Fermi-Walker: Showing Rotation in Plane of 4-Accel & 4-Vel

[kent davidge]

Is it difficult to show that a Fermi-Walker "rotation" happens only in the plane formed by a particle four-acceleration and four-velocity?

 

[PeterDonis]

Is it difficult to show that a Fermi-Walker "rotation" happens only in the plane formed by a particle four-acceleration and four-velocity?

Since it follows directly from the definition of Fermi-Walker transport, I would say no.

 

[kent davidge]

Since it follows directly from the definition of Fermi-Walker transport, I would say no.

I can't see directly from the definition. So I am trying to prove/show it. I guess we need to show that a vector in the plane formed by the four-acceleration and four-velocity, when rotated, still lies in the same plane. Correct?

 

[PeterDonis]

I can't see directly from the definition.

The definition is $D_F X = 0$, where $D_F$ is the Fermi derivative and $X$ is a vector field. (I'm leaving out indexes since just looking schematically at the definition is enough.) The Fermi derivative along a worldline with tangent vector $U$ and proper acceleration $A = \nabla_U U$ is

$$
D_F X = \nabla_U X - \left( X \cdot A \right) U + \left( X \cdot U \right) A
$$

If we set $D_F X = 0$, we get

$$
\nabla_U X = \left( X \cdot A \right) U - \left( X \cdot U \right) A
$$

which just says that the covariant derivative of $X$ along $U$, which is what you are calling "Fermi-Walker rotation", lies in the plane spanned by $U$ and $A$.

 

[PeterDonis]

I guess we need to show that a vector in the plane formed by the four-acceleration and four-velocity, when rotated, still lies in the same plane. Correct?

No. You can Fermi-Walker transport any vector you like, even if it doesn't lie in the plane spanned by $U$ and $A$. But the covariant derivative of that vector, if it's Fermi-Walker transported, will lie in the plane spanned by $U$ and $A$; i.e., that's the plane it will "rotate" in.

 

[vanhees71]

You can find a derivation of Fermi-Walker transport in terms of old-fashioned Ricci calculus here:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[kent davidge]

You can find a derivation of Fermi-Walker transport in terms of old-fashioned Ricci calculus here:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

I find it hard to show that an infinitesimal Lorentz boost in the $u-a$ plane gives as a result your equation 1.8.5.

 

[vanhees71]

This is discussed in the previous section (following Eq. (1.7.7)).