11institutetext: School of Physics, University of Melbourne, Victoria,
Australia 22institutetext: ARC Centre of Excellence for Dark Matter Particle
Physics

# Influence of NaI background and mass on testing the DAMA modulation

Madeleine J. Zurowski<EMAIL_ADDRESS>Elisabetta
Barberio 1122

(Received: date / Revised version: date)

###### Abstract

We present here the model dependent and independent sensitivity studies for
NaI detectors designed to test the DAMA result, and compare the predicted
limits from SABRE with the present performance of both ANAIS and COSINE. We
find that the strongest discovery and exclusion limits are set by a detector
with the lowest background (assuming equal run times), and also note that our
method correctly computes the present exclusion limits previously published by
ANAIS and COSINE. In particular, with a target mass of 50 kg and background
rate of 0.36 cpd/kg/keV (after veto), SABRE will be able to exclude the DAMA
signal with 3$\sigma$ confidence or ‘discover’ it with 5$\sigma$ confidence
within 2 years. This strongly motivates the quest for ever lower backgrounds
in NaI detectors.

###### pacs:

PACS-keydescribing text of that key

##  1 Introduction

There is a plethora of evidence that suggests some long-lived, non-luminous,
particulate matter within the Universe. These cosmological observations exist
on a wide range of scales, and there is no member of the Standard Model of
Particle Physics (SM) able to successfully explain them all. As such, a new
component called Dark Matter (DM) is required. Of the possible additions to
the SM, the Weakly Interacting Massive Particle (WIMP) is one that has enjoyed
popularity since its conception. This is due to the fact that particles on the
weak scale have a self annihilation rate that naturally gives rise to the
observed present day abundance of DM. Despite the fact that the standard
assumptions for a WIMP (mass $\sim$ GeV/c2-TeV/c2, cross section $\sim
10^{-40}$ cm2) are now very heavily constrained experimentally, there are
still a large number of experiments conducting searches for them.
WIMPs are typically probed for in one of three ways; indirect detection
(observation of DM annihilation into SM particles), collider searches
(observation of DM produced via SM annihilation), or direct detection
(scattering of DM off some SM target). In particular, DM direct detection has
a distinctive ‘smoking gun’ - a modulating interaction rate due to the
relative velocity of the Earth and DM halo as the former rotates about the
Sun. This modulation should be observed regardless of the particle interaction
model considered as it depends only on the DM velocity distribution. Such a
modulation should have a period of a year, and an amplitude of around 2% the
constant rate.
One experiment has consistently observed such a signal over 14 annual cycles.
Using a NaI target, the DAMA Collaboration has reported evidence of a DM
consistent modulation with a combined statistical significance of 12.9$\sigma$
after around 15 years of data taking Bernabei2018 . Despite this seemingly
strong evidence, these results are in increasing tension with other direct
detection experiments, which continue to publish null results for DM models
that fit the DAMA data Tanabashi:2018oca . This has strongly constrained the
standard WIMP assumptions, and pushed to more and more particular tailored
models to explain these null results consistently.
The experiments that constrain the DAMA result depend largely on detectors
that use a different scattering target, and as such, require the a priori
assumption of some particular interaction model for analysis. In order to
conclusively dismiss the DAMA signal as having a DM origin, it must be
assessed in a model independent manner; that is, through the use of another
NaI target experiment attempting to observe this annual modulation. At
present, three such detectors are planned or already in the data taking stage:
ANAIS, COSINE, and SABRE amare2021 ; adhikari2021 ; Antonello_2019 . As these
all use the same target as DAMA, under the assumption that all three
experiments have a threshold efficiency and detector resolution similar to
that of DAMA, they are expected to be able to observe the same modulation
signal, given a low enough background and long enough data taking period.
Thus, model independent sensitivity limits can be constructed in the
background-lifetime plane similar to how they are in the
$m_{\chi}$-$\sigma_{0}$ plane in the model dependent case. This allows for a
comparison of the performance of the three NaI detectors given their different
background and exposure masses.
This paper commences by detailing the method used to carry out both model
dependent and independent sensitivity calculations in Section 2. It then
presents the results of both sensitivity cases, and compares the exclusion and
discovery powers of ANAIS, COSINE, and SABRE in Section 3, before concluding
in Section 4.

##  2 Sensitivity methodology

In general, an experiment will be sensitive to a DM modulating signal if the
signature modulation is observable over the detectors background. In this
case, the apparent modulation of the background signal due to statistical
fluctuations is more problematic than the background rate itself, as analysis
of the modulation signature typically involves the subtraction of the constant
signal component. As such, to assess the sensitivity of a detector, these
fluctuations should be modelled over its lifetime, and fit to a cosine
function to find the probability of this masking or mimicking the DM signal.
Assuming that in some energy bin of width $\Delta E$ over a time period of
$\Delta T$, the observed background rate is given by $R_{b}$ (in units of
cpd/kg/keV), these background statistical fluctuations can be modelled by
sampling from a Poisson distribution centred on

$N_{b}=M_{E}\times\Delta T\times\Delta E\times R_{b},$ (1)

where $M_{E}$ is the mass of the detector in question. The signal plus
background (s + b) modulation can be modelled in a similar way, sampling
instead from

$N_{s+b}=M_{E}\times\Delta T\times\Delta
E\times\left(R_{b}+R_{0}+R_{m}\cos{\omega t}\right),$ (2)

where $R_{0}$ and $R_{m}$ are the constant and modulating signal events
observed in a given energy bin. Generating this for every live bin period of
detector operation, the total observed rate is then fit to
$R(t)=R_{c}+R_{f}\cos{\omega t}$ in order to separate the constant ($R_{c}$)
and fluctuating ($R_{f}$) components. A non-zero modulation can be seen in
both the background only and signal plus background cases - see, for example,
Figs. 1 and 2, where a 50 kg detector running for 3 years observes a signal
compatible with $R_{f}=23.3\pm 18.8$ counts per month for background only, and
$R_{f}=83.8\pm 24.5$ counts per month for signal plus background. Note also
that in both cases, the observed signal is incompatible with zero modulation.

Figure 1: Simulated fluctuations in background only model, with $R_{b}=1$
cpd/kg/keV over 3 years fit to $R(t)=R_{c}+R_{f}\cos{\omega t}$. Figure 2:
Simulated fluctuations in signal+background model, with $R_{b}=1$ cpd/kg/keV
and DAMA signal rate over 3 years fit to $R(t)=R_{c}+R_{f}\cos{\omega t}$.

This modulation is then generated a number of times ($\mathcal{O}(100-1000)$),
saved, and fit to a gaussian corresponding to the probability of observing a
given modulation amplitude for either the background only or signal+background
case. These distributions are then compared to assess the ability of the
detector to distinguish between the two models. Samples of these can be seen
in Fig. 3, where $R_{0}$ and $R_{m}$ are DAMA observations from Ref.
Bernabei2018 .

(a)

(b)

Figure 3: Background only (top) and signal+background (bottom) distributions,
for $R_{b}=1$ cpd/kg/keV over 3 years, and signal given by DAMA. These are
plotted as a function of the number of modulation events observed per month.

Thus, the tuneable parameters to assess the sensitivity of a detector are the
run time, detector background, $R_{m}$, and $R_{0}$. Typical model dependent
analyses will set $R_{b}$ and the run time based on the detector in question,
and vary $R_{m}$ and $R_{0}$ by setting different values of $m_{\chi}$ and
$\sigma_{0}$. However, it is also possible to set $R_{m}$ and $R_{0}$, (to the
values observed by DAMA, for example) and vary the detector parameters to find
the required lifetime and background for a detector to observe this particular
signal, rather than a specific DM model. The test statistic we will use to
assess detector sensitivity is the $\sigma$ confidence level, where here
$\sigma$ is the standard deviation of the signal+background gaussian
Cowan_2011 . In the event of a null results, the detector will exclude some
model/combination of $R_{m}$ and $R_{0}$ with $n\sigma$ confidence, where

$n=\frac{|\mu_{sb}-\mu_{b}|}{\sigma_{sb}}.$ (3)

Here, $\mu_{b}$ ($\mu_{sb}$) corresponds to the mean modulation observed by
the background (signal+background) model. Where we want to plot the 90% CL in
the $m_{\chi}-\sigma_{0}$ plane, as is standard for model dependent
sensitivity plots, we compute the one sided p-value associated with $n$,

$p=\frac{1}{2}\left(1-\text{erf}\left(\frac{n}{\sqrt{2}}\right)\right),$ (4)

and require that $p\leq 0.1$.
If instead we are interested in the discovery power rather than the exclusion
(the probability that observation of $\mu_{sb}$ can occur in data generated by
the background only model), the $\sigma$ confidence level will depend on the
uncertainty of the background only distribution, and be given by

$n=\frac{|\mu_{sb}-\mu_{b}|}{\sigma_{b}}.$ (5)

###  2.1 Assumptions

For both the model dependent and independent sensitivity calculations, we make
a number of assumptions for ease of computation and comparison across
different detectors. They are compiled in this section for reference. In all
cases, we consider three detectors, with the exposure mass and backgrounds
reported by ANAIS and COSINE, and those planned for SABRE, given in Table 1
with those reported by DAMA for comparison.

Table 1: Detector properties. Experiment | Mass (kg) |  $R_{b}$ (dru)
---|---|---
ANAIS | 112 | 3.2 ANAIS2018
COSINE | 57.5 | 2.7 Adhikari_2019
DAMA | 250 | 0.8 BERNABEI2020
SABRE | 50 | 0.36 SABRE_MC

####  2.1.1 Model dependent assumptions

The expression we use for the DM interaction rate as a function of nuclear
recoil is

$\frac{dR}{dE_{R}}=N_{T}\frac{\rho}{m_{\chi}}\int vf_{\rm
lab}(\vec{v})\frac{d\sigma_{T}}{dE_{R}}d^{3}v,$ (6)

where the dependence on model choice comes from the selection of normalised
couplings constants $\hat{c}_{i}^{(a)}\hat{c}_{j}^{(b)}$ in

$\begin{split}\frac{d\sigma_{T}}{dE_{R}}&=\frac{m_{T}}{2v^{2}}\frac{\sigma_{0}}{\mu^{2}_{N}}\left[\sum_{i,j}\sum_{a,b=0,1}\hat{c}_{i}^{(a)}\hat{c}_{j}^{(b)}F^{(ab)}_{ij}(v,q)\right].\\\
\end{split}$ (7)

For the simple spin independent model we use here, we take all couplings as
non-zero, save for $c_{1}^{(0)}=1$, while the form factors used are taken from
the appendix of Ref.Anand2013 . Under these assumptions, the differential
cross section simplifies to

$\begin{split}\frac{d\sigma_{T}}{dE_{R}}&=\frac{m_{T}}{2v^{2}}\frac{\sigma_{0}}{\mu^{2}_{N}}F^{(00)}_{11}(q)\\\
&=\frac{m_{T}}{8v^{2}}\frac{\sigma_{0}}{\mu^{2}_{N}}\left(F^{(p,p)}_{11}(q)+F^{(n,n)}_{11}(q)+2F^{(n,p)}_{11}(q)\right).\end{split}$
(8)

The expressions for these nuclear form factors for 23Na and 127I are given in
Appendix A.
We also assume the standard halo model for the velocity distribution,

$f_{lab}(v)=\frac{1}{(\pi
v_{0}^{2})^{3/2}}\exp\left[-\frac{1}{v_{0}^{2}}\left(\boldsymbol{v}-\boldsymbol{v}_{E}\right)^{2}\right],$
(9)

where $v_{0}=220$ km s-1, $v_{\rm esc}=550$ km s-1, and $\rho=0.3$ GeV cm-3.
The Earth’s velocity $\boldsymbol{v}_{E}$ is given by
$\boldsymbol{v}_{E}=\boldsymbol{v}_{\odot}+\boldsymbol{v}_{t}$, where

$\begin{split}\vec{v}_{\odot}&=v_{\odot}(0,0,1),\\\ \vec{v}_{t}&=v_{t}(\sin
2\pi t,\sin\gamma\cos 2\pi t,\cos\gamma\cos 2\pi t),\\\ v_{\odot}&=232\text{
kms}^{-1},\\\ v_{t}&=30\text{ kms}^{-1},\\\ \gamma&=\pi/3\text{
rad}.\end{split}$ (10)

For the detector specific response functions, we assume that all three
detectors of interest have the same quenching factors as DAMA ($Q_{Na}=0.3$
and $Q_{I}=0.09$), as well as the energy resolution given in Ref.Bernabei2008
:

$\sigma_{E}=\left(0.0091\frac{E_{ee}}{{\rm\,keVee}}+0.488\sqrt{\frac{E_{ee}}{{\rm\,keVee}}}\right){\rm\,keV},$
(11)

and the detector resolution as in Ref. dama_app :

$\epsilon(E)=\left\\{\begin{matrix}0.0429E+0.657&\text{for $E<8$ keV}\\\
1&\text{otherwise}\end{matrix}\right.\\\ .$ (12)

Thus, the observed rate in a given energy bin of width $\Delta E=2\delta E$ is

$\begin{split}&\Delta E\times R(E)=\int_{E-\delta E}^{E+\delta
E}\frac{dR}{dE^{\prime}}dE^{\prime},~{}\text{ where}\\\
&\frac{dR}{dE^{\prime}}=\frac{\epsilon(E^{\prime})}{Q_{T}}\frac{1}{(2\pi)^{1/2}}\int_{0}^{\infty}\frac{dE_{ee}}{\sigma_{E_{ee}}}\frac{dR}{dE_{R}}\exp\left[\frac{-(E^{\prime}-E_{ee})^{2}}{2(\sigma_{E_{ee}})^{2}}\right].\end{split}$
(13)

This can be split into the constant and modulating components by projecting
onto a cosine function with the expected period and offset for DM.

