Geometrical qualities of the generalised Schwarzschild spacetimes

Geometrical qualities of the generalised Schwarzschild spacetimes
Orchidea Maria Lecian
Sapienza University of Rome,
Rome, Italy.
International Cosmology, Astronomy and Astrophysics
Conference (ICAA-2023)
Virtual Meeting
07 December 2023
International Cosmology, Astronomy and Astrophysics Conference (ICAA-2023) Virtual Meeting
Geometrical qualities of the generalise

Abstract
The geoemtrical qualities of the generalised Schwarzschild spacetimes are
newly studied. The method of the infinite-redshift surfaces is newly
applied; more in detail, the analytical radii are found, after which the new
analytical expressions of the physical horisons are reconducted.
The parameter spaces of the models are newly analysed and newly
constrained.
In the case of the Nariai spacetime, the Schwarzschild radius and the
(positive) cosmological constant are found to obey new analytical
constraints which refine the results of [S.A. Hayward, K. Nakao, T.
Shiromizu, A cosmological constant limits the size of black holes, Phys.
Rev. D 49, 5080(1994)].
The further cases of the Schwarzschild-antideSitter spacetime, the
generalised Schwarzschild spacetimes with a linear term, the generalised
Schwarzschild spacetimes with a linear term and a cosmological constant,
and the Mannheim-Kazanas-inspired spacetimes are newly studied
accordingly.
The coordinates-singularity-avoiding coordinates extensions are newly
analytically found.

The applied analytical methods allow one to newly analytically
demonstrate that, in the generalised Schwarzschild spacetimes, the
allowed regions of the parameter spaces are those in which the role of the
Schwarzschild radius is modified only slightly.
The weak-field limits are studied.
The embedding diagrams are provided with.
The Astrophysical characterisation is proposed.
The quantum regimes are envisaged.
The geometrical objects are calculated.
[O.M. Lecian, in preparation.]

Summary
Generalised Schwarzschild spacetimes:
• methodology:
- study of the infinite-redshift surface grr = 0;
- study of the physical horisons;
- study of the coordinates-singularity-avoiding coordinates extensions.
• generalised Schwarzschild spacetimes:
- generalised Schwarzschild spacetimes with a cosmological-constant term,
- generalised Schwarzschild spacetimes with a linear term,
- generalised Schwarzschild spacetimes with a linear term and a cosmological-constant
term,
- Mannheim-Kazanas-isnpired spacetimes.
• geometrical objects;
• weak-field limit.
• quantum regime.
• cosmological implementation;
• Astrophysical characterisation.
• new constraints on the parameter spaces.

Introduction
The geometrical qualities of the generalised Schwarzschild spacetimes are
analysed, as well as those of the Mannheim-Kazanas-inspired spacetimes.
The methods of the infinite-redshift-surfaces analysis allow one to acquire
new items of information about the analytical expressions of the radii
characterising the spacetimes. The analytical expressions of the physical
horisons are newly provided with.
The coordinates-singularity-avoiding coordinates extensions allow one to
retrieve information about the geometrical features of the spacetimes.
New constraints on the parameter spaces are found, from which the
analytical constraints on the parameters which qualify the generalised
spacetimes are newly established.

The quantum regimes allows one to envisage the quantum features of the
blackholes.
The weak-field limit allows one to introduce some of the features of the
Astrophysical characterisations.
A complete Astrophysical characterisation of the spacetimes ensures the
direct physical interpretation of the schemes, and allows for the scrutiny
of the blackhole shadows, from which the further physical aspects are
studied.

The infinite-redshift-surfaces method
the infinite-redshift surfaces are defined after the solution(s) of
grr = 0
- usually aimed at the characetrisation of the redshift functions of the
corresponding blackholes.
D. Gregoris, Y.C. Ong, B. Wang, A critical assessment of black hole solutions with a
linear term in their redshift function, Eur. Phys. J. C 81, 684 (2021).

The Schwarzschild-deSitter spacetimes
ds2
= c2

1 −
rs
r
−
Λ
3
r2

dt2
−
dr2
1 − rs
r − Λ
3 r2
− r2
dθ2
− r2
(sinθ)2
dϕ2
M ≤
p
1/9Λ
Hayward S. A., Nakao K., Shiromizu T. A cosmological constant limits the size of
black holes. Phys. Rev. D 1994, 49, 5080–5085.

parameterisation of the analytical radii
R1 =
2
√
Λ
cos
θ̃
3
,
R2 =
2
√
Λ
cos

θ̃
3
+
2π
3
#
,
R3 =
2
√
Λ
cos

θ̃
3
+
4π
3
#
,
with θ = −3
√
ΛM
S. Akcay, R.A. Matzner, The Kerr-de Sitter Universe, Class. Quantum Grav. 28,
085012 (2011).

Nariai coordinates extension
coordinates-singularity-avoiding coordinates extension
∂v
∂t
=
∂u
∂ρ
,
∂u
∂t
=
∂v
∂ρ
,
u = ekρ/T
ch
kt
T
,
v = ekρ/T
sh
kt
T
,
ρ = r + 2Mln
r
2M
− 1

,
with T−1
=
p
Λ/3
u2
+ v2
= e2kρT
Λ
k = 32M2
H. Nariai, On the Kruskal-Type Representation of Schwarzschild-de Sitter’s
Spacetime, Hiroshima Univ., Takehara (Japan). Research Inst. for Theoretical
Physics, Report number: RRK 86-13 (1986).

The Schwarzschild spacetimes with a
cosmological-constant term
ds2
= c2
1 − rs
r − k2r2

dt2
− 1
(1− rs
r −k2r2
)
dr2
− r2
dθ2
− r2
(sin θ)2
dϕ2
analytical radii
r1 =
1
6
(βk2
2 )(
1/3)
k2
+
2
(βk2
2 )1/3
,
r2 = −
1
12
(βk2
2 )(
1/3)
k2
−
1
(βk2
2 )1/3
+ i
√
3
2

1
6
(βk2
2 )1/3
k2
−
2
(βk2
2 )1/3

,
r3 = −
1
12
(βk2
2 )(
1/3)
k2
−
1
(βk2
2 )1/3
− i
√
3
2

1
6
(βk2
2 )1/3
k2
−
2
(βk2
2 )1/3

,
β ≡ 12
√
3
r
α
k2
− 108rs
α ≡ 27k2r2
s − 4

existence of the analytical radii
• 27k2r2
s −4
k2 ≥ 0
the results from [S.A. Hayward, K. Nakao, T.Shiromizu, A cosmological
constant limits the size of black holes, Phys. Rev. D 49, 5080 (1994)] are
therefore since here refined;
• r1 is found to be well-defined in the new intervals
− ∞ k2 0,
k2
4
27r2
s
.
• denominators be different from zero
12
√
3
q
27k2r2
s −4
k2
− 108rs ̸= 0.
• request of the denominator of the function multiplying the imaginary
unit be different from zero requires
k2
2

√
3
q
27k2r2
s −4
k2
− 9rs

̸= 0.

physical horisons
function multiplying the imaginary unit vanishes:
ra =
1
6
(βk2
2 )(1/3)
k2
+
2
(βk2
2 )1/3
,
rb = −
1
12
(βk2
2 )(1/3)
k2
−
1
(βk2
2 )1/3
.

comparison with eneralised Schwarzschild spacetimes with a
linear term and a cosmological-constant term
r1 =
1
6
B1/3
k2
+
2
3
+3k2
2
k2B1/3
,
r2 = −
1
12
B1/3
k2
−
1
3
3k2
k2B1/3
+ i
√
3
2

B1/3
6k2
−
2
3
3k2
k2B1/3

,
r3 = −
1
12
B1/3
k2
−
1
3
3k2
B1/3
− i
√
3
2

B1/3
6k2
−
2
3
3k2
k2B1/3

,
B ≡ 12
√
3
p
+27k2rs − 4k2k2 − 108k2
2 rs.

Generalised Schwarzschild spacetimes with a linear
term
ds2 = c2 1 − rs
r − k1r

dt2 − 1
(1−rs
r
−k1r)
dr2 −r2dθ2 −r2(sin θ)2dϕ2
analytical radii
r1 =
1
2
1 +
√
1 − 4k1rs
k1
,
r2 =
1
2
1 −
√
1 − 4k1rs
k1
.
existence of the radii
1 − 4k1rs 0

Coordinates-singularity-avoiding coordinates extension
• new expression of the new radial coordinate ρ
ρ − ρ0 ≃
r + rs ln

r
rs
− 1

(1 − k1r)

1 − r
rs

(1 + k1r)
;
• radial coordinate ρ newly rewritten as
ρ = r + rs ln

r
rs
− 1

+ O(r2
) + O(k2
1 ) + R3 + R4,
with O(r2
) + O(k2
1 ) the remainders of the series expansions;
the further remainders are
R3 = O r6
s
ln r−rs
rs
r − rs
!
;
R4 = O

r5
s ln
r − rs
rs

.
• new initial value of the new radial coordinate ρ newly found
ρ0 = 0.

