The klein gordon field in two-dimensional rindler space-time - smcprt
1. The Klein-Gordon field in two-dimensional Rindler space-time
Ferdinand Joseph P. Roa
Independent physics researcher
rogueliknayan@yahoo.com
Abstract
The Klein-Gordon scalar in the background of two-dimensional Rindler space-time is considered in this exercise. In an
informal way without resorting to methods of dimensional reduction, a two-dimensional action for the Klein-Gordon
scalar is written with the said background and obtaining from this action the equation of motion for the scalar field. The
equation of motion is solvable exactly in this two-dimensional space-time using imaginary time. In imaginary time, the
solution is oscillatory with a given frequency that corresponds to an integral number.
Keywords: Coordinate Singularity, Series Solution
1 Introduction
The Schwarzschild metric[1]
๐๐2
= โ ๐๐๐ก2
+ ๐ ๐๐2
๐2 ( ๐๐2
+ ๐ ๐๐2
๐ ๐๐2 )
๐ = ๐โ1
= 1 โ
2๐บ๐ ๐
๐
(1)
as expressed in the standard coordinates has a (coordinate) singularity[2] at ๐๐ป = 2๐บ๐๐. This can be seen
crudely from the fact that ๐ = โ at ๐ = ๐๐ป. This coordinate value of ๐ at which one piece of the metric is
singular defines a horizon[2, 3] that puts bounds to (1), confining it in a portion of space-time where this
metric in that form is sensible. That is, in rough language say (1) is for all those regions of space-time
where ๐ > ๐๐ป and dipping below ๐๐ป can no longer be covered by the given metric as expressed in that
form.
Figure 1: This is the space-time graph on rt-plane.
๐๐ก
๐๐
= ยฑ
1
1 โ
2๐บ๐ ๐
๐
(2)
On a space-time graph where one can draw a light-cone bounded by the intersecting lines whose
slopes are given by (2), it can be superficially shown that the region at ๐ > ๐๐ป is not causally connected to
that at ๐ < ๐๐ป . This is so since asymptotically the light-cone closes as ๐๐ป is approached from the right. As
the light-cone closes there can be no way of connecting a time-like particleโs past to its supposed future
along a time-like path that is enclosed by the light-cone. So any coordinate observer wonโt be able to
construct a causal connection between the past and the future for a time-like particle falling into that region
๐ < ๐๐ป .
However, such singularity is only a coordinate one specific to the form (1) since expressing the
same metric in suitable coordinates will remove the said coordinate singularity.
For example, from the standard coordinates (๐ก, ๐, ๐, ๐) we can change (1) into
๐๐2 = โ ๐๐๐ขฬ2 + 2๐๐ขฬ๐๐ + ๐2 ( ๐๐2 + ๐ ๐๐2 ๐ ๐๐2 ) (3)
using the Eddington-Finkelstein coordinate
๐ขฬ = ๐ก + ๐ โ (4)
with the Regge-Wheeler coordinate
2. ๐ โ = ๐ + 2๐บ๐ ๐ ๐๐(
๐
2๐บ๐ ๐
โ 1)
(5)
Noticeable in (3) is that none of the metric components goes infinite at ๐๐ป so the singularity ( ๐ =
โ at ๐ = ๐๐ป) in (1) is not a case in (3).
In this paper, we deal with the Klein- Gordon scalar as dipped very near the horizon but not
having completely fallen into those regions at ๐ < ๐ ๐ป. We consider that near the horizon we can make the
substitution[2]
๐ฅ2
8๐บ๐ ๐
= ๐ โ 2๐บ๐ ๐
(6)
Hence, in approximate form we write (1) as
๐๐2 โ โ( ๐ ๐ฅ)2 ๐๐ก2 + ๐๐ฅ2 + ๐๐
2 ๐ฮฉ2 (7)
๐ =
1
4๐บ๐ ๐
where ๐๐ = 1/2 ๐ is the approximate radius of a two-sphere ๐2
: ๐ ๐
2 ๐ฮฉ2
and we think of the (3+1)-
dimensional space-time ascribed to metric (7) as a product of a two-dimensional Rindler space-time
๐๐(๐ )
2
โ โ( ๐ ๐ฅ)2 ๐๐ก2 + ๐๐ฅ2 (8)
and that of the two-sphere. We give to this two dimensional Rindler space-time the set of coordinates
๐ฅ ๐
= {๐ฅ0
= ๐ก, ๐ฅ1
= ๐ฅ} (9.1)
with t as the real time to be transcribed into an imaginary time by
๐ก โ ๐ = โ๐๐ก (9.2)
2 Two-dimensional action
In this Rindler space-time we just write a two dimensional action for our scalar field
๐ ๐ถ = โซ ๐๐ฅ2
โโ๐ (1
2
๐ ๐๐
(๐ ๐ ๐๐)( ๐ ๐ ๐๐) +
1
2
๐2
๐ ๐ถ
2) (10)
The metric components in this action are those belonging to the two-dimensional Rindler space-time given
by metric form (8). This is rather an informal way without having to derive it from an original 3 + 1
dimensional version that would result into (10) through the process of a dimensional reduction[4] with the
Rindler space-time as background. Anyway, on the way the solution exists for the resulting equation of
motion to be obtained from the given action and this solution is oscillatory as taken in the imaginary time.