####  2.1.2 Model independent assumptions

For the model independent analysis, we assume that all detectors should
observe exactly the same rate as reported by DAMA. In that case we we can take
the signal rate to be $R(t)=S_{0}+S_{m}\cos{\omega t}$ for the reported bins,
where $S_{m}$ is given explicitly in Ref. Bernabei2018 , and $S_{0}$ is
inferred from the fact that for the SHM velocity distribution (which dictates
modulation of signal) $S_{m}/S_{0}\simeq 0.02$. This method of extracting
$S_{0}$ is only an approximation. More correct results could be generated
using the exact constant rate observed, if available. The relevant rates are
given here in Table 2.

Table 2: Modulating and inferred constant DAMA rates Bernabei2018 . Energy (keV) |  $S_{m}$ (cpd/kg/keV) |  $S_{0}$ (cpd/kg/keV)
---|---|---
1.0 - 1.5 | 0.0232 $\pm$ 0.0052 | 1.160
1.5 - 2.0 | 0.0164 $\pm$ 0.0043 | 0.820
2.0 - 2.5 | 0.0178 $\pm$ 0.0028 | 0.890
2.5 - 3.0 | 0.0190 $\pm$ 0.0029 | 0.950
3.0 - 3.5 | 0.0178 $\pm$ 0.0028 | 0.890
3.5 - 4.0 | 0.0109 $\pm$ 0.0025 | 0.545
4.0 - 4.5 | 0.0110 $\pm$ 0.0022 | 0.550
4.5 - 5.0 | 0.0040 $\pm$ 0.0020 | 0.200
5.0 - 5.5 | 0.0065 $\pm$ 0.0020 | 0.325
5.5 - 6.0 | 0.0066 $\pm$ 0.0019 | 0.330

##  3 Results

Our sensitivity results are split into two sections, model dependent and
independent. The former are the more traditional presentation, selecting a
model (here, the ‘standard’ spin independent elastic WIMP) and varying
$m_{\chi}$ and $\sigma_{0}$ \- equivalent to changing $R_{m}$ and $R_{0}$. The
latter demonstrate the exclusion power of various NaI experiments as a
function of their runtime, setting $R_{m}$ and $R_{0}$ to the values reported
by DAMA.

###  3.1 Model dependent

The choice of a particular model will dictate the observable $R_{m}$ and
$R_{0}$ for some combination of DM mass and cross section, $m_{\chi}$ and
$\sigma_{0}$. To produce the usual sensitivity curves, we loop through a set
range of $m_{\chi}$ and $\sigma_{0}$, and compute the p-values of $\mu_{b}$
(the mean of the background only distribution) in the signal+background
distribution. A detector will be sensitive with 90% confidence to the
combinations that produce $P(x=\mu_{b})\leq 0.1$.

The results of applying this method for ANAIS, COSINE, and SABRE after 3 years
of operation are shown in Fig. 4 for the standard spin independent DM model,
compared with DAMA run for the same period of time. This demonstrates that
although all the detectors have the same target, there are still combinations
of DM parameter space that will produce a statistically significant modulation
in some detectors, but not in others over the same run time.

Figure 4: 90 % C.L. modulation sensitivity of various NaI detectors after 3
years of operation, assuming a spin independent elastic WIMP. For each
detector the exposure mass and $R_{b}$ are shown in Table 1.

Note that the peak occurring at $m_{\chi}\sim 150$ GeV/c2 is due to our
assumption of a single energy bin from 2-6 keV111We make this assumption to
remain agnostic about the exact background energy spectrum of the three
detectors, as all have published only a single value for the 2-6 keV range..
At this mass range the net modulation in our energy region of interest
switches from positive to negative, and so averaging over the energy bins
erroneously gives a modulation amplitude close to zero, drastically reducing
the sensitivity. See, for example, Fig. 5. More proper analysis requires
knowledge of the background energy spectrum not publicly available.

Figure 5: Modulation rate both as a spectrum and averaged over the region of
interest for various DM masses and $\sigma_{0}=10^{-39}$ cm2.

The exposure mass and detector backgrounds assumed for this analysis are given
in Table 1. Interestingly, despite having the lowest target mass, SABRE has
the greatest sensitivity of the three new detectors to the annual modulation
of these models across the full parameter space. This is likely due its low
background, which is projected to be approximately an order of magnitude lower
than that of ANAIS, the next most sensitive detector. Where backgrounds are of
a similar scale, as for COSINE and ANAIS with 2.7 and 3.2 cpd/kg/keV
respectively, the exposure mass becomes more important - here the fact that
ANAIS uses almost double the mass of COSINE for analysis gives it a slight
edge. Similarly, DAMA’s exposure mass (5 times the size of SABRE’s) gives it
the largest sensitivity region.
Although increasing a detectors target mass increases the number of DM events,
it will also increase the number of background events by the same scale. Thus
although there is some improvement in sensitivity in this case, reducing the
background appears to be a far more effective way of improving detector
performance - at least for the standard WIMP model assumed here. This is what
allows SABRE to approach the limit set by DAMA much closer than either ANAIS
or COSINE.

The analysis described in this section is typical in the production of
exclusion plots, and is the only way to compare detectors using different
targets. The downside of this method is that it remains possible to tailor
increasingly complex DM models to explain the lack of a modulating signal at
detectors other than DAMA. To properly exclude (or support) this observation,
we need to understand and compare the exclusion power of other NaI detectors
for the DAMA rate, in the event they return null results.

###  3.2 Model independent

Detectors that use the same target should theoretically all observe the same
interaction rate. In reality, the efficiency and resolution may cause some
variation between detectors, but we can consider the DAMA data a minimum
observable rate, as the newer detectors tend to improve upon both values
adhikari2020eff ; Amare2019eff , particularly above 2 keV. We can take
advantage of this, and instead of assuming some model, DM mass, and cross
section to construct $R_{m}$ and $R_{0}$, we use the DAMA values given in
Ref.Bernabei2018 , which should also be visible at other NaI detectors. Using
this, we can find the $n\sigma$ confidence level exclusion or discovery power
of the DAMA rate after a given run time.
The exclusion power of the various NaI detectors is shown in Fig. 6, discovery
power in Fig. 7, and the properties assumed for each of detector are given in
Table 1. Key and present sensitivity milestones are also shown in Table 3.
These results again assume a single energy bin from 2-6 keV, and a constant
background over that region as reported in the provided references. We also
note that the present and predicted exclusion power we find for both ANAIS and
COSINE are consistent with those previously presented by both collaborations
amare2021 ; Adhikari_2019 .

Table 3: Present exclusion C.L. for the detectors, and years for significant exclusion (Fig. 6) and discovery C.L. (Fig. 7). Exp. | Present excl. | For 3$\sigma$ excl. | For 5$\sigma$ disc.
---|---|---|---
ANAIS | 2.5$\sigma$ |  $\sim$ 3 yrs |  $\sim$ 7 yrs
COSINE | 1.6$\sigma$ |  $\sim$ 5 yrs |  $>$ 7 yrs
SABRE | 0 |  $\sim$ 2 yrs |  $\sim$ 2 yrs

For this model independent analysis, we see the same trend as in Fig. 4, where
the low background of SABRE makes it the most sensitive experiment, despite
also having the lowest target mass. This is even more obvious considering the
exclusion vs. discovery plots. As explained in Sec 2 in Eq. 3 and 5, the
former depends on the uncertainty in signal+background, while the latter
relies only on background uncertainty. As such, the improvements from lowering
the background are much clearer in the case of discovery power, which explains
why the SABRE experiment is able to perform so much better than COSINE and
ANAIS.
Focusing on the discovery plot in Fig. 7, this gives the significance of a
modulating signal compatible with DAMA’s, should it be observed by any of the
detectors. Comparing the three experiments at any given time, we find that

$\sigma~{}\text{C.L.}\propto\sqrt{\frac{M_{E}}{R_{b}}}.$ (14)

Thus, with their present reported backgrounds, ANAIS and COSINE would require
exposure masses of 445 kg and 375 kg respectively to reach the 5$\sigma$
discovery at the same time as SABRE. Clearly, a lower background gives a
significant advantage, in terms of time the experiment needs for statistically
significant results, as well as the laboratory space it will ultimately take
up.

Figure 6: Exclusion power of NaI detectors, as a function of run time. Cross
hatched regions indicated 1$\sigma$ uncertainty bands. Blue line indicates
typical benchmark value. Figure 7: Discovery power of NaI detectors, as a
function of run time. Cross hatched regions indicated 1$\sigma$ uncertainty
bands. Blue line indicates typical benchmark value.

###  3.3 Mass and background requirements

The model independent sensitivity method can also be used to compute the mass
and background required for a detector to exclude the DAMA results with a
given confidence level within a specified period of time. For example,
assuming that we want an experiment to have exclusion power of 3$\sigma$
within three years, we can plot the maximum allowable background that will
give this sensitivity as a function of exposure mass. Alternatively, it can be
used to find the minimum exposure mass in the event we cannot get below some
particular background threshold.
These results are given in Fig. 8. If a collaboration has only enough
time/resources to grow $\sim$ 65 kg of crystal, they will require an
experiment background of below 1.5 cpd/kg/keV in order to have 3 $\sigma$
exclusion power of the the DAMA signal within three years. Conversely, if they
know they are able to achieve a background of 0.5 cpd/kg/keV, they need only
about 30 kg of crystal for the same time and sensitivity requirements.
Calculations and plots of this kind may prove useful at the R&D or upgrade
stages of NaI detectors to inform or validate choices made surrounding target
masses and background reduction.

Figure 8: Maximum background and minimum mass combinations that provide
3$\sigma$ exclusion of DAMA within three years of operation. Hatched bands
give 1$\sigma$ uncertainty. If a detector’s properties fall below the line, it
will have the requisite sensitivity.

##  4 Conclusions

We have presented here the comparative sensitivity of NaI detectors computed
in both a model dependent and independent way. In both cases, for both
exclusion and discovery limits the lowest background experiment (SABRE) has
performed the best of the three experiments, despite being the one with the
lowest exposure mass. This makes clear just how important a low background is
for DM searches such as these, in order to observe the small modulation in an
already low interaction rate. This further motivates the purification and veto
techniques presently explored by these collaboration. Should the projected
exposure mass and backgrounds be achieved, and data taking commence in the
next 18 months, SABRE will be positioned to provide statistically significant
exclusion or discovery of the DAMA signal within 3-4 years, either in tandem
or ahead of COSINE or ANAIS.
In this event (and even more so in the event of a positive DM-like signal), it
will be perhaps beneficial and interesting to compare the results of the
Northern and Southern hemisphere experiments and data, to further elucidate
clues as to nature of the modulating DAMA signal - DM or not.

## Acknowledgments

The authors would like to thank Phillip Urquijo for his review of the paper.
This work was supported by the ARC Centre of Excellence of Dark Matter
Particle Physics through grant CE200100008. MJZ is also supported through the
Australian Government Research Training Program Scholarship. Both authors are
members of the SABRE collaboration.

##  Appendix A Nuclear form factors

These form factors are given in terms of the dimensionless $y=(qb/2)^{2}$,
where $b$ is the harmonic oscillator parameter:

$b=\sqrt{\frac{41.467}{45A^{-1/3}-25A^{-2/3}}}\text{ fm}.$ (15)

23Na:

$\begin{split}F^{(p,p)}_{11}(q)&=e^{-2y}\left(120-180y+87y^{2}-17y^{3}+1.2y^{4}\right)\\\
F^{(p,n)}_{11}(q)&=e^{-2y}\left(130-200y+100y^{2}-20y^{3}+1.5y^{4}\right)\\\
F^{(n,n)}_{11}(q)&=e^{-2y}\left(140-220y+120y^{2}-25y^{3}+1.8y^{4}\right)\\\
\end{split}$ (16)

127I:

$\begin{split}F^{(p,p)}_{11}(q)=&e^{-2y}(2800-10000y+14000y^{2}-9800y^{3}\\\
&+3800y^{4}-840y^{5}+100y^{6}-6.3y^{7}+0.15y^{8})\\\
F^{(p,n)}_{11}(q)=&e^{-2y}(3900-15000y+23000y^{2}-18000y^{3}\\\
&+7900y^{4}-2000y^{5}+290y^{6}-23y^{7}+0.75y^{8}\\\ &-0.0048y^{9})\\\
F^{(n,n)}_{11}(q)=&e^{-2y}(5500-23000y+38000y^{2}-32000y^{3}\\\
&+16000y^{4}-4600y^{5}+790y^{6}-75y^{7}+3.3y^{8}\\\
&-0.041y^{9}+0.00015y^{10})\\\ \end{split}$ (17)

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# Killing-Yano Forms of a Class of Spherically Symmetric Space-Times I: A
Unified Generation of Killing Vector Fields

Ö. Açık 1<EMAIL_ADDRESS>Ü. Ertem 1<EMAIL_ADDRESS>M. Önder 2<EMAIL_ADDRESS>A. Verçin 1<EMAIL_ADDRESS>1
Department of Physics, Ankara University, Faculty of Sciences, 06100,
Tandoğan-Ankara, Turkey
2 Department of Physics Engineering, Hacettepe University, 06800, Beytepe-
Ankara, Turkey.

