- #1
Phrak
- 4,267
- 6
A vector in special relativity is the quantity:
[tex]V = V^\mu \hat{e_\mu}[/tex]
On a change of coordinates, the basis vectors co-vary with the coordinate derivatives:
[tex]\hat{e_\mu'} = \frac{\partial x_\mu'}{\partial x_\mu} \hat{e_\mu}[/tex]
The vector elements are the opposite. They are said to be contravariant.
[tex]V'_\mu = \frac{\partial x_\mu}{\partial x_\mu'} V_\mu[/tex]
All basic stuff.Factually speaking, the vector itself is unchanged under a coordinate transformation; factually, the vector is invariant. The vector does not vary. A vector is unchanged under the Lorentz group. The coordinate system changes but the vector does not. That's why its called relativity! The immutable vector is unchanged by a change in coordinate system. It's elements and basis are not.
But the standard language is different. We say, "A vector is Lorentz Covariant". This is shorthand. It really means "The bases of a vector V is Lorentz Covariant." Through whatever route, this grammatical error has become standard. It is wrong-speak, but there it is.
However:-
From time to time I may slip and use the factual language rather than the standard shorthand without thinking. I would appreciate, if in the future this happens again, I am not again insulted over this and implied to be ignorant or stupid or pretentious, ad nauseum, in the subject matter of this relativity forum by either mentors, members or those having any other label.
.
[tex]V = V^\mu \hat{e_\mu}[/tex]
On a change of coordinates, the basis vectors co-vary with the coordinate derivatives:
[tex]\hat{e_\mu'} = \frac{\partial x_\mu'}{\partial x_\mu} \hat{e_\mu}[/tex]
The vector elements are the opposite. They are said to be contravariant.
[tex]V'_\mu = \frac{\partial x_\mu}{\partial x_\mu'} V_\mu[/tex]
All basic stuff.Factually speaking, the vector itself is unchanged under a coordinate transformation; factually, the vector is invariant. The vector does not vary. A vector is unchanged under the Lorentz group. The coordinate system changes but the vector does not. That's why its called relativity! The immutable vector is unchanged by a change in coordinate system. It's elements and basis are not.
But the standard language is different. We say, "A vector is Lorentz Covariant". This is shorthand. It really means "The bases of a vector V is Lorentz Covariant." Through whatever route, this grammatical error has become standard. It is wrong-speak, but there it is.
However:-
From time to time I may slip and use the factual language rather than the standard shorthand without thinking. I would appreciate, if in the future this happens again, I am not again insulted over this and implied to be ignorant or stupid or pretentious, ad nauseum, in the subject matter of this relativity forum by either mentors, members or those having any other label.
.
Last edited: