Question

In paperPhysRevD.24.297, the authors follow the treatment on the Rindler horizon (0803.2001) to find a mode expansion on the Schwarzschild horizon. But The paper find a coordinate(t,ξ,y,ϕ)(t,ξ,y,ϕ)to decompose the metric near the horizon region asRindler×S2Rindler×S2. But to treat the quantum effect measured by a physical observer at a point of null-infinity, a more reasonable choice is(u,v,xA)(u,v,xA).They perform a mode expansion of the Maxwell field as eqs (4.34),(A.5). That is very unlike the result in flat spacetime QFTAμ∼ϵαμeipxaα(p).Aμ∼ϵμαeipxaα(p).Or, say, the absence of polarization vector in the mode expansion. Thus my problem is Can we find a mode expansion of the gauge field likeAμ∼ϵαμV(ω,p)aα(p)Aμ∼ϵμαV(ω,p)aα(p)under the Schwarzschild background by using(u,v,xA)(u,v,xA)frame?

Options:
A) Yes, a mode expansion of the gauge field Aμ ∼ ϵαμV(ω,p)aα(p) can be obtained using the (u,v,xA) frame.
B) No, the (u,v,xA) frame does not allow for a mode expansion of the gauge field Aμ in the Schwarzschild background.
C) A different frame, distinct from (u,v,xA), is better suited for deriving a mode expansion of the gauge field Aμ in the Schwarzschild background.
D) There is insufficient information to determine whether a mode expansion of the gauge field Aμ can be achieved using the (u,v,xA) frame in the Schwarzschild background.

Answer 1

The (u,v,xA) frame does not allow for a mode expansion of the gauge field Aμ in the Schwarzschild background.

Step-by-step explanation:

The authors of the paperPhysRevD.24.297 use the (u,v,xA) frame to perform a mode expansion of the Maxwell field in the Schwarzschild background. This mode expansion is represented by equations (4.34) and (A.5) in the paper. The mode expansion in the Schwarzschild background is different from the mode expansion in flat spacetime QFTAμ ∼ ϵαμeipxaα(p).Aμ ∼ ϵμαeipxaα(p) because it includes the term V(ω,p)aα(p) instead of a polarization vector ϵαμ.

Therefore, the answer to your question is B) No, the (u,v,xA) frame does not allow for a mode expansion of the gauge field Aμ in the Schwarzschild background.