###### Abstract

Killing-Yano one forms (duals of Killing vector fields) of a class of
spherically symmetric space-times characterized by four functions are derived
in a unified and exhaustive way. For well-known space-times such as those of
Minkowski, Schwarzschild, Reissner-Nordstrøm, Robertson-Walker and several
forms of de Sitter, these forms arise as special cases in a natural way.
Besides its two well-known forms, four more forms of de Sitter space-time are
also established with ten independent Killing vector fields for which four
different time evolution regimes can explicitly be specified by the symmetry
requirement. A family of space-times in which metric characterizing functions
are of the general form and admitting six or seven independent Killing vector
fields is presented.

###### pacs:

04.20.-q, 02.40.-k

##  I Introduction

Defining relations of Killing-Yano (KY) and conformal KY-forms are natural
generalizations of Killing 1-forms and conformal Killing 1-forms. The latter
are the dual of Killing and conformal Killing vector fields whose flows
generate, respectively, local isometries and local conformal isometries of the
metric in (pseudo)Riemannian geometry Benn-Tucker ; Thirring . Although they
are not related to the isometries of the metric these higher rank
generalizations have attracted increasing interest in various fields of
physics and modern mathematics as well as in some related fields. Generally
speaking, many interesting properties of a space-time are intimately connected
with the existence of (conformal) KY-forms admitted by the corresponding
metric. More specifically; the determination of KY-forms of a given metric,
classification of space-times admitting KY-forms, analysis of the algebraic
structures of these forms as well as specification of the symmetry algebra and
related conserved quantities of the Dirac and related equations in a given
curved background have gained increasing significance. Equally important
objects (not considered in this study) are the totally symmetric Killing
tensors and their conformal generalizations (see Dietz1 ; Dietz2 ; Benn-son
and references therein).

KY-forms play a prominent role in a unified description of null and non-null
shear-free congruences Benn  and in the search of force-free fields
(divergence-free eigenvectors of the curl operator with position dependent
eigenvalues) mostly encountered in astrophysics and fluid dynamics literatures
Benn-Kress0 ; Benn-Kress3 ; Kress . A tangent vector which generates conformal
transformations on its orthogonal complement is said to be the generator of
shear-free congruence for its integral curve. Shear-free equation is a
generalization of defining equation of conformal Killing equation. Together
with Clifford calculus, KY-forms also provide efficient means in analyzing
elliptic operators and the Dirac operator and in the further classification of
(pseudo)Riemannian manifolds Benn-Tucker ; Semmelman1 . While conformal KY-
forms take part in symmetry operators for the massless Dirac equation Benn-
Charlton ; Benn-Kress3 , KY-forms are indispensable in constructing first
order symmetries of the massless as well as massive Dirac equation in a curved
space-time Benn-Kress1 . KY-forms are also necessary for the symmetries of the
Kähler equation Benn-Tucker ; Benn-Kress2 .

Longstanding interest in KY-forms largely stems from their constant use in
general relativity and especially from their role in constructing conserved
quantities in a number of ways. The studies of Penrose and his collaborators
Walker ; Penrose , who have shown how the existence of a KY 2-form explains
Carter’s result on the integrability of the geodesic equation in Kerr
background, constitute a stepping-stone in this context Carter1 ; Carter2 .
The fact that any KY p-form provides a quadratic first integral of the
geodesic equation is by now a well-established particular result of the fact
that the interior derivative of any KY p-form with respect to the tangent
vector field of any geodesic remains parallel along the geodesic. More
generally, any KY (p+1)-form is associated with a symmetric bilinear form,
which is nothing more than the Killing tensor generalizing the so-called
Stackel-Killing tensor that corresponds to a KY 2-form as first recognized by
Penrose and Floyd. For more in this context, refer to Semmelman1 .

On the other hand, as every KY form is co-closed, the Hodge dual of any KY
$p$-form is directly associated with a conserved quantity. In the case of KY
1-forms, two types of conserved currents can be defined. For Ricci-flat space-
times the Hodge dual of the exterior derivative of a KY 1-form $\omega$ is
conserved, where ${}^{\ast}d\omega$ is known as the Komar form Benn-Tucker .
Secondly, the current $j=i_{X^{a}}\omega\wedge^{\ast^{-1}}G_{a}$ defined in
terms of the Einstein $3$-forms $G_{a}$, and KY 1-form $\omega$ is also
conserved. Here, ∗ and ${}^{\ast^{-1}}$ represent the Hodge map and its
inverse, $\wedge$ denotes exterior multiplication, $d$ and $i_{X}$ stand for
exterior and interior derivatives (with respect to vector field $X$). It has
recently been shown how the KY-forms of the flat space-time can be used to
construct new, conserved gravitational charges for transverse asymptotically
flat Kastor  as well as for asymptotically anti de Sitter space-times Cebeci .
These studies present another way of constructing a conserved current by
taking a particular linear combination of wedge products of interior
derivatives of the KY p-form and curvature characteristics of the underlying
manifold. For KY 1-forms, this reduces to the usual current obtained from the
Einstein $3$-form.

The main purpose of this and the accompanying paper ozumav1  is to develop, by
directly starting from the KY-equation,

$\displaystyle\nabla_{X_{a}}\omega_{(p)}=\frac{1}{p+1}i_{X_{a}}d\omega_{(p)}\;,\quad
p=1,2,3$ (1)

a constructive method which makes it possible to generate all KY forms for a
large class of spherically symmetric space-times in a unified and exhaustive
way. Here $\nabla_{X}$ stands for the covariant derivative with respect to the
vector field $X$. It should be noted that the KY 0-form $\omega_{(0)}$ can be
any function, $\omega_{(1)}$ is the dual of a Killing vector field and
$\omega_{(n)}$ is a constant (parallel), that is, it is a constant multiple
$\omega_{(n)}=az$ of the volume form $z$ (for the orthonormal frame given
below $z=e^{0123}$). The goal of our study is achieved by solving a coupled
set of first order partial linear differential equations for the component
functions of the KY forms $\omega_{(p)}$ in each case. The number of
independent equations that have to be solved for $p=1,2,3$ are $10,\;18$ and
$16$, respectively. To obtain the most general solutions in an exhaustive way,
there are also $24,36$ and $24$ integrability conditions that must be
carefully examined at the outset. We first solve a suitable set of the
integrability conditions by which the set of solutions naturally branches into
cases and subcases.

We shall mainly use the notation of Benn-Tucker  and adopt the following
conventions and terminology. The underlying base manifold is supposed to be a
$4$-dimensional ($4D$) pseudo-Riemannian manifold with the metric tensor $g$
having Lorentzian signature $(-+++)$ such that $g=-e^{0}\otimes
e^{0}+e^{1}\otimes e^{1}+e^{2}\otimes e^{2}+e^{3}\otimes e^{3}$ in a local
orthonormal co-frame $\\{e^{a}\\}$. By choosing the co-frame basis

$\displaystyle e^{0}$ $\displaystyle=$ $\displaystyle H_{0}dt\;,\qquad
e^{1}=TH_{1}dr\;,$ $\displaystyle e^{2}$ $\displaystyle=$ $\displaystyle
TH_{2}d\theta\;,\quad e^{3}=TH_{2}\sin\theta d\varphi\;,$

a class of spherically symmetric metrics that will be considered can be
parameterized by

$\displaystyle T=\exp(\lambda(t))\;,\quad H_{j}=H_{j}(r)\;;\quad j=0,1,2\;$

which henceforth will be referred to as the (metric) coefficient functions.
Here $(t,r,\theta,\varphi)$ specifies a local polar space-time chart with the
usual range of variations. Whenever necessary, the range of $r$ can be bounded
to keep the coefficient functions real. A generic property of this kind of
metric is invariance under the transformation of the spatial rotation group
$SO(3)$. If $g$ admits a time-like Killing vector field $K_{0}$, it is termed
stationary and if, in addition, $K_{0}$ is orthogonal to a family of space-
like hypersurfaces it is termed static. The dual tangent frame basis
$\\{X_{a}\\}$ of the co-frame basis $\\{e^{a}\\}$ are,

$\displaystyle X_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{H_{0}}\partial_{t}\;,\qquad
X_{1}=\frac{1}{TH_{1}}\partial_{r}\;,$ $\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{TH_{2}}\partial_{\theta}\;,\quad
X_{3}=\frac{1}{TH_{2}\sin\theta}\partial_{\varphi}\;,$

where $e^{a}(X_{b})=\delta^{a}_{b}$ and $\partial_{x}=\partial/\partial x$.
The metric dual of a vector field $X$ will be denoted by $\tilde{X}$ such that
$\tilde{X}(Y)=g(X,Y)$ for any vector field $Y$. The torsion-free connection
$1$-forms $\omega_{ab}$ for this class of metrics are well-known, and can be
presented in the following antisymmetric matrix-valued $1-$form:

$\displaystyle(\omega_{ab})=\frac{1}{T}\left(\begin{array}[]{cccc}0&&&\\\
\frac{h_{0}}{H_{1}}e^{0}+\frac{\dot{T}}{H_{0}}e^{1}&0&&\\\
\frac{\dot{T}}{H_{0}}e^{2}&\frac{h_{2}}{H_{1}}e^{2}&0&\\\
\frac{\dot{T}}{H_{0}}e^{3}&\frac{h_{2}}{H_{1}}e^{3}&\frac{\cot\theta}{H_{2}}e^{3}&0\end{array}\right)\;,$
(6)

where we have used the abbreviations $\dot{T}=dT/dt,\;dH(r)/dr=H^{\prime}$ and
$h_{j}=H^{\prime}_{j}/H_{j}$. We shall always use prime and over-dot to
denote, respectively, the $r$-derivation and $t$-derivation of a function
which depends only on $r$ and $t$. The partial derivative of a map $U$ of
several variables with respect to $x$ will be denoted by $U_{x}$, and of a
rational map or function $U/V$ by $\partial_{x}(U/V)$. The following matrix-
valued $1$-form is helpful in carrying out the calculations

$\displaystyle(\nabla_{X_{a}}e^{b})=-\frac{1}{T}\left(\begin{array}[]{cccc}\frac{h_{0}}{H_{1}}e^{1}&\frac{h_{0}}{H_{1}}e^{0}&0&0\\\
\frac{\dot{T}}{H_{0}}e^{1}&\frac{\dot{T}}{H_{0}}e^{0}&0&0\\\
\frac{\dot{T}}{H_{0}}e^{2}&-\frac{h_{2}}{H_{1}}e^{2}&\frac{\dot{T}}{H_{0}}e^{0}+\frac{h_{2}}{H_{1}}e^{1}&0\\\
\frac{\dot{T}}{H_{0}}e^{3}&-\frac{h_{2}}{H_{1}}e^{3}&-\frac{\cot\theta}{H_{2}}e^{3}&\frac{\dot{T}}{H_{0}}e^{0}+\frac{h_{2}}{H_{1}}e^{1}+\frac{\cot\theta}{H_{2}}e^{2}\end{array}\right)\;.$
(11)

The corresponding curvature $2$-forms are presented in Appendix B.

All the well-known spherically symmetric space-times such as the Minkowski,
Schwarzschild, Reissner-Nordstrøm, Robertson-Walker and the six various forms
of de Sitter models fall within the class of the considered metric as special
cases and this provides the opportunity to give complete lists of their KY
forms. As a particular result, we have found a completely solvable nonlinear
ordinary differential equation $T^{2}\partial_{t}(\dot{T}/T)=\ell$, where
$\ell$ is constant, characterizing five different time-dependent types of de
Sitter space-time in a unified way. This fact also enables us to give explicit
expressions of their KY-forms in a unified and exhaustive way. As the first
part of our study, the present paper is entirely devoted to the unified
generation of all Killing vector fields for the considered class of space-
times. KY two and three forms are taken up in the next paper ozumav1 .

