtaza (talk) 11:01, 24 February 2024 (UTC)[reply]

First up, I'm a physicist with an incomplete and probably not too solid a background in the relevant mathematics, so I may be shaky in my use of terminology and incomplete in my perspective on these things. With that out of the way, back to pretending that I understand something. A vector is simply defined as a member of a vector space. It is defined by how it behaves, not by what it is. For instance you can show that the ordered quadruples (I called them matrices above, but ordered quadruples is better) with the appropriate definition of addition and multiplication by real numbers form a vector space. Therefore ${\displaystyle \mathbb {R} ^{4}}$ is a vector space and the ordered quadruples are vectors. Add the appropriate definition for the metric and you can show that ${\displaystyle \mathbb {R} ^{4}}$ with that metric ${\displaystyle \eta (\cdot ,\cdot )}$ is a Minkowski space. Next comes the physical assumption that space-time with events and separations between events can be described as a Minkowski space (if you insist in thinking about the origin, maybe as a class of Minkowski spaces because the origin is physically irrelevant by virtue of the relativity principle). Thus we have two spaces that are both Minkowski, therefore these two spaces are isomorphic, therefore we can map space-time to ${\displaystyle \left(\mathbb {R} ^{4},\eta (\cdot ,\cdot )\right)}$, and it will turn out that we can do that in infinitely many ways (through Lorentz transformations). Maybe more directly to your question: (1) Mathematically, a Minkowski space is an abstract space with certain properties. (2) Physically, in special relativity, Minkowski space is space-time, the vectors are separations between events. I guess that on this abstract level, four-momenta form yet another Minkowski space. (3) ${\displaystyle \left(\mathbb {R} ^{4},\eta (\cdot ,\cdot )\right)}$ is another, extremely useful, Minkowski space. This is of course the main mathematical structure to apply the theory.

I don't think, actually, that you need to worry too much about these things (subtleties?) while learning, but it's good be aware that they exist. Note incidentally that in quantum mechanics it is more common to say things like "Spin states form a Hilbert space", implying that Hilbert space is an abstract mathematical structure that can be filled with physical content in various ways. In relativity "Minkowski space" more uniquely associated with a particular physical context, viz. space-time. --Wrongfilter (talk) 12:05, 24 February 2024 (UTC)[reply]

@Wrongfilter

Can you please tell what η(.,.) means? Taabibtaza (talk) 14:07, 24 February 2024 (UTC)[reply]

See Minkowski space#Scalar product. Double sharp (talk) 14:58, 24 February 2024 (UTC)[reply]

I find it convenient to think of Minkowski space as having one imaginary coordinate (timelike), the others real (spacelike). —Tamfang (talk) 21:57, 9 March 2024 (UTC)[reply]

Obviously it'd only be pure hydrogen-1 before ignition. Assuming no mass is added by stellar companion or any other means.Sagittarian Milky Way (talk) 16:59, 24 February 2024 (UTC)[reply]

Helium would make it a bit more opaque by adding absorption lines. Also HeH⁺ would add opacity. More energy could be derived from pure hydrogen. A pure hydrogen star could form bigger than when mixed with helium. Graeme Bartlett (talk) 21:21, 24 February 2024 (UTC)[reply]

And note that some Helium (and possibly a trace of Lithium) was formed by the 'Big Bang' (the last I heard), so a completely pure Hydrogen star is never going to have happened. {The poster formerly known as 87.81.230.195} 176.24.45.226 (talk) 22:03, 24 February 2024 (UTC)[reply]

Yes I was aware. This does bring up the question of 1st generation stars vs. non-existent ones that are exactly the same except they ignited with exactly 100% isotopes of H and He in 1st gen star ratios. Could current tech tell them apart and from what distance? Sagittarian Milky Way (talk) 22:26, 24 February 2024 (UTC)[reply]

Can current technology tell existent and non-existent starts apart? That is an interesting issue.  --Lambiam 04:56, 25 February 2024 (UTC)[reply]

All stars have spectral lines and specs: Mass, luminosity and magnitude, radius, variability, isotope and element ratios, color indexes like blue minus V and U minus blue... Different starting compositions would have different specs or spectra at the same age and starting masses, the question is how easy could they be told apart with 6 orders of magnitude less non-1st row element content than the Sun vs infinite orders of magnitude less. Sagittarian Milky Way (talk) 06:46, 25 February 2024 (UTC)[reply]

Are there any flying arthropods that are not insects? —Mahāgaja · talk 22:13, 24 February 2024 (UTC)[reply]

Assuming you mean powered flight, then no according to Flying and gliding animals. If you accept gliders, then see the bridge-spider as an example of a spider which uses threads in the wind to travel.-gadfium 23:03, 24 February 2024 (UTC)[reply]

Thanks! I do indeed mean true flight (like bats) not gliding (like flying squirrels). —Mahāgaja · talk 08:24, 25 February 2024 (UTC)[reply]

The human pelvis that has an anteroposterior diameter exceeding the transverse diameter qualifies a human as an anthropod. Human-powered flight has been possible since 1961 or possibly much earlier. Philvoids (talk) 05:06, 26 February 2024 (UTC)[reply]

An anthropod is not an arthropod. ←Baseball Bugs ^{What's up, Doc?} carrots→ 07:00, 26 February 2024 (UTC)[reply]

But there is a good chance that spiders are hitching rides on helicopters and planes. Graeme Bartlett (talk) 09:20, 26 February 2024 (UTC)[reply]

BB is correct. An arthropod is invertebrate animal of the phylum Arthropoda, characterized by a chitinous exoskeleton and multiple jointed appendages. Humans do not qualify. Philvoids (talk) 12:37, 26 February 2024 (UTC)[reply]

The first words of our Insect flight article: "Insects are the only group of invertebrates that have evolved wings and flight". Alansplodge (talk) 21:29, 26 February 2024 (UTC)[reply]

ah but it doesn't say no other has evolved wings or flight. Loophole! —Tamfang (talk) 20:45, 27 February 2024 (UTC)[reply]

no, unless you count spiders (theyre arachnids, not insects) 'ballooning' themselves in the air. :P mushi^{( ? )} 16:40, 29 February 2024 (UTC)[reply]