- #1
Shirish
- 244
- 32
I'm reading about the geometry of spacetime in special relativity (ref. Core Principles of Special and General Relativity by Luscombe). Here's the relevant section:
-----
Minkowski space is a four-dimensional vector space (with points in one-to-one correspondence with those of ##\mathbb{R}^4##) spanned by one timelike basis vector, ##\vec e_t##, and three spacelike basis vectors, ##\vec e_x, \vec e_y, \vec e_z##. While any four linearly independent vectors can constitute a basis, in IRFs we require time to be orthogonal to space.
-----
I'm not clear what is meant by this. By "time", does the author mean the ##[1,0,0,0]## basis vector and by "space", does he mean the ##[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]## basis vectors? If that's the case, this is no different from the Euclidean ##\mathbb{R}^4## case - I could've just said that "we require ##x## to be orthogonal to the other dimensions", where by "##x##" I mean the corresponding standard basis vector. So I'm guessing that's not what the author meant.
If by "time" and "space" the author isn't referring to corresponding standard basis vectors, then I'm not sure what's stopping me from taking an arbitrary basis of ##\mathbb{R}^4## as a basis of Minkowski Space as well. Why do we require "time" to be orthogonal to "space"? (whatever the quoted words mean in this context) What is even meant by the statement?
Is there a proper mathematical justification for this?
-----
Minkowski space is a four-dimensional vector space (with points in one-to-one correspondence with those of ##\mathbb{R}^4##) spanned by one timelike basis vector, ##\vec e_t##, and three spacelike basis vectors, ##\vec e_x, \vec e_y, \vec e_z##. While any four linearly independent vectors can constitute a basis, in IRFs we require time to be orthogonal to space.
-----
I'm not clear what is meant by this. By "time", does the author mean the ##[1,0,0,0]## basis vector and by "space", does he mean the ##[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]## basis vectors? If that's the case, this is no different from the Euclidean ##\mathbb{R}^4## case - I could've just said that "we require ##x## to be orthogonal to the other dimensions", where by "##x##" I mean the corresponding standard basis vector. So I'm guessing that's not what the author meant.
If by "time" and "space" the author isn't referring to corresponding standard basis vectors, then I'm not sure what's stopping me from taking an arbitrary basis of ##\mathbb{R}^4## as a basis of Minkowski Space as well. Why do we require "time" to be orthogonal to "space"? (whatever the quoted words mean in this context) What is even meant by the statement?
Is there a proper mathematical justification for this?