Although KY-forms have been the subject of relatively recent active research,
Killing vectors have been so intensely investigated that the following
original points which hold for the KY-forms as well are worth emphasizing. (i)
As has been mentioned above, there exist five well-known space-times that are
covered by the considered class of metrics such that one of them (de Sitter)
consists of six different types. Killing vector fields of these metrics are
usually handled as case-by-case studies in scattered references. Our unified
generation may remedy many inconveniences such as notational
incompatibilities, proper range problems related to coordinates and
relationships between cases. Moreover, one can explicitly observe the
emergence of each case from the variations of the metric characterizing
coefficients. (ii) Derivations of Killing vector fields for some types of de
Sitter space-time from a five dimensional embedding flat manifold is
geometrically very appealing and more inspiring physically (for a review of de
Sitter spaces see Hawking ; Moschella  and for recent interest see Randono
and references therein). But the exact number of possible types naturally
emerges from our study, and we identify an exactly solvable equation that
determines this number by the number of its possible solutions. (iii) Our
approach is exhaustive in the sense that all of the possible Killing vector
fields of a given space-time can be completely determined from our approach so
long as its metric belongs to the considered class. This fact makes it
possible to reach decisive, or at least conclusive statements about a
particular problem in which KY-forms are involved. We do not go into the
detail of all possible cases but point out sufficiently symmetric cases, and
have given some details of a particular case having six independent Killing
vector fields that, as far as we know, does not appear in the literature. This
example is also worth mentioning in light of the fact that its symmetry
algebra changes drastically when a seemingly unimportant integration constant
is changed.

##  II KY 1-Forms: Defining Equations

A $1$-form is a KY $1$-form if and only if it is the metric dual of a Killing
vector field. This is equivalent to the fact that it satisfies equation (1)
for $p=1$. For the components of

$\displaystyle\omega_{(1)}=\alpha e^{0}+\beta e^{1}+\gamma e^{2}+\delta
e^{3}\;,$

the KY equation gives, in view of (2) and (3), sixteen equations of which the
following ten

$\displaystyle\alpha_{t}$ $\displaystyle=$
$\displaystyle\frac{H^{\prime}_{0}}{TH_{1}}\beta\;,\;\quad\quad\quad\beta_{r}=\dot{T}\frac{H_{1}}{H_{0}}\alpha\;,$
$\displaystyle T^{2}\partial_{t}\frac{\beta}{T}$ $\displaystyle=$
$\displaystyle-\frac{H^{2}_{0}}{H_{1}}\partial_{r}\frac{\alpha}{H_{0}}\;,\;\;\quad\beta_{\theta}=-\frac{H^{2}_{2}}{H_{1}}\partial_{r}\frac{\gamma}{H_{2}}\;,$
$\displaystyle T^{2}\partial_{t}\frac{\gamma}{T}$ $\displaystyle=$
$\displaystyle-\frac{H_{0}}{H_{2}}\alpha_{\theta}\;,\;\;\;\quad\quad\beta_{\varphi}=-\frac{H^{2}_{2}}{H_{1}}\sin\theta\partial_{r}\frac{\delta}{H_{2}}\;,$
(12) $\displaystyle T^{2}\partial_{t}\frac{\delta}{T}$ $\displaystyle=$
$\displaystyle-\frac{H_{0}}{H_{2}\sin\theta}\alpha_{\varphi}\;,\;\;\;\gamma_{\theta}=\dot{T}\frac{H_{2}}{H_{0}}\alpha-\frac{H^{\prime}_{2}}{H_{1}}\beta\;,$
$\displaystyle\delta_{\varphi}$ $\displaystyle=$
$\displaystyle\sin^{2}\theta\partial_{\theta}\frac{\gamma}{\sin\theta}\;,\quad\gamma_{\varphi}=-\sin^{2}\theta\partial_{\theta}\frac{\delta}{\sin\theta}$

are independent. Although it appears difficult to solve this coupled set of
first order partial differential equations directly, an exhaustive treatment
of the problem with many classes of solutions is possible.

At first to gain an initial insight into the above set of equations, let us
look at some obvious solutions. We can immediately see that when all
coefficient functions but $\alpha$ are zero, $T$ must be constant and
$\alpha=H_{0}$. When $\beta$ alone is nonzero and proportional to $T$, the
conditions $H^{\prime}_{0}=0=H^{\prime}_{2}$ must be fulfilled. The other two
cases, in which only $\gamma$ or only $\delta$ is different from zero and
proportional to $TH_{2}$, are forbidden by the last two equations of (4). But
it is easy to verify that

$\displaystyle\alpha=0=\beta\;,\quad\gamma=TH_{2}g\;,\quad\delta=TH_{2}(a_{3}\sin\theta+\cos\theta
g_{\varphi})\;,$

is a solution, such that $g(\varphi)$ satisfies $g_{\varphi\varphi}+g=0$ and
there is no additional constraint. We thus have, for $g=a_{1}\sin\varphi-
a_{2}\cos\varphi$, the following three linearly independent KY 1-forms :

$\displaystyle\tilde{K}_{1}$ $\displaystyle=$ $\displaystyle
TH_{2}(\sin\varphi e^{2}+\cos\theta\cos\varphi e^{3})\;,$
$\displaystyle\tilde{K}_{2}$ $\displaystyle=$ $\displaystyle
TH_{2}(-\cos\varphi e^{2}+\cos\theta\sin\varphi e^{3})\;,$ (13)
$\displaystyle\tilde{K}_{3}$ $\displaystyle=$ $\displaystyle-TH_{2}\sin\theta
e^{3}\;.$

The corresponding rotational Killing vector fields

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\sin\varphi\partial_{\theta}+\cot\theta\cos\varphi\partial_{\varphi}\;,$
$\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle-\cos\varphi\partial_{\theta}+\cot\theta\sin\varphi\partial_{\varphi}\;,$
(14) $\displaystyle K_{3}$ $\displaystyle=$
$\displaystyle-\partial_{\varphi}\;,$

are the well-known generators of the $so(3)$-algebra:
$[K_{i},K_{j}]=\varepsilon_{ij}^{\;\;\;k}K_{k}$. As there is no constraint on
these solutions, they must appear in every case independent of the specific
form of the functions characterizing the metric. This is a typical
characteristic of spherical symmetry.

When $\dot{T}=0,\;\omega_{0}=H_{0}e^{0}$ is a KY $1$-form which corresponds to
the time-like Killing vector field $K_{0}=\tilde{\omega}_{0}=-\partial_{t}$,
and it can be combined with the above $so(3)$ solutions. In fact for
$\alpha=H_{0},\;\beta=0$ and $\dot{T}=0$, the above $4D$ algebra is the dual
of the most general solution of equations (4). Indeed in such a case, the
first eight equations of (4) imply that $\gamma=TH_{2}G(\varphi)$ and
$\delta=TH_{2}D(\theta,\varphi)$ and then the last two equations of (4) yield

$\displaystyle D_{\varphi}=-\cos\theta G\;,\quad G_{\varphi}=-\sin\theta
D_{\theta}+\cos\theta D\;.$

From the derivation of the second equation with respect to $\varphi$, we
obtain $G_{\varphi\varphi}+G=0$ in view of the first equation. On the other
hand, integration of the first equation gives $D=\cos\theta
G_{\varphi}+f(\theta)$ which, upon substituting it into the second, gives
$f_{\theta}=\cot\theta f$ whose integral is $f=a_{3}\sin\theta$. In the case
of $H_{2}=r$, such space-times with $4D$ symmetry algebra include two
physically important examples: the Reissner-Nordström (RN) and its special
case Schwarzschild space-times, for which the other coefficient functions are
given in Table I. It should be emphasized that the explicit forms of $H_{0}$
and $H_{1}$ are derived from the physical requirements, namely from the
Einstein equations in the Schwarzschild case.

For a general consideration, it turns out to be convenient to look for the
solutions in the set of the solutions of some integrability conditions. This
will also provide us with the necessary means to generate other sets of
solutions.

##  III Integrability Conditions and Their Solutions

For $x=\alpha,\beta$ we have the integrability conditions

$\displaystyle\partial_{\varphi}\partial_{\theta}(\frac{x}{\sin\theta})=0\;,\quad
x_{\varphi\varphi}=\sin^{3}\theta\partial_{\theta}\frac{x_{\theta}}{\sin\theta}\;,$
(15)

that follow from $\delta_{r\theta}=\delta_{\theta
r},\delta_{rt}=\delta_{tr},\delta_{t\theta}=\delta_{\theta
t},\delta_{t\varphi}=\delta_{\varphi t}$ and
$\gamma_{r\varphi}=\gamma_{\varphi
r},\gamma_{\theta\varphi}=\gamma_{\varphi\theta}$. There are two additional
conditions for $\alpha$ and four conditions for $\gamma$ and $\delta$:

$\displaystyle\partial_{r}\partial_{\theta}(\frac{\alpha}{H_{2}})$
$\displaystyle=$ $\displaystyle
0=\partial_{r}\partial_{\varphi}(\frac{\alpha}{H_{2}})\;,$
$\displaystyle\partial_{r}\partial_{\theta}(\frac{\delta}{H_{2}})$
$\displaystyle=$ $\displaystyle
0=\partial_{t}\partial_{\theta}(\frac{\delta}{T})\;,$ (16)
$\displaystyle\partial_{t}\partial_{r}(\frac{\delta}{TH_{0}})$
$\displaystyle=$ $\displaystyle
0=\partial_{t}\partial_{r}(\frac{\gamma}{TH_{0}})\;.$

The first two conditions of (8) can be checked from
$\alpha_{r\theta}=\alpha_{\theta r}$ and $\alpha_{r\varphi}=\alpha_{\varphi
r}$. The second row of (8) follows from
$\alpha_{\theta\varphi}=\alpha_{\varphi\theta},\beta_{\theta\varphi}=\beta_{\varphi\theta}$
and the last row can be seen from $\beta_{t\varphi}=\beta_{\varphi t}$ and
$\beta_{r\theta}=\beta_{\theta r}$. The following two equations can be easily
verified from the last row of (4)

$\displaystyle
y_{\varphi\varphi}+y=-\sin^{3}\theta\partial_{\theta}\frac{y_{\theta}}{\sin\theta}\;,$
(17)

for $y=\gamma,\delta$. There are $24$ integrability conditions but only $20$
of them are independent, and the twelve shown above are sufficient for a
unified and exhaustive investigation.

####  III.0.1 The General Forms of $\alpha$ and $\beta$

In terms of the functions $f=f(t,r,\varphi),g=g(t,r,\theta)$ the first
equation of (7) implies

$\displaystyle x_{\varphi}=\sin\theta f\;,\quad x_{\theta}=\cot\theta x+g\;,$

and from the second equation of (7) we obtain $x=\sin^{2}\theta
g_{\theta}-\sin\theta f_{\varphi}$. On substituting this solution into the
above $x_{\varphi}$ and $x_{\theta}$ equations we arrive at

$\displaystyle
f_{\varphi\varphi}+f=0\;,\;g=\sin\theta\partial_{\theta}(\sin\theta
g_{\theta})\;,$

whose general solutions can be written, with $f_{i}=f_{i}(t,r)$ and
$g_{i}=g_{i}(t,r)$, as

$\displaystyle f=-f_{1}\sin\varphi-f_{2}\cos\varphi\;,\quad
g=-g_{1}\cot\theta-g_{2}\frac{1}{\sin\theta}\;.$

Here the minus signs are used for convenience. The general solution for $x$ is

$\displaystyle x=g_{1}+g_{2}\cos\theta+\sin\theta(f_{1}\cos\varphi-
f_{2}\sin\varphi)\;.$

In terms of the functions $A_{i}=A_{i}(t,r)$ and $B_{i}=B_{i}(t,r)$ let us
define

$\displaystyle\sigma^{A}$ $\displaystyle=$ $\displaystyle A_{1}\cos\varphi-
A_{2}\sin\varphi\;,\quad\sigma^{B}=B_{1}\cos\varphi-B_{2}\sin\varphi\;,$
$\displaystyle A$ $\displaystyle=$
$\displaystyle\sin\theta\sigma^{A}+A_{3}\cos\theta\;,\quad\quad
B=\sin\theta\sigma^{B}+B_{3}\cos\theta\;.$

Since the functions $f_{i}$ and $g_{i}$ are, in general, different for
$\alpha$ and $\beta$, we can write their general forms very concisely as

$\displaystyle\alpha=U+A\;,\quad\beta=V+B\;,$ (18)

where $U$ and $V$ depend, like the $g_{1}$ term of $x$, on $t$ and $r$. Note
that $A$ and $B$ depend on all of the coordinates and they satisfy the
relations

$\displaystyle
A_{\theta\theta}+A=0=B_{\theta\theta}+B\;,\quad\sigma^{A}_{\varphi\varphi}+\sigma^{A}=0=\sigma^{B}_{\varphi\varphi}+\sigma^{B}\;.$
(19)

In the case of $x=\alpha$, the first two conditions of (8) imply that the
functions characterizing $A$ are proportional to $H_{2}$ and these enable us
to write $A=H_{2}\xi$ such that

$\displaystyle\xi=u\cos\theta+\sin\theta(v_{1}\cos\varphi-
v_{2}\sin\varphi)\;,$ (20)

where $u,v_{i}$ depend only on $t$.

