Klein-Gordon Field in Two-Dimensional Rindler Space-Time
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]
ππ2
= β πππ‘2
+ π ππ2
+ π2 ( ππ2
+ π ππ2
π ππ2 )
π = πβ1
= 1 β
2πΊπ π
π
(1)
as expressed in the standard coordinates has a (coordinate) singularity[2] at ππ» = 2πΊππ. (Note that in this
entire document we take the square of the speed of light as unity, π2
= 1 and whenever consistency in
units needed, we insert its appropriate value.) This can be seen crudely from the fact that π = β at π =
ππ». This coordinate value of π 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 π > ππ» and dipping below ππ» 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.
ππ‘
ππ
= Β±
1
1 β
2πΊπ π
π
(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 π > ππ» is not causally connected to
that at π < ππ» . This is so since asymptotically the light-cone closes as ππ» is approached from the right. As
the light-cone closes there can be no way of connecting a time-like particleβs past to its supposed future
along a time-like path that is enclosed by the light-cone. So any coordinate observer wonβt be able to
construct a causal connection between the past and the future for a time-like particle falling into that region
π < ππ» .
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 (π‘, π, π, π) we can change (1) into
ππ2 = β πππ’Μ2 + 2ππ’Μππ + π2 ( ππ2 + π ππ2 π ππ2 ) (3)
using the Eddington-Finkelstein coordinate
π’Μ = π‘ + π β (4)
with the Regge-Wheeler coordinate
3. 3 The solution
The differential equation (12.1) is satisfied by a series solution of the following form
π(π) =
1
β π₯
β
1
π₯ π
( π π exp( ππ₯) + π π exp(βππ₯) )
π
π=0
(13)
This series solution corresponds to an integral number π and the series stops at the ππ‘β term. (As
a cautionary let us not confuse π with π. The latter is the mass of our scalar field in units of per length.)
The (π + 1)π‘β term and all other higher terms vanish as the π π+1 and π π+1 coefficients are terminated.
That is, π π+1 = π π+1 = 0. Each coefficient π π is given by this recursion formula
π π =
(2π β 1)2 β (2π β1 π πΈ)2
8ππ
π πβ1
(14.1)
and each π π by
π π = β
(2π β 1)2
β (2π β1
π πΈ)2
8ππ
π πβ1
(14.2)
These formulas are defined for all π β₯ 1 and with these the vanishing of those π + 1 coefficients would
imply that
π πΈ(π) = (π +
1
2
)π (14.3)
π = 0, 1, 2,3, β¦ , πππ‘ππππ (14.4)
π = (4πΊππ)
β1
(14.5)
(There in (14.3) we have relabeled π as π and this includes zero as one of its parameter values.
With the inclusion of zero, the lowest vanishing (π + 1) coefficients given π = 0 would be π1 = 0 and
π1 = 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 π0 and π0 that are non-zero.)
Given (14.3), we can write the differential equation (12.1) as
π1
2 π(π) +
1
π₯
π1 π(π) β
1
4π₯2
(2π + 1)2 π(π) = π2 π(π)
(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 π
π( π) = π΄πππ π(π) π + π΅π πππ(π) π
(14.7)
Here we have identified the separation constant π πΈ as the angular frequency π in the solution above. That
is, π πΈ(π) = π(π), and given (14.3), we find that this oscillatory solution has an angular frequency that
corresponds to an integral number π.
We can choose to set π0 = π0 so that π π = (β1) π π π and at π = 0 , implying π1 = 0 and π1 =
0, we have
π(0) =
2π0 πππ βππ₯
β π₯
(15.1)
with π(0) = π /2 .
Going back to the recursion relations (14.1) and (14.2), we must take note that these fail when the scalar
field is massless since these are singular at π = 0. In the massless case we may consider π0 = π0, so as a
consequence, π π = (β1) π
π π and
π(π) =
2π0
β π₯
+
2
β π₯
β
π π
π₯ π
π
π =2
(15.2)
So in the massless case we apply (14.6) with π = 0 to (15.2) to get a form of constraint on the coefficients
π π and this is given by
β [2 ( π +
1
2
)
2
β 2 ( π +
1
2
)
2
]
π
π =0
π π π₯βπβ5/2
= 0
(15.3)
4. From this constraint we can choose only one coefficient ππ to be non-zero, while all other coefficients
π π β π to be zero. Example, if we choose π0 to be the only non-zero, then π = 0, and all other coefficients
π π β 0 to be zero. For every non-zero coefficient ππ, there corresponds an angular frequency in the form of
(14.3). Eventually then, following this condition in the constraint, series (15.2) would only be made up of a
term with the non-zero coefficient,
π(π) =
2π π
β π₯
1
π₯ π
(15.4)
This satisfies the differential equation in the same form (14.6) for a given integral value of π with π = 0.
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 π πΈ(π) = π(π), given (9.2) seems to
have values that correspond to integral values of π. 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 π. We also have a curious result as manifest in (14.3) that in
the imaginary time, the scalar field can oscillate at frequencies that are odd multiples of the surface gravity
over four pi, π(π) = (2π + 1) π /4π, where surface gravity is π = π3
/4πΊπ π.