Taking the variation of (10) in terms of the variation of our classical scalar ๐๐๏ would yield the
equation of motion
1
โโ๐
๐๐(โโ๐ ๐ ๐๐
๐ ๐ ๐๐) โ ๐2
๐๐ = 0
(11.1)
or with (8) as the said background we have explicitly
๐1
2
๐๐ +
1
๐ฅ
๐1 ๐๐ โ
1
๐ 2 ๐ฅ2
๐0
2
๐๐ = ๐2
๐๐
(11.2)
We take that the solution is variable separable, ๐๐ = ๐( ๐ฅ) ๐(๐ก)๏ so that (11.2) could be written into
two independent equations
1
๐
( ๐1
2
๐ +
1
๐ฅ
๐1 ๐) โ
1
๐ 2 ๐ฅ2
๐ ๐ธ
2
= ๐2
(12.1)
and
1
๐
๐0
2
๐ = โ
1
๐
๐๐
2
๐ = ๐ ๐ธ
2
๐ ๐ธ
2
> 0 (12.2)
Later, the constant ๐ ๐ธ is to be identified as the angular frequency ๐ in the imaginary time (9.2).
3. 3 The solution
The differential equation (12.1) is satisfied by a series solution of the following form
๐(๐) =
1
โ ๐ฅ
โ
1
๐ฅ ๐
( ๐ ๐ exp( ๐๐ฅ) + ๐ ๐ exp(โ๐๐ฅ) )
๐
๐=0
(13)
This series solution corresponds to an integral number ๐ and the series stops at the ๐๐กโ term. The
(๐ + 1)๐กโ term and all other higher terms vanish as the ๐ ๐+1 and ๐ ๐+1 coefficients are terminated. That
is, ๐ ๐+1 = ๐ ๐+1 = 0. Each coefficient ๐ ๐ is given by this recursion formula
๐ ๐ =
(2๐ โ 1)2
โ (2๐ โ1
๐ ๐ธ)2
8๐๐
๐ ๐โ1
(14.1)
and each ๐ ๐ by
๐ ๐ = โ
(2๐ โ 1)2
โ (2๐ โ1
๐ ๐ธ)2
8๐๐
๐ ๐โ1
(14.2)
These formulas are defined for all ๐ โฅ 1 and with these the vanishing of those ๐ + 1 coefficients would
imply that
๐ ๐ธ(๐ ) = (๐ +
1
2
)๐ (14.3)
๐ = 0, 1, 2, 3, โฆ , ๐๐๐ก๐๐๐๐ (14.4)
๐ = (4๐บ๐๐)
โ1
(14.5)
(There in (14.3) we have relabeled ๐ as ๐ and this includes zero as one of its parameter values.
With the inclusion of zero, the lowest vanishing (๐ + 1) coefficients given ๐ = 0 would be ๐1 = 0 and
๐1 = 0 so that all other higher terms with their corresponding coefficients vanish. Then in this particular
case, the series solution only has the 0th terms with their coefficients ๐0 and ๐0 that are non-zero.)
Given (14.3), we can write the differential equation (12.1) as
๐1
2 ๐(๐) +
1
๐ฅ
๐1 ๐(๐) โ
1
4๐ฅ2
(2๐ + 1)2 ๐(๐) = ๐2 ๐(๐)
(14.6)
whose solution is in the series form (13).
Put simply, the differential equation (12.2) has as solution the following function of the imaginary
time ๐
๐( ๐) = ๐ด๐๐๐ ๐(๐) ๐ + ๐ต๐ ๐๐๐(๐) ๐
(14.7)
Here we have identified the separation constant ๐ ๐ธ as the angular frequency ๐ in the solution above. That
is, ๐ ๐ธ(๐ ) = ๐(๐), and given (14.3), we find that this oscillatory solution has an angular frequency that
corresponds to an integral number ๐.
We can choose to set ๐0 = ๐0 so that ๐ ๐ = (โ1) ๐
๐ ๐ and at ๐ = 0 , implying ๐1 = 0 ๏ and ๐1 =
0, we have
๐(0) =
2๐0 ๐๐๐ โ๐๐ฅ
โ ๐ฅ
(15.1)
with ๐(0) = ๐ /2 .
4 Conclusions
Taking the Klein-Gordon field as a classical scalar, we have shown that its two-dimensional equation of
motion in Rindler space-time has a series solution that can terminate at a certain term. As a consequence of
this termination the angular frequency (14.3) with the identification ๐ ๐ธ(๐ ) = ๐(๐), given (9.2) seems to
have values that correspond to integral values of ๐. This is seemingly suggestive that the classical scalar
field can already appear quantized in terms of its angular frequency or can have a spatial mode given by
(13) that corresponds to an integral value of ๐.
4. References
[1] J. Foster, J. D. Nightingale, A SHORT COURSE IN GENERAL RELATIVITY, 2nd
edition copyright
1995, Springer-Verlag, New York, Inc.,
[2] P. K.Townsend, Blackholes โ Lecture Notes, http://xxx.lanl.gov/abs/gr-qc/9707012
[3] S. M. Carroll, Lecture Notes On General Relativity, arXiv:gr-qc/9712019
[4] Kaluza-Klein Theory, http://faculty.physics.tamu.edu/pope/ihplec.pdf
[5] R. MacKenzie, Path Integral Methods and Applications, arXiv:quant-ph/0004090v1