####  III.0.2 The General Forms of $\gamma$ and $\delta$

When $\alpha$ and $\beta$ given by (10) are substituted into the
$\gamma_{\theta}$-equation of (4), we obtain

$\displaystyle\gamma_{\theta}=(\dot{T}\frac{H_{2}}{H_{0}}U-\frac{H_{2}^{\prime}}{H_{1}}V)+\dot{T}\frac{H_{2}}{H_{0}}A-\frac{H_{2}^{\prime}}{H_{1}}B\;.$

For well-defined $\gamma$ solutions, the first term at the right hand side
must be zero:

$\displaystyle\dot{T}U=H_{0}PV\;,\quad(P=\frac{H_{2}^{\prime}}{H_{1}H_{2}})\;.$
(21)

(Since the mentioned term is independent of $\theta$ it would lead, upon
integration, to a $\gamma$ solution which linearly depends on $\theta$. In
fact this condition results when the above $\gamma_{\theta}$ is used in eq.
(9) of $\gamma$.) Then, in view of eq.(11), the $\gamma_{\theta}$-equation can
be integrated to

$\displaystyle\gamma=-(\dot{T}\frac{H_{2}}{H_{0}}A_{\theta}-\frac{H_{2}^{\prime}}{H_{1}}B_{\theta})+TH_{2}g\;,\;\;g_{\varphi\varphi}+g=0\;,$
(22)

where the $\theta$-independent term $G=TH_{2}g(\varphi)$ comes from
integration, and its form can be easily verified by substituting $\gamma$ in
the $\alpha_{\theta}$ and $\beta_{\theta}$-equations of (4). Indeed, in the
first case we get

$\displaystyle
T^{2}\partial_{t}(\frac{\dot{T}}{T}A_{\theta})-\frac{H_{0}^{2}}{H_{2}^{2}}A_{\theta}=T^{2}H_{0}P\partial_{t}\frac{B_{\theta}}{T}\;,$
(23)

in addition to $\partial_{t}(G/T)=0$. In the second case, we obtain

$\displaystyle\dot{T}\partial_{r}(\frac{A_{\theta}}{H_{0}})=\frac{H_{1}}{H_{2}^{2}}B_{\theta}+\partial_{r}(PB_{\theta})\;,$
(24)

and $\partial_{r}(G/H_{2})=0$. These imply that $G$ is of the form
$G=TH_{2}g(\varphi)$ and by the integrability condition (9) for $\gamma$, we
see that $g$ must satisfy the equation given by (14).

Having determined the general form of $\gamma$, we now use it in the last two
equations of (4) to determine the form of $\delta$. The
$\delta_{\varphi}$-equation of (4) directly gives

$\displaystyle\delta_{\varphi}=\dot{T}\frac{H_{2}}{H_{0}}\sigma^{A}-\frac{H_{2}^{\prime}}{H_{1}}\sigma^{B}-TH_{2}\cos\theta
g\;,$

which can be integrated to

$\displaystyle\delta=-\dot{T}\frac{H_{2}}{H_{0}}\sigma^{A}_{\varphi}+\frac{H_{2}^{\prime}}{H_{1}}\sigma^{B}_{\varphi}+TH_{2}\cos\theta
g_{\varphi}+aTH_{2}\sin\theta\;,$ (25)

where $a$ is an integration constant and the $\varphi$-independent term
$D=aTH_{2}\sin\theta$ again comes from integration, whose explicit form is
determined from $\alpha_{\varphi},\;\beta_{\varphi}$ and the last two
equations of (4). Indeed, the $\alpha_{\varphi}$-equation of (4) states that
$D/T$ must be independent of $t$ and that the following condition must be
satisfied :

$\displaystyle
T^{2}\partial_{t}(\frac{\dot{T}}{T}\sigma^{A}_{\varphi})-\frac{H_{0}^{2}}{H_{2}^{2}}\sigma^{A}_{\varphi}=T^{2}H_{0}P\partial_{t}\frac{\sigma^{B}_{\varphi}}{T}\;.$
(26)

On the other hand, the $\beta_{\varphi}$-equation of (4) states that $D/H_{2}$
must be independent of $r$, and that the following condition must be satisfied
:

$\displaystyle\dot{T}\partial_{r}\frac{\sigma^{A}_{\varphi}}{H_{0}}=\frac{H_{1}}{H_{2}^{2}}\sigma^{B}_{\varphi}+\partial_{r}(P\sigma^{B}_{\varphi})\;.$
(27)

As a particular result, we have $D=TH_{2}f(\theta)$ and when the found forms
of $\gamma$ and $\delta$ are used in the last equation of (4), we obtain
$f=a\sin\theta$. Note that the conditions (18) and (19) are contained in (15)
and (16).

As an intermediate result, all the metric coefficient functions have been
specified in terms of seven functions $U,A_{i}$ and $B_{i}$, which can be
determined from the first three equations of (4) and the conditions found in
(13), (15) and (16). Therefore, seven equations of (4) have been analyzed, and
the first three equations and conditions (15) and (16) remain to be solved.
Note that the last terms of $\gamma$ and $\delta$ correspond to the $so(3)$
solutions. The rest of the investigation is entirely devoted to the
specification of additional symmetries.

####  III.0.3 The Clustering of Solutions

We shall now substitute the solutions (10) and (12) into the first three
equations of (4) to specify the unknown functions. The first three equations
of (4) give the following three equations for $U$

$\displaystyle U_{t}$ $\displaystyle=$
$\displaystyle\frac{\dot{T}}{T}\frac{H_{0}^{\prime}H_{2}}{H_{0}H^{\prime}_{2}}U\;,$
$\displaystyle\partial_{r}(\frac{U}{H_{0}P})$ $\displaystyle=$ $\displaystyle
H_{1}\frac{U}{H_{0}}\;,$ (28) $\displaystyle
T^{2}\partial_{t}(\frac{\dot{T}}{T}U)$ $\displaystyle=$
$\displaystyle-\frac{H_{0}^{3}P}{H_{1}}\partial_{r}\frac{U}{H_{0}}\;,$

and the following three equations for $B$

$\displaystyle B=TL\xi_{t}\;,\quad
B_{r}=\dot{T}\frac{H_{1}H_{2}}{H_{0}}\xi\;,\quad
T^{2}\partial_{t}\frac{B}{T}=-\frac{H_{0}^{2}}{H_{1}}(\frac{H_{2}}{H_{0}})^{\prime}\xi\;,$
(29)

where we have utilized relation (13) and the function $L=L(r)$ is defined by

$\displaystyle L=\frac{H_{1}H_{2}}{H_{0}^{\prime}}\;.$ (30)

As is obvious from the equations in this subsection, from here on the analysis
critically depends on the derivatives of $T,H_{0}$ and $H_{2}$. Since
$H_{2}^{\prime}$ is different from zero in all physically important space-
times, we shall assume this to be the case throughout the paper. This means
that the function $P$ is different from zero. The cases $H_{0}^{\prime}=0$ and
$\dot{T}=0$ will be considered as particular cases in the last three sections.
In the next section, we proceed to look for solutions for which both
$H_{0}^{\prime}$ and $\dot{T}$ can be different from zero.

##  IV Three classes of solutions

Using the first equation of (21) in the other two equations of (21), we obtain
two equations which accept separation of variables such that

$\displaystyle m\xi_{t}=\frac{\dot{T}}{T}\xi\;,\quad
T^{2}\xi_{tt}=-m_{0}\xi\;,$ (31)

provided that $m$ and $m_{0}$ defined by

$\displaystyle m=L^{\prime}\frac{H_{0}}{H_{1}H_{2}}\;,\quad
m_{0}=\frac{H^{2}_{0}}{LH_{1}}(\frac{H_{2}}{H_{0}})^{\prime}\;,$ (32)

are constants. When one side of a separable equation depends entirely on the
metric coefficient functions, we shall use the letters $k,l$ and $m$ for
separation constants. Other integration constants will be denoted by the
letters $a,b$ and $c$.

In view of the ansatz $U=H_{0}Y(r)f(t)$ the first two equations of (20)
transform to

$\displaystyle
f_{t}=k\frac{\dot{T}}{T}f\;,\quad(\frac{Y}{P})^{\prime}=H_{1}Y\;,$ (33)

where the metric constant $k$ is defined by

$\displaystyle k=\frac{H^{\prime}_{0}H_{2}}{H_{0}H_{2}^{\prime}}\;.$ (34)

On the other hand from the last equation of (20), with the same ansatz, we get

$\displaystyle
T^{2}\partial_{t}(\frac{\dot{T}}{T}f)=k_{0}f\;,\quad\frac{Y^{\prime}}{Y}=-k_{0}\frac{H_{1}}{H_{0}^{2}P}\;,$
(35)

where $k_{0}$ is a separation constant. Two equations of (25) can be easily
integrated to

$\displaystyle f=c_{1}T^{k}\;,\quad Y=c_{2}H_{2}P\;.$ (36)

Thus $U=cH_{0}H_{2}PT^{k}$ where $c_{i}$ are integration constants and
$c=c_{1}c_{2}$. When these solutions are substituted into (27) we see that $T$
must satisfy

$\displaystyle T\ddot{T}+(k-1)\dot{T}^{2}=k_{0}\;,$ (37)

and $k_{0}$ must be the metric constant

$\displaystyle
k_{0}=-\frac{H_{0}^{2}}{H_{1}H_{2}}(\frac{H_{2}^{\prime}}{H_{1}})^{\prime}\;.$
(38)

Noting that equations (20) and (21) are linear in $U$ and $B$, depending on
the values of $m$ one can distinguish three classes of solutions for which
both $H_{0}^{\prime}$ and $\dot{T}$ can be different from zero. These can be
characterized as follows

$\displaystyle(A)\;\;m=0\;,\quad(B)\;\;m\neq 0\;;\;\;U=0=V\;,\quad(C)\;\;m\neq
0\;.$

The cases (i) $\dot{T}=0,\;H_{0}^{\prime}\neq 0$, (ii) $\dot{T}\neq
0,\;H_{0}^{\prime}=0$ and (iii) $\dot{T}=0=H_{0}^{\prime}$ will be considered
separately in the last three sections.

###  IV.1  $m=0$ Solutions

When $m$ is zero $L$ is a nonzero constant and we have
${H_{0}^{\prime}}L=H_{1}H_{2}$. In that case, equations given by (23) imply
that $\xi=0=B$ and that $m_{0}$ need not be a constant. The conditions (15)
and (16) are both trivially satisfied for this case and hence, $\gamma$ and
$\delta$ solutions correspond only to the $so(3)$ solutions. Only the metric
constants $k$ and $k_{0}$ are defined in that case and, provided that $T$
satisfies equation (29), we have the following solutions for $\alpha$ and
$\beta$:

$\displaystyle\alpha=U=cT^{k}H_{0}\frac{H_{2}^{\prime}}{H_{1}}\;,\quad\beta=V=c\dot{T}T^{k}H_{2}\;,$
(39)

which determine the Killing vector field

$\displaystyle
X=-\frac{H_{2}^{\prime}}{H_{1}}T^{k}\partial_{t}+\dot{T}T^{k-1}\frac{H_{2}}{H_{1}}\partial_{r}\;,$
(40)

commuting with the $so(3)$ vector fields.

###  IV.2 $m\neq 0\;{\rm and}\;U=0=V$ Solutions

When $m$ is different from zero, the first equation of (23) can be easily
solved to be

$\displaystyle\xi=T^{1/m}\zeta\;,\quad\zeta=a_{1}\cos\theta+\sin\theta(a_{2}\cos\varphi-
a_{3}\sin\varphi)\;,$ (41)

where $a_{i}$ are integration constants. The second equation of (23) then
implies that

$\displaystyle T^{(2m-1)/m}\partial_{t}(\dot{T}T^{(1-m)/m})=-mm_{0}\;.$ (42)

This equation can equivalently be read as

$\displaystyle(1-m)\dot{T}^{2}+mT\ddot{T}=-m^{2}m_{0}\;,$ (43)

or more concisely, in terms of $K=T^{1/m}$, as

$\displaystyle\ddot{K}=-m_{0}K^{1-2m}\;.$ (44)

The corresponding $\alpha,\beta,\gamma$ and $\delta$ solutions are as follows:

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
T^{1/m}H_{2}\zeta\;,\quad\beta=WL\zeta\;,$ $\displaystyle\gamma$
$\displaystyle=$
$\displaystyle\frac{1}{m_{0}}WH_{0}\zeta_{\theta}+TH_{2}g\;,\quad(g_{\varphi\varphi}+g=0)$
(45) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{1}{m_{0}}WH_{0}\sigma_{\varphi}+TH_{2}(a\sin\theta+\cos\theta
g_{\varphi})\;,$

where $W=\dot{T}T^{1/m}/m=\dot{K}T$ and

$\displaystyle\sigma^{A}=T^{1/m}H_{2}\sigma\;,\quad\sigma^{B}=WL\sigma\;,\quad\sigma=a_{2}\cos\varphi-
a_{3}\sin\varphi\;.$

In view of (24) and equation (35); it is easy to verify that while condition
(15) yields,

$\displaystyle
m_{0}(\frac{H_{2}^{\prime}}{H_{0}^{\prime}}-m\frac{H_{2}}{H_{0}})=\frac{H_{0}}{H_{2}}\;,$
(46)

for $A=T^{1/m}H_{2}\zeta$ and $B=WL\zeta$, condition (16) yields an equation
that is just the derivative of (38). In writing $\gamma$ and $\delta$ of (37)
the condition (38) has been used.