• new expression of T newly found
T−1
=
1
2
rs2
k2
1
1 − k2
1 r2
s
i.e. such that its expansions respects the orders of infinitesimals of r2
in
the coordinates extensions.
• new further constraint on the linear r from the expression of
the new radial coordinate is newly found
k1rs
1 − k112r2
s
≡ 1.

new qualities of k1
• here k1 newly fixed after the requests imposed on the new radial
variable. The coordinates extensions therefore hold after having provided
that the modification(s) of the Schwarzschild radius be small.
• the set the coordinates extension allows one to find the physical
horison without posing k1 = 0:
from zero − th order in r of the coordinates extension,
• new condition newly found
(r − rs)(1 − k1r) ̸= 0
• the role of the Schwarzschild radius is newly modified.

Generalized Schwarzschild spacetimes with a
linear term and a cosmological constant
ds2 = c2 1 − rs
r − k1r − k2r2

dt2+
− 1
(1−rs
r
−k1r−k2r2
)
dr2 − r2dθ2 − r2(sin θ)2dϕ2
analytical radii
r1 =
1
6
b1/3
k2
+
2
3
k2
1 + 3k2
2
k2b1/3
−
1
3
k1
k2
,
r2 = −
1
12
b1/3
k2
−
1
3
k2
1 + 3k2
k2b1/3
−
1
3
k1
k2
+ i
√
3
2

b1/3
6k2
−
2
3
k2
1 + 3k2
k2b1/3

,
r3 = −
1
12
b1/3
k2
−
1
3
k2
1 + 3k2
b1/3
−
1
3
k1
k2
− i
√
3
2

b1/3
6k2
−
2
3
k2
1 + 3k2
k2b1/3

.
b ≡
12
√
3
p
4rsk3
1 − k2
1 + 18k1k2rs + 27k2rs − 4k2k2 −8k3
1 −108k2
2 rs −36k1k2

• denominator different from zero:
new constraints of the Schwarzschild radius
rs ̸=
1
54k2
2

4
35
√
3
q
−105k2
1 − 315k2

k2
1 + 3k2
2
3

− 4k3
1 − 18k1k2

,
rs ̸=
1
54k2
2

−
4
35
√
3
q
−105k2
1 − 315k2

k2
1 + 3k2
2
3

− 4k3
1 − 18k1k2

,
⇒ new condition on the parameters k1 and k2
k2 ̸= −
1
3
k2
1 .

physical horisons
• discussion of r2 and of r3

12
√
3
q
4rs k3
1 − k2
1 + 18k1k2rs + 27k2rs − 4k2k2 − 8k
3
1 − 108k
2
2 rs − 36k1k2
2
3
− 4k
2
1 − 12k2 = 0,
⇒ the new constraint on the Schwarzschild radius is newly found
rs = −
1
27
2(k2
1 + 3k2)3/2
+ 2k3
1 + 9k1k2
k2
2
.

Coordinates-singularity avoiding coordinates extension
• hypotheses that the cosmological-constant term should modify the
Schwarzschild term not strongly, and that the the k1 term induce a
modification of next order;
• new differential
dρ ≡
dr
(1 + k1r + k2r2) 1 − rs
r

(1 + k1r − k2r2)
integrates as the
• new radial variable with new initial value newly found
ρ − ρ0 ≃
1
4
rs (k1 + 2k2rs )(k2
1 − k2 + k1k2rs)
k2(k2r2
s + k1rs + 1)2
+ r + rs ln

r
rs
− 1

+ R1(k1, k2, rs ) + R2(k1, k2, rs ).
with the remainders
R1(k1, k2, rs ) ≡ O(k2
2 ),
R2 ≃
P
i fi k1, k2, rs , 1
q
k2
1
−4k2
!
O(r2
)
with fi the pertinent set of functions.

• new condition k2r2
s + k1rs + 1 ̸= 0, that is
k2 ̸= −
1 + k1rs
r2
s
.
• from the new differential for the radial variable, new constraints on rS
newly found as
1
4
rs
k1
k2
k2
1 − k2 + rsk1k2
1 + k1rs + k2r2
s
= 1;
• new definition of T newly found found as
T−1
= 8
k2
rs
1 + rsk1 + r2
s k2
(k2
1 − 2k2)(k2
1 − k2 + rsk1k2)
⇒ new conditions on the parameter space newly found.

