- #1
MManuel Abad
- 40
- 0
Hi there, Physics lovers!
I've got some questions for you!
Denoting by
(1) [tex]ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=c^{2}d\tau^{2}[/tex]
the interval (and [tex]\tau[/tex] the proper time) and using the signature (+---), we have that the equations of motion for a free particle are:
(2) [tex]\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma^{\mu}_{\nu\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\lambda}}{d\tau}=0[/tex]
; the so called "geodesic (timelike) equations", (those capital Gamma are the Christoffel symbols).
Obviously we have the relation:
(3) [tex]g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=c^{2}[/tex]
'Cuz the LHS of (3) gives us the square of the magnitude of the 4-velocity, and that's always PLUS c2
(FIRST QUESTION: using the -+++ signature we get that LHS of (3) is equal to -c2, don't we?)
Everything is fine. But then I get in trouble. Using an action based on an auxiliary field I found in some place, I get that for a MASSLESS particle, the equations of motion are:
(4) [tex]g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=0[/tex]
; which people call the "geodesic (lightlike) equations".
SECOND (set of) QUESTION(s): So, we have TWO geodesic equations? (2) for massive particles and (4) for massless particles? If that's so, why does (2) doesn't involve mass m??
THIRD (set of) QUESTION(s):
And why is LHS of (4) equal to zero? isn't that the square of the magnitude of the 4-velocity for a massless particle? shouldn't it be c2?? I thought EVERY PARTICLE (massive or massless) travels in spacetieme with 4-speed equal to that of light!
FOURTH QUESTION:
I've read in many places:
"Well, obviously we get from (4) that:
(5) [tex]ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=0[/tex]
Sorry, but that's not obvious to me. Could someone derive it for me? Please? I know that (5) would imply that the massless particle travels at the speed of light c, even if LHS of (4) is equal to zero, so I THINK that would solve question 4. But I just don't know how to get (5) from (4).
Well, that's all, folks. I hope you help me. Thank you so much since now :)
I've got some questions for you!
Denoting by
(1) [tex]ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=c^{2}d\tau^{2}[/tex]
the interval (and [tex]\tau[/tex] the proper time) and using the signature (+---), we have that the equations of motion for a free particle are:
(2) [tex]\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma^{\mu}_{\nu\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\lambda}}{d\tau}=0[/tex]
; the so called "geodesic (timelike) equations", (those capital Gamma are the Christoffel symbols).
Obviously we have the relation:
(3) [tex]g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=c^{2}[/tex]
'Cuz the LHS of (3) gives us the square of the magnitude of the 4-velocity, and that's always PLUS c2
(FIRST QUESTION: using the -+++ signature we get that LHS of (3) is equal to -c2, don't we?)
Everything is fine. But then I get in trouble. Using an action based on an auxiliary field I found in some place, I get that for a MASSLESS particle, the equations of motion are:
(4) [tex]g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=0[/tex]
; which people call the "geodesic (lightlike) equations".
SECOND (set of) QUESTION(s): So, we have TWO geodesic equations? (2) for massive particles and (4) for massless particles? If that's so, why does (2) doesn't involve mass m??
THIRD (set of) QUESTION(s):
And why is LHS of (4) equal to zero? isn't that the square of the magnitude of the 4-velocity for a massless particle? shouldn't it be c2?? I thought EVERY PARTICLE (massive or massless) travels in spacetieme with 4-speed equal to that of light!
FOURTH QUESTION:
I've read in many places:
"Well, obviously we get from (4) that:
(5) [tex]ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=0[/tex]
Sorry, but that's not obvious to me. Could someone derive it for me? Please? I know that (5) would imply that the massless particle travels at the speed of light c, even if LHS of (4) is equal to zero, so I THINK that would solve question 4. But I just don't know how to get (5) from (4).
Well, that's all, folks. I hope you help me. Thank you so much since now :)