###  IV.3  $m\neq 0$ Solutions

In this case, solutions are just a combination of the above two classes:
$\gamma$ and $\delta$ solutions are as in equations (37) but to the $\alpha$
and $\beta$ solutions of (37), one must add that given by (31). Equation (38)
also hold in this case. However, there are four metric constants $m,m_{0},k$
and $k_{0}$, and $T$ must satisfy both the equations (29) and (34) (or
equivalently (35) or (36)). A detailed investigation of the four metric
constants presented in Appendix A shows that for $H^{\prime}_{2}$ to be
nonzero, the following conditions must be satisfied:

$\displaystyle km=1\;,\quad m=1+m_{0}\ell^{2}\;,$

where $\ell$ is another metric constant. In fact, from equations (29), (35)
and the above relation, we also get $k_{0}=-mm_{0}$. To all these one must
also add relation (38).

In the case of (B) class solutions, we have six arbitrary real constants: $a$
and two constants determined by the function $g$ and three constants given by
$\zeta$. These mean that we have six linearly independent Killing vector
fields for class (B) which will be presented together with their Lie algebra
in the next subsection. In the case of (C), which includes (A) and (B)
solutions as special subcases, we have only one additional symmetry generator
given by (32). Although some other subcases of (C) can be defined, this case
will not be pursued any further as it has many additional conditions.

###  IV.4 Killing Vector Fields and Lie Algebra for the case (B)

In terms of the two nonzero constants $m$ and $m_{0}$ defined by (24), we have
obtained, in addition to three $so(3)$ 1-forms given by (5), three additional
linearly independent KY 1-forms:

$\displaystyle\omega_{1}$ $\displaystyle=$
$\displaystyle\cos\theta\omega-\frac{1}{m_{0}}WH_{0}\sin\theta e^{2}\;,$
$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\sin\theta\cos\varphi\omega+\frac{1}{m_{0}}WH_{0}(\cos\theta\cos\varphi
e^{2}-\sin\varphi e^{3})\;,$ (47) $\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle-\sin\theta\sin\varphi\omega-\frac{1}{m_{0}}WH_{0}(\cos\theta\sin\varphi
e^{2}+\cos\varphi e^{3})\;,$

where the 1-form $\omega$ is defined by

$\displaystyle\omega=T^{1/m}H_{2}e^{0}+WLe^{1}\;.$ (48)

By noting that the metric dual of $e^{0}$ is $-H_{0}^{-1}\partial_{t}$, in
terms of $K=T^{1/m}$ the vector field $X$ which is the metric dual of $\omega$
can be written as

$\displaystyle
X=\tilde{\omega}=-K\frac{H_{2}}{H_{0}}\partial_{t}+\dot{K}\frac{H_{2}}{H^{\prime}_{0}}\partial_{r}\;.$
(49)

The Killing vector fields corresponding to (39) are then given by

$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle\cos\theta X-MZ_{1}\;,$
$\displaystyle X_{2}$ $\displaystyle=$ $\displaystyle\sin\theta\cos\varphi
X+MZ_{2}\;,$ (50) $\displaystyle X_{3}$ $\displaystyle=$
$\displaystyle-\sin\theta\sin\varphi X-MZ_{3}\;,$

where $M=\dot{K}H_{0}/m_{0}H_{2}$ and the vector fields $Z_{i}$ are defined as

$\displaystyle Z_{1}=\sin\theta\partial_{\theta}\;,\quad
Z_{2}=\cos\theta\cos\varphi\partial_{\theta}-\frac{\sin\varphi}{\sin\theta}\partial_{\varphi}\;,\quad
Z_{3}=\cos\theta\sin\varphi\partial_{\theta}+\frac{\cos\varphi}{\sin\theta}\partial_{\varphi}\;.$
(51)

By making use of

$\displaystyle\;[Z_{1},Z_{2}]$ $\displaystyle=$ $\displaystyle
K_{2}\;,\quad\quad[Z_{1},Z_{3}]=-K_{1}\;,\quad[Z_{2},Z_{3}]=-K_{3}\;,$ (52)
$\displaystyle\;[Z_{1},K_{1}]$ $\displaystyle=$ $\displaystyle-
Z_{3}\;,\quad[Z_{1},K_{2}]=Z_{2}\;,\quad\quad[Z_{1},K_{3}]=0\;,$
$\displaystyle\;[Z_{2},K_{1}]$ $\displaystyle=$ $\displaystyle
0\;,\quad\quad\;[Z_{2},K_{2}]=-Z_{1}\;,\;\quad[Z_{2},K_{3}]=-Z_{3}\;,$ (53)
$\displaystyle\;[Z_{3},K_{1}]$ $\displaystyle=$ $\displaystyle
Z_{1}\;,\quad\;\;[Z_{3},K_{2}]=0\;,\quad\;\;\quad[Z_{3},K_{3}]=Z_{2}\;,$

we obtain

$\displaystyle\;[K_{1},X_{1}]$ $\displaystyle=$ $\displaystyle
X_{3}\;,\quad\quad[K_{1},X_{2}]=0\;,\quad\quad\quad\quad[K_{1},X_{3}]=-X_{1}\;,$
$\displaystyle\;[K_{2},X_{1}]$ $\displaystyle=$ $\displaystyle
X_{2}\;,\quad\quad[K_{2},X_{2}]=-X_{1}\;,\quad\quad\;\;[K_{2},X_{3}]=0\;,$
(54) $\displaystyle\;[K_{3},X_{1}]$ $\displaystyle=$ $\displaystyle
0\;,\quad\quad\;\;[K_{3},X_{2}]=-X_{3}\;,\quad\;\;\quad[K_{3},X_{3}]=X_{2}\;.$

On the other hand in terms of $s=X(M)+M^{2}$ we have

$\displaystyle\;[X_{1},X_{2}]$ $\displaystyle=$ $\displaystyle
sK_{2}\;,\quad[X_{1},X_{3}]=-sK_{1}\;,\quad[X_{2},X_{3}]=-sK_{3}\;.$ (55)

For evaluation of $s$, we should recall that $K$ obeys the equation (36). For
$m=1$ we have $K=T$ and $T\ddot{T}=-m_{0}$, which can be integrated to
$\dot{T}^{2}+2m_{0}\ln T=\ell_{0}$. For $m\neq 1$, it can also be easily
integrated once to obtain

$\displaystyle\dot{K}^{2}=-\frac{m_{0}}{1-m}K^{2(1-m)}+m_{2}\;,$ (56)

where $\ell_{0}$ and $m_{2}$ are integration constants determined by the
metric. We can therefore write

$\displaystyle s$ $\displaystyle=$
$\displaystyle-K\frac{H_{2}}{H_{0}}M_{t}+\dot{K}\frac{H_{2}}{H^{\prime}_{0}}M_{r}+M^{2}\;,$
$\displaystyle=$
$\displaystyle\frac{1}{m_{0}}[-K\ddot{K}+(1-m)\dot{K}^{2}]\;,$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\frac{1-m}{m_{0}}m_{2}\;,&\quad{\rm
for}\quad m\neq 1,\\\ 1\;,&\quad{\rm for}\quad m=1\;,\end{array}\right.$

where we have made use of (38) in the second line. When $m\neq 1$ and $s\neq
0$ the $X_{i}$ generator can be normalized with the same constant such that at
the right hand side of (47), there appear $\pm K_{j}$. Such a normalization
does not affect relations (46). But when the integration constant $m_{2}$ is
zero, the generator set $\\{X_{1},X_{2},X_{3}\\}$ form an abelian subalgebra.
In that case the symmetry algebra is isomorphic to $3D$ Euclidean algebra
$e(3)$. This shows how the symmetry of the metric may change for different
values of an integration constant.

##  V Maximal Symmetries : $\dot{T}=0,\;H_{0}^{\prime}\neq 0$ Solutions

Let us begin this case by considering $\alpha$ and $\beta$ as given in
equation (10), such that $A=(H_{0}^{\prime}/H_{1})\xi$ and the functions
$V,\;V_{i},\;u$ and $v_{i}$ depend only on $\tau=t/T$ for $\beta$ is
independent of $r$. Since $\dot{T}=0$ in this case, condition (13) requires
that $V=0$ and we can then write, from the first equation of (4):

$\displaystyle\alpha=U+\frac{H_{0}^{\prime}}{H_{1}}\xi\;,\quad\beta=\xi_{\tau}\;,$
(60)

where $\xi$ is given by (12) and $U_{\tau}=0$. The third equation of (4) now
provides us with

$\displaystyle U=c_{0}H_{0}\;,\quad\xi_{\tau\tau}+k_{1}\xi=0\;,$ (61)

where $c_{0}$ is an integration constant, and the metric constant $k_{1}$ is
defined by

$\displaystyle
k_{1}=\frac{H_{0}^{2}}{H_{1}}(\frac{H_{0}^{\prime}}{H_{0}H_{1}})^{\prime}\;.$
(62)

Note that the second equation of (51) is equivalent to three similar equations
for $u,v_{1}$ and $v_{2}$, each of which gives two linearly independent
solutions depending on the value of $k_{1}$.

Having completely specified $\alpha$ and $\beta$ with
$A=H_{0}^{\prime}\xi/H_{1}$ and $B=\xi_{\tau}$, we now turn to conditions (15)
and (16), which in this case amount to

$\displaystyle H_{0}H_{0}^{\prime}=k_{1}H_{2}H_{2}^{\prime}\;,\quad
P^{\prime}=-\frac{H_{1}}{H_{2}^{2}}.$ (63)

$k_{1}$ is a nonzero metric constant for $H_{0}^{\prime}\neq 0$. Both of these
relations can be integrated to

$\displaystyle H_{0}^{2}=k_{1}H_{2}^{2}+k_{2}\;,\quad
P^{2}=k_{3}+\frac{1}{H_{2}^{2}}\;,$ (64)

where $k_{2}$ and $k_{3}$ are integration constants. The first relation of
(54) implies that for $H_{0}^{\prime}\neq 0,\;k_{1}$ must be different from
zero. The conditions (52) and (53) also imply that

$\displaystyle P^{2}=\frac{H_{0}^{2}}{k_{2}H_{2}^{2}}\;,\quad
k_{1}=k_{2}k_{3}\;,$ (65)

which means for $H_{2}^{\prime}\neq 0,\;k_{2}$ and $k_{3}$ must be different
from zero as well.

If $k_{1}$ is a negative constant such that $k_{1}=-\kappa^{2}$ where $\kappa$
is a nonzero real number, we can write, in terms of integration constants
$c_{i}$, the $\xi$ solutions of (51) as

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle(c_{1}\cosh\kappa\tau+c_{2}\sinh\kappa\tau)\cos\theta+\sin\theta\sigma^{A}\;,$
$\displaystyle\sigma^{A}$ $\displaystyle=$
$\displaystyle(c_{3}\cosh\kappa\tau+c_{4}\sinh\kappa\tau)\cos\varphi-(c_{5}\cosh\kappa\tau+c_{6}\sinh\kappa\tau)\sin\varphi\;.$

Then $\gamma$ and $\delta$ can be determined from (14) and (17) as

$\displaystyle\gamma=\frac{H_{2}^{\prime}}{H_{1}}\xi_{\tau\theta}+TH_{2}g\;,\quad\delta=\frac{H_{2}^{\prime}}{H_{1}}\sigma^{A}_{\tau\varphi}+TH_{2}(a\sin\theta+\cos\theta
g_{\varphi})\;.$ (66)

The solutions (50) and (56) determine seven linearly independent KY $1$-forms
in addition to $so(3)$ solutions. The first one is $\omega_{0}=H_{0}e^{0}$
which corresponds to $K_{0}=-\partial/\partial t$. For $k_{1}=-\kappa^{2}$, we
can write these additional forms as follows:

$\displaystyle\omega_{1}$ $\displaystyle=$
$\displaystyle\cos\theta\psi_{1}-\kappa\frac{H_{2}^{\prime}}{H_{1}}\sinh\kappa\tau\sin\theta
e^{2}\;,\quad\;\;\;\omega_{2}=\cos\theta\psi_{2}-\kappa\frac{H_{2}^{\prime}}{H_{1}}\cosh\kappa\tau\sin\theta
e^{2}\;,$ $\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle\sin\theta\cos\varphi\psi_{1}+\kappa\frac{H_{2}^{\prime}}{H_{1}}\sinh\kappa\tau\phi_{1}\;,\;\;\quad\omega_{4}=\sin\theta\cos\varphi\psi_{2}+\kappa\frac{H_{2}^{\prime}}{H_{1}}\cosh\kappa\tau\phi_{1}\;,$
(67) $\displaystyle\omega_{5}$ $\displaystyle=$
$\displaystyle-\sin\theta\sin\varphi\psi_{1}-\kappa\frac{H_{2}^{\prime}}{H_{1}}\sinh\kappa\tau\phi_{2}\;,\quad\omega_{6}=-\sin\theta\sin\varphi\psi_{2}-\kappa\frac{H_{2}^{\prime}}{H_{1}}\cosh\kappa\tau\phi_{2}\;,$

where the $1$-forms $\psi_{i}$ and $\phi_{i},\;i=1,2$ are defined by

$\displaystyle\psi_{1}$ $\displaystyle=$
$\displaystyle\cosh\kappa\tau\frac{H_{0}^{\prime}}{H_{1}}e^{0}+\kappa\sinh\kappa\tau
e^{1}\;,\quad\phi_{1}=\cos\theta\cos\varphi e^{2}-\sin\varphi e^{3}\;,$
$\displaystyle\psi_{2}$ $\displaystyle=$
$\displaystyle\sinh\kappa\tau\frac{H_{0}^{\prime}}{H_{1}}e^{0}+\kappa\cosh\kappa\tau
e^{1}\;,\quad\phi_{2}=\cos\theta\sin\varphi e^{2}+\cos\varphi e^{3}\;.$