Experimental validation: data analyses
• The features of the terms corresponding to the fluid, which are worked
out from the linear-term effects, have been scrutinised within the
experimental framework of COBE-Planck for the deSitter case;
A. Sarkar and B. Ghosh, Constraining the quintessential α-attractor inflation through
dynamical horizon exit method, Phys. Dark Univ. 41, 101239 (2023).
• The potentiality of the determinations of the values of the linear term
and of that of the cosmological constant have been recently envisaged in
the observational analyses of CMB, gravitational waves, dark matter and
dark radiation form string cosmology.
M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo and I. Zavala,
String Cosmology: from the Early Universe to Today, [arXiv:2303.04819 [hep-th]].

Mannheim-Kazanas-inspired spacetimes
ds2 = c2

1 − β(2−3βγ)
r − 3βγ + γr − kr2

dt2+
− 1

1−
β(2−3βγ)
r
−3βγ+γr−kr2
dr2 − r2dθ2 − r2(sinθ)2dϕ2
analytical radii
r1 =
1
3
γ
k
+
2
3
γ2
+ 3k − 9βγk
F1/3
+
1
6
F1/3
6k
,
r2 =
1
3
γ
k
−
1
3
γ2
+ 3k − 9βγk
F1/3
+
1
2
i
√
3

1
6k
1/3
F1/3
,
r3 =
1
3
γ
k
−
1
3
γ2
+ 3k − 9βγk
F1/3
−
1
2
i
√
3

1
6k
1/3
F1/3
,
F ≡ 324β2
γ2
k2
− 108βγ2
k + 8γ3
+ 12
√
3G1/2
− 216βk2
+ 36γk
G ≡ 343β4
γ2
k2
− 54β3
γ3
k − 324β3
γk2
+ 3β2
γ4
+ 54β2
γ2
k +
108β2
k2
− 2βγ3
− γ2
− 4k

G ≥ 0:
• either
k ≤ γ2
(βγ−1)
9β2γ2−12βγ+4 : in this case, the request of non-vanishing
cosmological-constant term k newly implies
γ ̸= 0,
βγ − 1 ̸= 0,
βγ ̸=
2
3
.
The case k 0 is defined as βγ − 1 0;
while the case k 0 is obtained as βγ − 1 0;

• or
k ≥
1
27
3βγ + 1
β2
,
which is assured after
3βγ + 1 ̸= 0
β ̸= 0,
The case k 0 is allowed after 3βγ + 1 0, while the case k 0
happens as 3βγ + 1 0.
• G = 0 allows one to define new families of spacetimes.

denominators F1/3 ̸= 0
new constraints on k
k ̸= 1
3
γ2
3βγ−1.

the two physical horisons
• r1 0:
- either
k 0 → k ka, ka =
γ2(βγ − 1)
9β2γ2 − 12βγ + 4
;
⇒
r1a = β
3βγ − 2
βγ − 1
.

- or
k 0, k kb, kb =
1
27
3βγ + 1
β2
;
⇒
r1b = 3β.
r1b 0 ⇒
new the qualification of β as
β 0

the physical horisons
new condition on k newly found
k = 1
3
γ2
3βγ−1
R1 =
1
2
3βγ − 1
γ2

41/3

γ3
(3βγ−1)2
√
3βγ − 1(β − 1)
1/3
+ 2γ
,
R2 =
3βγ − 1
γ
.

• request that the physical horisons should have a positive value,
the new constraints on γ newly found:
- either γ 1
3β 0;
- or γ 0.

comparisons
in [P.D. Mannheim, D. Kazanas, Exact vacuum solution to conformal
Weyl gravity and galactic rotation curves, Astrophys. Jour. 342, 635
(1989)]
1) β → 0 (or β negligible) is considered; and
2) βγ → 0 is considered;
in the present work, these instances are newly found to be forbidden.

