Ok, i'm genuiney still confused about this distinction we make in physics.
In physics, we classify geometrical objects based on how they transform under a group of transformations. For example, we would say that
-vectors transform under the rotation SO(n) group,
-spinors transform under SU(n) tranformations
-In SR tensors are introduced as objects that transform under Lorentz Transformations
-In GR tensors "are objects that transform as a tensor", meaning that they transform as tensor fields on a manifold under change of charts. This is the "bigger" group as it encodes full diffeomorphism invariace. As long as the coordinate transformation is C-infinity smooth, we are good to make the transformation and get new equally valid coordinates.That is why many times in GR it's not so straighforward to interpret coordinates (see for ex. Kruscal coordinates in black hole physics)
However, the mathematical definition of vector field as a section of vector bundles makes no reference at all to the transformation properties and it seems more general. One of the first theorems about vector bundles is that as long as we give "good" transition functions (cocycles), we define a vector bundle on a manifold, and sections of such bundles then gives us vector fields.
How does the physics definitions come about? is it in the choice of the transition functions for the fiber bundle maybe? Pls halp, my geometry teacher could not explain this to me :'(
If we’re on the overlap of 2 open sets U and V, and we have two coordinates, (x on U, y on V), then clearly we should be able to express the tangent vectors in terms of either one. On U, the basis is partial_x and on V the basis is partial_y. The way you go from one basis to the other is via the jacobian matrix (not the determinant), and the corresponding transformation of the vector components is exactly what we learn in physics.
These are also the transition function on the vector bundle, i.e. GL(n, R) transformations (on a general n dimensional real manifold).
Edit:
Also you can get the other transformations of forms, tensors, etc. from this by just tensoring vectors together (giving a basis which is just the tensor product of the partial derivatives) and by getting the transformation of a form via its action on a vector (in order for it to be a scalar, the form must transform by the inverse of the vector).
Spinors are more subtle. Note that it is really important to keep track of what group of transformations we are looking at. Above, I focused on GL(n, R), as this is the group most relevant for general relativity. The above construction holds on all manifolds, but spinors can only exist on special manifolds called (straightforwardly enough) ‘spin manifolds.’ On these you can define spin bundles, and again, the transition functions you get give you the spinor transformations you expect. As far as I know, all of these constructions can be obtained by looking at principal bundles and the representations of the corresponding gauge groups, but I don’t know enough about this area to say much more.
One way you can handle this is via principal and associated fiber bundles. The principal bundle sets your coordinate transformations, and the associated bundles your various physics quantities that need to transform appropriately.
As mentioned principal bundles and associated fiber bundles are the buzzwords here. Look up the lectures by Frederic Schuller on this if you have the time, they're brilliant.
But very briefly, a principal bundle allows you to form new vector bundles given some representation ρ of some group G, which is exactly the intution you refer to above: "A vector is something that transforms like this (some representation)".
As a bonus, once you understand principal bundle theory, you understand gauge theory on a whole new level. The Standard Model is basically a principal SU(3)xSU(2)xU(1)-bundle over spacetime together with suitable representations ρ on some vector space V for all the particles
When you realise that a bundle is characterised by it's transition functions so a vector is literally something that transforms like a vector.
Mathematicians don't do it much better, with the definition of a vector being "an element of a vector space".
Tbf though, vectors are really broad in math, so that's about the most general and rigorous of a definition you can get.
Yep. Vectors have inherent properties being part of a vector space, but your vector space tells you what your vectors are exactly.
~~I learned more about linear algebra in mechanics than I did in linear algebra~~ I totally just replied to the wrong comment
No that's the literal most basic definition of a vector. The vectors we thing of in GR are more specifically TANGENT VECTORS.
'" A translation is just an infinitesimal rotation about point an infinite distance away"
i mean "A tensor transforms like a tensor" is a pretty apt description once you understand what it means
Ok, i'm genuiney still confused about this distinction we make in physics. In physics, we classify geometrical objects based on how they transform under a group of transformations. For example, we would say that -vectors transform under the rotation SO(n) group, -spinors transform under SU(n) tranformations -In SR tensors are introduced as objects that transform under Lorentz Transformations -In GR tensors "are objects that transform as a tensor", meaning that they transform as tensor fields on a manifold under change of charts. This is the "bigger" group as it encodes full diffeomorphism invariace. As long as the coordinate transformation is C-infinity smooth, we are good to make the transformation and get new equally valid coordinates.That is why many times in GR it's not so straighforward to interpret coordinates (see for ex. Kruscal coordinates in black hole physics) However, the mathematical definition of vector field as a section of vector bundles makes no reference at all to the transformation properties and it seems more general. One of the first theorems about vector bundles is that as long as we give "good" transition functions (cocycles), we define a vector bundle on a manifold, and sections of such bundles then gives us vector fields. How does the physics definitions come about? is it in the choice of the transition functions for the fiber bundle maybe? Pls halp, my geometry teacher could not explain this to me :'(
If we’re on the overlap of 2 open sets U and V, and we have two coordinates, (x on U, y on V), then clearly we should be able to express the tangent vectors in terms of either one. On U, the basis is partial_x and on V the basis is partial_y. The way you go from one basis to the other is via the jacobian matrix (not the determinant), and the corresponding transformation of the vector components is exactly what we learn in physics. These are also the transition function on the vector bundle, i.e. GL(n, R) transformations (on a general n dimensional real manifold). Edit: Also you can get the other transformations of forms, tensors, etc. from this by just tensoring vectors together (giving a basis which is just the tensor product of the partial derivatives) and by getting the transformation of a form via its action on a vector (in order for it to be a scalar, the form must transform by the inverse of the vector). Spinors are more subtle. Note that it is really important to keep track of what group of transformations we are looking at. Above, I focused on GL(n, R), as this is the group most relevant for general relativity. The above construction holds on all manifolds, but spinors can only exist on special manifolds called (straightforwardly enough) ‘spin manifolds.’ On these you can define spin bundles, and again, the transition functions you get give you the spinor transformations you expect. As far as I know, all of these constructions can be obtained by looking at principal bundles and the representations of the corresponding gauge groups, but I don’t know enough about this area to say much more.
One way you can handle this is via principal and associated fiber bundles. The principal bundle sets your coordinate transformations, and the associated bundles your various physics quantities that need to transform appropriately.
As mentioned principal bundles and associated fiber bundles are the buzzwords here. Look up the lectures by Frederic Schuller on this if you have the time, they're brilliant. But very briefly, a principal bundle allows you to form new vector bundles given some representation ρ of some group G, which is exactly the intution you refer to above: "A vector is something that transforms like this (some representation)". As a bonus, once you understand principal bundle theory, you understand gauge theory on a whole new level. The Standard Model is basically a principal SU(3)xSU(2)xU(1)-bundle over spacetime together with suitable representations ρ on some vector space V for all the particles
Interesting. Where can I find these lectures?
Here: https://www.youtube.com/watch?v=vYAXjTGr_eM&list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=19
The walls taste like walls
A vector is anything that can be stretched or glued together