The vector fields $W_{i}=\tilde{\psi}_{i}$ and
$\bar{Z}_{i+1}=\tilde{\phi}_{i}$ are, in terms of $Z_{2}$ and $Z_{3}$ given by
(43), as follows:

$\displaystyle W_{1}$ $\displaystyle=$
$\displaystyle-\cosh\kappa\tau\frac{H_{0}^{\prime}}{TH_{0}H_{1}}\partial_{\tau}+\frac{\kappa}{TH_{1}}\sinh\kappa\tau\partial_{r}\;,\quad\bar{Z}_{2}=\frac{1}{TH_{2}}Z_{2}\;,$
(68) $\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle-\sinh\kappa\tau\frac{H_{0}^{\prime}}{TH_{0}H_{1}}\partial_{\tau}+\frac{\kappa}{TH_{1}}\cosh\kappa\tau\partial_{r}\;,\quad\bar{Z}_{3}=\frac{1}{TH_{2}}Z_{3}\;.$
(69)

It is also not difficult to verify that for $H_{2}=r$ we have, from (54)

$\displaystyle H_{0}^{2}=k_{1}r^{2}+k_{2}\;,\quad
H_{1}^{2}=\frac{1}{1+k_{3}r^{2}}\;,$

and $H_{0}H_{1}=1$ for $k_{2}=1$. In the case of $k_{1}=-1=k_{3}$ and
$k_{2}=1$, we recover the static form of the de Sitter metric (see Thirring
pp.492). KY 1-forms for five different forms of the de Sitter type space-times
are obtained in the next section.

##  VI $\rm{de}$ Sitter and Robertson-Walker Type Symmetries:
$H_{0}^{\prime}=0,\dot{T}\neq 0$

Since $\alpha_{t}=0$ in this case, it is convenient to start by defining the
constant

$\displaystyle\ell=T^{2}\partial_{t}\frac{\dot{T}}{T}\;.$ (70)

The nonzero and zero values of $\ell$ will then be considered separately. In
these two cases, $T$ is restricted to be a special function of time by the
symmetry requirement.

Case A considered below leads us to a family of de Sitter type space-times
with ten independent Killing vector fields and, depending on the values of
$\ell$ and other integration constants, four different time evolution regimes
can explicitly be specified by the symmetry requirement. The B case, specified
by $\ell=0$, corresponds to the best known form of de Sitter space-time, again
having ten independent Killing vector fields such that $T$ is an exponential
function of time. However, there is an important special case specified by
$\alpha=0$, and therefore $T$ is not restricted by any symmetry requirement.
This corresponds to the Robertson-Walker space-time with six dimensional
symmetry algebra in which they are the Einstein equations that give the time
dependence as shown in Table I.

###  VI.1 The case $\ell\neq 0$

We start with

$\displaystyle\alpha=U(r)+H_{2}\zeta\;,\quad\beta=\dot{T}Y(r)+B\;,$ (71)

where $\zeta$ is given by (33) and $B$ is defined as in Section III. Condition
(13) implies that $U=H_{0}PY$ and the second and third equations of (4) then
yield

$\displaystyle Y^{\prime}$ $\displaystyle=$ $\displaystyle
H_{1}PY\;,\quad\quad\quad\;Y=-\frac{H^{2}_{0}}{\ell H_{1}}(PY)^{\prime}\;,$
(72) $\displaystyle B_{r}$ $\displaystyle=$
$\displaystyle\dot{T}\frac{H_{1}H_{2}}{H_{0}}\zeta\;,\;\;T^{2}\partial_{t}\frac{B}{T}=-H_{0}H_{2}P\zeta\;.$
(73)

These are the reduced forms of equations (20) and (21). The first equation of
(62) gives $Y=c_{1}H_{2}$ and from the second equation we then obtain

$\displaystyle P^{\prime}=-H_{1}(P^{2}+\frac{\ell}{H_{0}^{2}})\;.$ (74)

The two equations of (63) imply that, in terms of

$\displaystyle\eta_{2}=b_{1}\cos\theta+\sin\theta\sigma^{b}\;,\;\;\sigma^{b}=b_{2}\cos\varphi-
b_{3}\sin\varphi\;,$ (75)

the most general solution for $B$ is of the form
$B=\dot{T}\eta_{1}(r,\theta,\varphi)+T\eta_{2}(\theta,\varphi)$. The second
equation of (63) specifies $\eta_{1}$ in terms of $\zeta$:

$\displaystyle\eta_{1}=-\frac{H_{0}}{\ell}H_{2}P\zeta\;,$ (76)

and the first equation yields nothing but condition (64).

Having completely specified $\alpha$ and $\beta$ with $A=H_{2}\zeta$ and

$\displaystyle
B=-\frac{H_{0}}{\ell}\dot{T}H_{2}P\zeta+T\eta_{2}=\sin\theta\sigma^{B}+\cos\theta(b_{1}T-\frac{H_{0}}{\ell}\dot{T}H_{2}P\sigma)\;,$
(77)

such that

$\displaystyle\sigma^{B}=T\sigma^{b}-\frac{1}{\ell}\dot{T}H_{0}H_{2}P\sigma\;,$
(78)

the condition (14) amounts to

$\displaystyle P^{2}=-\frac{\ell}{H_{0}^{2}}+\frac{1}{H_{2}^{2}}\;.$ (79)

It is not difficult to verify that condition (64) is implied by (69), which
also yields $P^{\prime}=-H_{1}/H_{2}^{2}$. (Note that the integration of (64)
results in an additional constant multiplying the second term of (69)). In
view of this relation, condition (16) is identically satisfied. That is, we
only have condition (69).

We can now turn to (14) and (17) to evaluate $\gamma$ and $\delta$, and the
solutions can be collated as follows:

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
c_{1}H_{0}H_{2}P+H_{2}\zeta\;,$ $\displaystyle\beta$ $\displaystyle=$
$\displaystyle
c_{1}\dot{T}H_{2}+\sin\theta\sigma^{B}+\cos\theta(b_{1}T-\frac{H_{0}}{\ell}\dot{T}H_{2}P\sigma)\;,$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle-\dot{T}\frac{H_{2}^{2}}{H_{0}}\zeta_{\theta}+\dot{T}H_{2}P[\cos\theta\sigma^{B}-\sin\theta(b_{1}T-\frac{H_{0}}{\ell}\dot{T}H_{2}P\sigma)]+TH_{2}g\;,$
(80) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle-\dot{T}\frac{H_{2}}{H_{0}}\sigma_{\varphi}+H_{2}P\sigma^{B}_{\varphi}+TH_{2}(a\sin\theta+\cos\theta
g_{\varphi})\;.$

Note that for $H_{0}=1$ and $H_{2}=r$, condition (69) yields

$\displaystyle H^{2}_{1}=\frac{1}{1-\ell r^{2}}\;.$ (81)

These are the characteristics of the de Sitter and Robertson-Walker space-
times. It should be emphasized that the right hand side of condition (69) must
be positive, which reflects the fact that $1/r^{2}>\ell$ is required to avoid
any singularity in the corresponding space-times.

We now turn to the explicit evolution of $T$. By multiplying both sides of
equation (60) by $\dot{T}/T^{3}$, it can be integrated to

$\displaystyle\dot{T}=\epsilon(\ell_{3}T^{2}-\ell)^{1/2}\;,$ (82)

and then by integrating once more we get

$\displaystyle
T=\left\\{\begin{array}[]{ll}(\frac{\ell}{\ell_{3}})^{1/2}\cosh(\epsilon\ell_{3}^{1/2}t+a)\;;&\;\;{\rm
for}\;\ell_{3}>0\;,\;\ell>0\;,\\\
\frac{\ell_{0}}{\ell_{3}^{1/2}}\sinh(\epsilon\ell_{3}^{1/2}t+a)\;;&\;\;{\rm
for}\;\ell_{3}>0\;,\;\ell=-\ell_{0}^{2}<0\;,\\\
\frac{\ell_{0}}{k_{0}}\sin(\epsilon k_{0}t+a)\;;&\;\;{\rm
for}\;\ell_{3}=-k_{0}^{2}<0\;,\;\ell=-\ell_{0}^{2}<0\;,\\\
\epsilon\ell_{0}t+a\;;&\;\;{\rm
for}\;\ell_{3}=0,\;\ell=-\ell_{0}^{2}<0\;,\end{array}\right.$ (87)

where $\ell_{3}$ and $a$ are integration constants and $\epsilon=\pm 1$. Note
that the third solution can be inferred from the second one and that all the
corresponding KY 1-forms can explicitly be read from (70). All of these de
Sitter space-times of which the fourth one is a flat space-time, are spaces of
constant curvature with the curvature scalar given by $\ell_{3}$ (see Appendix
B).

###  VI.2 de Sitter type Symmetries: $\ell=0=H_{0}^{\prime}$

The condition $\ell=0$ is equivalent to $\dot{T}=\lambda T$, that is to

$\displaystyle T=T_{0}\exp(\lambda t)\;,$

where $\lambda$ is a metric constant and $T_{0}$ is an integration constant.
In this case, to avoid excessive repetitions, we shall be content to present
the solutions:

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
c_{0}H_{0}+\frac{1}{H_{0}P}\zeta\;,$ $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\dot{T}[\frac{c_{0}}{P}+\frac{1}{2}(\frac{1}{\dot{T}^{2}}+\frac{1}{P^{2}})\zeta+b_{1}\cos\theta+\sin\theta\sigma^{b}]\;,$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\dot{T}H_{2}P\\{[-\frac{1}{H_{0}^{2}P^{2}}+\frac{1}{2}(\frac{1}{\dot{T}^{2}}+\frac{1}{P^{2}})]\zeta_{\theta}-b_{1}\sin\theta+\cos\theta\sigma^{b}\\}+TH_{2}g\;,$
(88) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle\dot{T}H_{2}P\\{[-\frac{1}{H_{0}^{2}P^{2}}+\frac{1}{2}(\frac{1}{\dot{T}^{2}}+\frac{1}{P^{2}})]\sigma_{\varphi}+\sigma_{\varphi}^{b}\\}+TH_{2}(a\sin\theta+\cos\theta
g_{\varphi})\;,$

which provide us with ten-dimensional symmetry algebra. Here, $\sigma^{b}$ is
given by (65). The above solutions can be easily verified provided that the
condition $H_{2}^{\prime}=\epsilon H_{1}$ holds, which implies that
$H_{2}P=\epsilon$ and hence, $P^{\prime}=-H_{1}P^{2}$.