further cosmological implementation
• conformal Weyl gravity and perihelion precession:
- perihelion shift of planetary motion within the Mannheim-Kazanas
spacetimes within the framework of Weyl gravity,
- Solar-System qualities possibly originating from Λ;
J. Sultana, D. Kazanas, J. Levi Said, Conformal Weyl gravity and perihelion
precession, Phys. Rev. D 86, 084008 (2012).
• investigation of length scales as
rS
r2 ∼ γ ∼ 1
RH
;
P.D. Mannheim, D. Kazanas, Exact vacuum solution to conformal Weyl gravity and
galactic rotation curves, Astrophys. Jour. 342, 635 (1989)

new qualities of the cosmological implementations
• case β = 0 and the case β negligible are excluded in order for the
physical singularity r = 0 to be excluded for the possible values of the
horisons:
the parameters βγ ≃ 10−12
, the parameter γ accounting as a ’typical
expected flat rotation velocity’, and β must be set according to the new
constraints
- the parameter β is newly found to be necessarily β 0 from
r1b = 3β 0 in order to avoid the physical singularity r = 0;
- the new condition r1a 0 must be newly implemented.

Weak-field limit
From the metric element
gtt ≡ 1 −
rs
r
+ ψ(r)
the pertinent Christoffel symbol is calculated as
Γr
00 ≡
1
2
(1 −
rs
r2
+ ψ(r))(
rs
r
+
dψ
dr
).
generalised Schwarzschild spacetimes
gtt = 1 − rs
r − k1r − k2r2
:
the Newtonian gravitational potential Phi(r) descends form the rs
addends, the non-negligible modification terms descend form the k1
addends, and the addends containing the k2 are negligible. The new
analysis of the addend containing k2
1 − k2 is achieved.

Embedding diagrams
spherically-symmetric spacetimes:
the segment
dσ2
= grr (r)dr2
+ gφφ(r)dϕ2
is rewritten in a three-dimensional Euclidean space coordinatized in the
cylindrical coordinates as
dσ2
=

1 +

dz
dϱ
2

dϱ2
+ ϱ2
dφ2
with ϱ(r2
) = gφφ(r) ≥ 0.

generalised Schwarzschild spacetimes
embedding functions z(ϱ)ki
gϱϱ = [1 +
1
4
(
1
ϱ
−
rs
ϱ3/2
−
k1
√
ϱ
− k2)],
for which the limits of vanishing k1 and k2 are well-posed.

Figure: The φ = 0 sections of the embedding surfaces σ for the
Schwarzschild spacetime (yellow- light gray) and for the spacetime
k1 = 1/4rs, k2 = 0 (green- gray) in units M = 1; the difference in the
two R1212(σ)’s is appreciated. The role of the linear term in the
modification of the Schwarzschild radius is therefore outlined.

Perspective studies
Astrophysical characterisation
Schwarzschild–deSitter spacetimes, Reissner–Nordstrom–de Sitter
spacetimes and Kerr-deSitter spacetimes were compared as far as the
influence of the nowadays duty of the cosmological constant is concerned
about the properties of accretion discs orbiting black holes for quasars
and for the active galactic nuclei;
Z. Stuchlik, Influence of the relict cosmological constant on accretion discs, Mod.
Phys. Lett. A 20, 561 (2005).
Quantum implementation
Gibbons-Hawking temperature defined as
TGH (r∗
) ≡ 1
4π
h
dgtt
dr
i
r=r∗
,
i.e. to be evaluated ate the coordinate point r = r∗
:
- for generalised Schwarzschild spacetimes, the Gibbons-Hawking
temperature is investigated.
V.V. Kiselev, Quintessence and black holes, Class. Quant. Grav. 20, 1187 (2003).

Geometrical objects of the generalised Schwarzschild
spacetimes
ds2
= c2
(1 −
rs
r
+ ψ(r))dt2
−
1
(1 − rs
r + ψ(r))
dr2
− r2
dθ2
− r2
(sinθ)2
dϕ2
,
(1 − rs
r + ψ(r)) is the generalisation of the Schwarzschild element
depending on a function ψ(r) and on the parameters qualifying it.
• Ricci tensor Rµν
Rtt =
1
r2
[r − rs + rψ]

r
d2
ψ
dr2
+ 2
dψ
dr
#
,
Rrr = −
1
2 [r − rs + rψ]

r
d2
ψ
dr2
+ 2
dψ
dr
#
,
Rθθ = −ψ + r
dψ
dr
,
Rϕϕ =

−ψ + r
dψ
dr

(sinθ)
2
;
• Ricci scalar R
R =
1
r2

r
2 d2
ψ
dr2
+ 4r
dψ
dr
+ 2ψ
#
.

Thank You for Your attention.

Geometrical qualities of the generalised Schwarzschild spacetimes

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