Some details of spatial solution
Two-dimensional scalar action defined in the background of two-dimensional Rindler space-time
We have arbitrarily defined a two-dimensional scalar action (10), given the fundamental line element (7) of
the two-dimensional Rindler space-time. This action we explicitly write as
π πΆ = β« ππ‘ β« ππ₯
1
2
(β
1
π π₯
( ππ‘ ππ)2
+ π π₯ ( π π₯ ππ)2
+ π2
π π₯ π πΆ
2 )
(16)
As earlier stated this Rindler space-time is endowed with a set of coordinates (9.1). Thus, the classical
scalar can have two degrees of freedom and its motion is along π₯0
= π‘ (the time direction) and one spatial
direction π₯1
= π₯.
We will no longer (or perhaps in the much later portion of this draft) present here the details of varying this
action in terms of the variation of the classical scalar to arrive at the equation of motion (11.1) or as
explicitly given by (11.2).
The spatial series solution
In tackling the differential equation (11.2), we assumed a variable separable solution in the form of ππ =
π( π₯) π(π‘) so decomposing (11.2) into equations (12.1) and (12.2). Most of our effort here is to work on
the details involved in solving (12.1).
Given the variable separable solution, we were able to write the spatial part of (11.2) as (12.1) and re-
writing that here as
π₯2
π1
2
π + π₯ π1 π β ( π πΈ π β1)2
π = π2
π₯2
π
(17.1)
In considering the solution in series form as given by (13), we have actually decomposed π into two
separate components
π(π) = π1 + π2 = π1(π) + π2(π)
(17.2)
(Again, we must note that π here is an integral number and must not be confused with mass π of the
scalar field.) We plug (17.2) into (17.1) and collect like terms we then write two separate differential
equations from (17.1).
π₯2
π1
2
π1 + π₯ π1 π1 β ( π πΈ π β1)2
π1 = π2
π₯2
π1
π1 = π1(π)
(17.3)
5. where
π1 = π1(π) = π1( π)(π₯) π ππ₯ =
1
β π₯
β
1
π₯ π
π π exp( ππ₯)
π
π=0
(17.4)
while the other part is given by
π₯2
π1
2
π2 + π₯ π1 π2 β ( π πΈ π β1)2
π2 = π2
π₯2
π2
π2 = π2(π)
(17.5)
to which belongs the other solution
π2 = π2(π) = π2( π) (π₯) πβ ππ₯
=
1
β π₯
β
1
π₯ π π πexp(βππ₯)
π
π=0
(17.6)
We proceed from (17.3), given (17.4) to get the following differential equation for π1( π).
π₯2
π β²β²1( π) + 2ππ₯2
π β²1( π) + π₯π β²1( π) + ππ₯π1( π) β ( π πΈ π β1)2
π1( π) = 0
π πΈπΉ = π πΈ π β1
(17.7)
The form of π1( π) is already evident in (17.4) and substituting this solution in (17.3) would give us the
recurrence relations between the coefficients π π.
Proceeding, we write
β [( π +
1
2
)
2
β π πΈπΉ
2] π π π₯β(π+ 1/2)
π
π=0
β β 2ππ
π
π = 0
π π π₯β( πβ
1
2
)
= 0
(17.8)
In the first major summation, we make the shift π β (π β 1) so that (17.8) can be re-written into the
following form
β [(( π β
1
2
)
2
β π πΈπΉ
2) π πβ1 β 2πππ π ] π₯β(π β 1/2)
π
π=1
= 0
(17.9)
For this to be satisfied for all coefficients we must have the recurrence relations between adjacent
successive coefficients and such relations are already given by (14.1). The series solution (17.4) can be
terminated so that it will consist only of series of terms up to the mth place. This can be done by setting the
π π = π +1 coefficients to zero.
π π = π +1 =
(2π + 1)2
β (2π β1
π πΈ)2
8π(π + 1)
ππ = 0
(17.10)
As earlier stated in the conclusion the consequence of having a terminated series solution is that the
frequencies are odd multiples of the surface gravity over four pi
π(π) = (2π + 1)
π
4π
(17.11)
It should already be clear from (12.2) that π πΈ is the angular frequency π(π) in the imaginary time.
Given the terminated coefficient (17.10), we can slightly re-write (17.7) as
π₯2
π β²β²1( π) + 2ππ₯2
π β²1( π) + π₯π β²1( π) + ππ₯π1( π) =
(2π + 1)2
4
π1( π)
(17.12)
with
6. π1( π)(π₯) =
1
β π₯
β
π π
π₯ π
π
π=0
(17.13)
Test for π = 0:
π1(0)(π₯) =
π 0
β π₯
(18.1)
π₯π β²1(0) = β
1
2
π 0
β π₯
π₯2
π β²β²1(0) =
3
4
π 0
β π₯
(18.2.1)
π₯2 π β²1(0) = β
1
2
π 0β π₯ π₯π1(0) = π 0β π₯ (18.2.2)
It is relatively easy to see that (18.1) and the results in (18.2.1) and (18.2.2) satisfy the their given
differential equation
π₯2
π β²β²1(0) + 2ππ₯2
π β²1(0) + π₯π β²1(0) + ππ₯π1(0) =
1
4
π1(0)
(18.3)
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 , 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