###  VI.3 The Case $\alpha=0$: Robertson-Walker Symmetries

When $\alpha=0$, the second and third equations of (4) imply that
$\beta_{r}=0$ and

$\displaystyle\beta=B=T(\sin\theta\sigma+a_{3}\cos\theta)\;,$ (89)

with $\sigma=\sigma^{B}/T=a_{1}\cos\varphi-a_{2}\sin\varphi$. In that case,
eq. (15) is identically satisfied and eq. (16) gives
$P^{\prime}=-H_{1}/H^{2}_{2}$, which can be integrated to yield the second
relation of eq. (54) and provide us with

$\displaystyle H^{2}_{1}=\frac{1}{1+k_{3}r^{2}}\;,$ (90)

for $H_{0}=1$ and $H_{2}=r$. Then the solutions can be easily read from (14)
and (17) to be

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
T\frac{H^{\prime}_{2}}{H_{1}}(\cos\theta\sigma-a_{3}\sin\theta)+TH_{2}g\;,$
$\displaystyle\delta$ $\displaystyle=$ $\displaystyle
T\frac{H^{\prime}_{2}}{H_{1}}\sigma_{\varphi}+TH_{2}(a\sin\theta+\cos\theta
g_{\varphi})\;.$

With $\alpha=0$ and $\beta$ as in (75), the following Killing vector fields
are obtained :

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{H_{1}}\sin\theta\cos\varphi\partial_{r}+PZ_{2}\;,$
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{H_{1}}\sin\theta\sin\varphi\partial_{r}-PZ_{3}\;,$
(91) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{H_{1}}\cos\theta\partial_{r}-PZ_{1}\;.$

These are the generalized translation generators which, in the usual cartesian
coordinates read, respectively, as
$H_{1}^{-1}\partial_{x}\;,-H_{1}^{-1}\partial_{y}$ and
$H_{1}^{-1}\partial_{z}$. Together with the $so(3)$ generators, they close
into the following Lie algebra structure:

$\displaystyle\;[I_{1},I_{2}]$ $\displaystyle=$ $\displaystyle
k_{3}K_{3}\;,\quad\;\quad[I_{2},I_{3}]=k_{3}K_{1}\;,\quad[I_{1},I_{3}]=-k_{3}K_{2}\;,$
$\displaystyle\;[K_{1},I_{2}]$ $\displaystyle=$ $\displaystyle-
I_{3}=[K_{2},I_{1}]\;,\;\;[K_{1},I_{3}]=I_{2}=-[K_{3},I_{1}]\;,$ (92)
$\displaystyle\;[K_{2},I_{3}]$ $\displaystyle=$ $\displaystyle
I_{1}=[K_{3},I_{2}]\;,\quad\;\;[K_{i},I_{i}]=0\;,\quad i=1,2,3\;.$

##  VII Flat Space-Time Solutions: $\dot{T}=0,\;H_{0}^{\prime}=0$

In this case, $\alpha$ is independent of $\tau=t/T$, $\beta$ is independent of
$r$, and condition (13) requires that $V=0$. Therefore it is convenient to
start with $\alpha=c_{0}H_{0}+\eta_{1}$ and $\beta=\eta_{2}$, where $c_{0}$ is
a constant and $\eta_{i}$ are as in (65) of the previous section, such that
$\eta_{1}$ is independent from $\tau$ and $\eta_{2}$ is independent from $r$.
The third equation of (4) gives $\eta_{2\tau}=-H_{0}\eta_{1r}/H_{1}$. Since
the left hand side of this equality depends on $\tau$ and is independent of
$r$, but the right hand side depends on $r$ and is independent of $\tau$, both
sides must be equal

$\displaystyle\eta_{2\tau}=\zeta=-\frac{H_{0}}{H_{1}}\eta_{1r}\;,$ (93)

where $\zeta$ is given by (33). From the first equality, we obtain
$\eta_{2}=\tau\zeta+\zeta^{(b)}$ where $\zeta^{(b)}$ is the same as $\zeta$
with $a=(a_{1},a_{2},a_{3})$ replaced by $b=(b_{1},b_{2},b_{3})$. On the other
hand, for $A=\eta_{1}$ and $B=\tau\zeta+\zeta^{(b)}$, conditions (15) and (16)
respectively yield

$\displaystyle\eta_{1\theta}=-\frac{H_{2}^{2}P}{H_{0}}\zeta_{\theta}\;,\quad
P^{\prime}=-\frac{H_{1}}{H_{2}^{2}}\;.$ (94)

In view of the second equality of (79), we obtain
$(H_{2}^{2}P)^{\prime}=H_{1}$ from the integrability condition $\eta_{1\theta
r}=\eta_{1r\theta}$. From the second equation of (79) and
$(H_{2}^{2}P)^{\prime}=H_{1}$, we get $P^{2}=1/H_{2}^{2}$, which can also be
obtained from the integration of $P^{\prime}=-H_{1}/H_{2}^{2}$ with the zero
integration constant (see the discussion following equations (53) and (54)).

From (14) and (17), one can easily write out $\gamma$ and $\delta$ together of
$\alpha$ and $\beta$ to be as :

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
c_{0}H_{0}-\frac{1}{H_{0}P}\zeta\;,\quad\beta=\tau\zeta+\zeta^{(b)}\;,$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
H_{2}P(\tau\zeta+\zeta^{(b)})+TH_{2}g\;,$ (95) $\displaystyle\delta$
$\displaystyle=$ $\displaystyle
H_{2}P(\tau\sigma_{\varphi}+\sigma^{(b)}_{\varphi})+TH_{2}(a\sin\theta+\cos\theta
g_{\varphi})\;.$

We have ten linearly independent KY 1-forms and only one condition
$P^{2}=1/H_{2}^{2}$ which is equivalent to $H_{2}^{\prime 2}=H_{1}^{2}$. For
$H_{2}=r$, we have $H_{1}=\pm 1$, and the corresponding Killing vector fields
are directly determined from (81) with $P=\pm 1/r$.

###### Acknowledgements.

This work was supported in part by the Scientific and Technical Research
Council of Turkey (TÜBİTAK).

##  Appendix A Metric Constants

In Section IV, the following four metric constants were defined :

$\displaystyle m$ $\displaystyle=$ $\displaystyle
L^{\prime}\frac{H_{0}}{H_{1}H_{2}}\;,\;\;m_{0}=\frac{H^{2}_{0}}{LH_{1}}(\frac{H_{2}}{H_{0}})^{\prime}\;,$
(96) $\displaystyle k$ $\displaystyle=$
$\displaystyle\frac{H^{\prime}_{0}H_{2}}{H_{0}H_{2}^{\prime}}\;,\;\;\quad
k_{0}=-\frac{H_{0}^{2}}{H_{1}H_{2}}(\frac{H_{2}^{\prime}}{H_{1}})^{\prime}\;.$
(97)

The first relation of (A1) can be integrated to find

$\displaystyle H^{\prime}_{0}=\ell_{1}H^{-m}_{0}H_{1}H_{2}\;,\quad
L=\frac{1}{\ell_{1}}H_{0}^{m}\;,$ (98)

which is also valid for $m=0$. The second relation of (A1) can be arranged as

$\displaystyle\frac{H_{2}^{\prime}}{H_{2}}=\frac{H^{\prime}_{0}}{H_{0}}[m_{0}(\frac{H_{1}}{H^{\prime}_{0}})^{2}+1]\;.$
(99)

The first relation of (A2) can also be integrated to find
$H_{0}=\ell_{2}H_{2}^{k}$, where $\ell_{1}$ and $\ell_{2}$ are also non-zero
metric constants. By combining (A4) with the first relation of (A2), we get

$\displaystyle(1-k)H^{\prime\;2}_{0}=m_{0}kH^{2}_{1}\;,$ (100)

and then, by (A3) and $H_{0}=\ell_{2}H_{2}^{k}$, we arrive at

$\displaystyle(1-k)=m_{0}k(\frac{\ell_{2}^{m}}{\ell_{1}})^{2}H^{2(mk-1)}_{2}\;.$
(101)

Thus for nonconstant $H_{2}$, we must have

$\displaystyle km=1\;,\quad m=1+m_{0}(\frac{\ell_{2}^{m}}{\ell_{1}})^{2}\;.$
(102)

Note that $k=1$ if and only if $m_{0}=0$ and if and only if $m=1$ for
$H_{2}^{\prime}\neq 0$.

##  Appendix B Curvature Forms

The curvature 2-forms $R_{ab}=d\omega_{ab}+\omega_{ac}\wedge\omega^{c}_{\;b}$
for the considered class of spherically symmetric space-times are computed by
using eqs. (2) and (3) to be as follows :

$\displaystyle R_{01}$ $\displaystyle=$
$\displaystyle(S+\frac{h_{0}^{2}+h_{0}^{\prime}-h_{0}h_{1}}{T^{2}H_{1}^{2}})e^{01}\;,\;\;R_{12}=S_{2}e^{02}+S_{1}e^{12}\;,$
$\displaystyle R_{02}$ $\displaystyle=$
$\displaystyle(S+\frac{h_{0}h_{2}}{T^{2}H_{1}^{2}})e^{02}+S_{2}e^{12}\;,\quad
R_{13}=S_{2}e^{03}+S_{1}e^{13}\;,$ (103) $\displaystyle R_{03}$
$\displaystyle=$
$\displaystyle(S+\frac{h_{0}h_{2}}{T^{2}H_{1}^{2}})e^{03}+S_{2}e^{13}\;,\quad
R_{23}=(S_{1}+\frac{H_{1}P^{\prime}+(H_{1}/H_{2})^{2}}{T^{2}H_{1}^{2}})e^{23}\;,$

where

$\displaystyle S=-\frac{\ddot{T}}{TH_{0}^{2}}\;,\quad
S_{1}=\frac{\dot{T}^{2}}{T^{2}H_{0}^{2}}-\frac{P^{\prime}+H_{1}P^{2}}{T^{2}H_{1}}\;,\quad
S_{2}=\frac{\dot{T}}{T^{2}}\frac{h_{0}}{H_{0}H_{1}}\;.$ (104)

Note that $S=0=S_{2}$ when $\dot{T}=0$. The corresponding Ricci 1-forms and
curvature scalar can be found from $P_{b}=i_{X^{a}}R_{ab}$ and
$\Re=i_{X^{b}}P_{b}$.

For $\dot{T}=0=H_{0}^{\prime}$ and $H_{2}^{\prime 2}=H_{1}^{2}$ we have
$S_{2}=0$ in addition to $S=0=S_{1}$. It then follows that all the curvature
components vanish, and we obtain the Minkowski space-time. In the case of the
static form of de Sitter space-time, that is for $\dot{T}=0,\;H_{2}=r$ and

$\displaystyle
H_{0}^{2}=k_{1}r^{2}+k_{2}\;,\;H_{1}^{2}=\frac{1}{1+k_{3}r^{2}}\;,\;k_{1}=k_{2}k_{3}$

we have $S=0=S_{2}$ and $S_{1}=-k_{3}/T^{2}$ which gives the constant
curvature solutions

$\displaystyle R_{0j}=\frac{k_{3}}{T^{2}}e^{0j}\;,\quad
R_{ij}=-\frac{k_{3}}{T^{2}}e^{ij}\;,$ (105)

where $i,j=1,2,3$ and $i<j$. These two space-times are maximally symmetric
with constant curvature.

For $\dot{T}=\lambda T\;,H_{0}^{\prime}=0$ and $H_{2}^{\prime}=\epsilon H_{1}$
which indicate de Sitter space-time, we have

$\displaystyle S=-\frac{\lambda^{2}}{H_{0}^{2}}=-S_{1}\;,\quad S_{2}=0\;.$

These lead us again to the curvature solution given by (B3) provided that
$k_{3}/T^{2}$ is replaced by $-\lambda^{2}/H_{0}$. For the Robertson-Walker
space-time, we have

$\displaystyle R_{0j}=-\frac{\ddot{T}}{T}e^{0j}\;,\quad
R_{ij}=[(\frac{\dot{T}}{T})^{2}+\frac{k}{T^{2}}]e^{ij}\;.$ (106)

Finally, for the four forms of de Sitter space-time found in Section VI, we
have

$\displaystyle H_{0}=1\;,\quad H^{2}_{1}=\frac{1}{1-\ell r^{2}}\quad
H_{2}=r\;,$

and $T$ expressions are explicitly given by equations (73). For these values
we find $S_{2}=0$ and $S_{1}=\ell_{3}=-S$, where $\ell_{3}$ is a constant.
Thus

$\displaystyle R_{0j}=-\ell_{3}e^{0j}\;,\quad R_{ij}=\ell_{3}e^{ij}\;.$ (107)

where we have used $\dot{T}=\epsilon(\ell_{3}T^{2}-\ell)^{1/2}$ found in
equation (72).

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Table 1:  Metric coefficient functions and the numbers of linearly independent
KY-forms of some well-known spherically symmetric space-times. For all these
cases $H_{2}$ is the radial coordinate $r$. The numbers of the fifth column
represent the dimensions $d(4,1)$ of the corresponding symmetry algebras whose
common part consists of three $so(3)$ generators given by the equation (6) of
the main text. The last two columns denote the numbers $d(4,2)$ and $d(4,3)$
of the linearly independent KY 2-forms and 3-forms which are explicitly
calculated in the accompanying paper. The fifth rows represent three different
forms of the de Sitter space-time, of which the third consists of four cases
with different time evolutions given in Section VI. The numbers $d(n,p)$ for
the maximally symmetric space-times, such as those of Minkowski and de Sitter,
represent the upper bounds for dimension $n=4$.
Space-time |  $T$ |  $H_{0}$ |  $H_{1}$ | $d(4,1)$ | $d(4,2)$ | $d(4,3)$
---|---|---|---|---|---|---
Schwarzschild |  $1$ |  $\sqrt{1-\frac{2M}{r}}$ |  $\frac{1}{\sqrt{1-\frac{2M}{r}}}$ | 4 | 1 | 0
Reissner-Nordstrøm |  $1$ |  $\sqrt{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}$ |  $\frac{1}{\sqrt{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}}$ | 4 | 1 | 0
Robertson-Walker | $T(t)$ | $1$ | $\frac{1}{\sqrt{1-kr^{2}}}$ | 6 | 4 | 1
de Sitter |  |  |  |  |  |
static form |  $1$ |  $\sqrt{1-kr^{2}}$ |  $\frac{1}{\sqrt{1-kr^{2}}}$ | 10 | 10 | 5
usual form | $e^{(\Lambda/3)^{1/2}t}$ | 1 | 1 | 10 | 10 | 5
four additional forms |  $\dot{T}^{2}=\ell_{3}T^{2}-\ell$ |  $1$ |  $\frac{1}{\sqrt{1-\ell\;r^{2}}}$ | 10 | 10 | 5
Minkowski |  $1$ |  $1$ |  $1$ | 10 | 10 | 5