Mathematica Notebook for "Exact, non-singular black holes from a phantom DBI Field as primordial dark matter" https://arxiv.org/abs/2511.14047

Black Hole Solu ion o he DBI
Lag angian
S = ∫
d4x-gR
2κ2+Λ41-1+2X
Λ4
X is he Kine ic Ene gy o he DBI
ield
Λ is he in insic ene gy scale
Ou me ic signa u e is (-,+,+,+)
We a e going o assume he
ollowing me ic solu ion
ds2
=
- ( )d 2
+
d 2
( )
+
ρ2
( )
dθ2
+
sin2
θ
dϕ2
)
whe e ( )= 1+
3GM
a2
( -
2+a2
a
A cTan (
a
) )
ρ2
( ) = (
2
+
a2
)
This Ma hema ica no ebook uses
“DBI_ ile1.nb” as i s inpu
X is he Kine ic Ene gy o he DBI
ield
Λ is he in insic ene gy scale
Ou me ic signa u e is (-,+,+,+)
We a e going o assume he
ollowing me ic solu ion
ds2
=
- ( )d 2
+
d 2
( )
+
ρ2
( )
dθ2
+
sin2
θ
dϕ2
)
whe e ( )= 1+
3GM
a2
( -
2+a2
a
A cTan (
a
) )
ρ2
( ) = (
2
+
a2
)
This Ma hema ica no ebook uses
“DBI_ ile1.nb” as i s inpu
In he ollowing sec ion, we calcula e he Eins ein enso
componen s and he cu a u e in a ian s o he
space ime me ic
In[

]:=
$Assump ions =G∈Posi i eReals && M ∈Posi i eReals &&
a∈Posi i eReals && B ∈Posi i eReals && κ∈Posi i eReals;
De ining he coo dina e sys em:
In[

]:=
n=4;
coo d = { , θ,ϕ, };
[ ]=1+3*(G*M)
a^2 * -( ^2+a^2)
a*A cTana
;
ρ[ ] = Sq [ ^2+a^2];
De ining he me ic and i s in e se:
In[

]:=
me ic = {{ [ ]^(-1), 0, 0, 0},{0, ρ[ ]^2, 0, 0},
{0, 0, ρ[ ]^2 *Sin[θ]^2, 0},{0, 0, 0, - [ ]}};
in e seme ic =Simpli y[In e se[me ic]];
De ining he Ch is o el symbols:
In[

]:=
a ine :=a ine =Simpli y[
Table[(1/2)*Sum[(in e seme ic[[i, s]])*(D[me ic[[s, j]], coo d[[k]]]+
D[me ic[[s, k]], coo d[[j]]]-D[me ic[[j, k]], coo d[[s]]]),
{s, 1, n}],{i, 1, n},{j, 1, n},{k, 1, n}]]
2
DBI_ ile2.nb
In[

]:=
lis a ine :=Table[I [UnsameQ[a ine [[i, j, k]], 0],
{ToS ing[Γ[i, j, k]], a ine[[ i, j, k]]}],{i, 1, n},{j, 1, n},{k, 1, j}]
TableFo m[Pa i ion[Dele eCases[Fla en[lis a ine], Null], 2],
TableSpacing → {2, 2}];
De ining he Riemann and Ricci:
In[

]:=
iemann := iemann =Simpli y[Table[
D[a ine[[i, j, l]], coo d[[k]] ]-D[a ine[[i, j, k]], coo d[[l]] ]+
Sum[
a ine[[s, j, l]]×a ine[[i, k, s]]-a ine[[s, j, k]]×a ine[[i, l, s]],
{s, 1, n}],
{i, 1, n},{j, 1, n},{k, 1, n},{l, 1, n}] ]
lis iemann :=Table[I [UnsameQ[ iemann[[i, j, k, l]], 0],
{ToS ing[R[i, j, k, l]], iemann[[i, j, k, l]]}] ,
{i, 1, n},{j, 1, n},{k, 1, n},{l, 1, k -1}]
TableFo m[Pa i ion[Dele eCases[Fla en[lis iemann], Null], 2],
TableSpacing → {2, 2}];
In[

]:=
icci := icci =
Simpli y[Table[Sum[ iemann[[i, j, i, l]],{i, 1, n}],{j, 1, n},{l, 1, n}] ]
lis icci :=Table[I [UnsameQ[ icci[[j, l]], 0],
{ToS ing[R[j, l]], icci[[j, l]]}] ,{j, 1, n},{l, 1, j}]
TableFo m[Pa i ion[Dele eCases[Fla en[lis icci], Null], 2],
TableSpacing → {2, 2}];
Ricciscala =Simpli y[Sum[in e seme ic[[i, j]]× icci[[i, j]],
{i, 1, n},{j, 1, n}] ] // FullSimpli y
Ou [

]=
-2a5+21 a3G M +18 a G M 3-9GMa2+ 2 a2+2 2A cTana

a3a2+ 22
In[

]:=
(*TeXFo m[Ricciscala ]*)
De ining he Eins ein Tenso :
In[

]:=
eins ein :=eins ein =Simpli y[ icci -(1/2)Ricciscala *me ic]
lis eins ein :=Table[I [UnsameQ[eins ein[[j, l]], 0],
{ToS ing[G[j, l]], eins ein[[j, l]]}],{j, 1, n},{l, 1, j}]
TableFo m[Pa i ion[Dele eCases[Fla en[lis eins ein], Null], 2],
TableSpacing → {2, 2}] // FullSimpli y;
DBI_ ile2.nb
3
De ining he Cu a u e In a ian s (K e schmann scala , Ricci enso squa ed,
Ricci scala )
In[

]:=
Ricciscala =Simpli y[
Sum[in e seme ic[[i, j]]× icci[[i, j]],{i, 1, n},{j, 1, n}] ] // FullSimpli y
Ou [

]=
-2a5+21 a3G M +18 a G M 3-9GMa2+ 2 a2+2 2A cTana

a3a2+ 22
◼
We calcula e he Ricci scala a =0
Ricciscala a ze o =
Ricciscala /. A cTana
 → π
2-A cTan
a /. →0// FullSimpli y
Ou [

]=
-2 a +9GMπ
a3
◼
We calcula e he o m o he Ricci scala a → ∞
In[

]:=
Se ies[Ricciscala , { , In ini y, 6}] // FullSimpli y
Ou [

]=
-2 a2
4+36 a2G M
5 5+4 a4
6+O1
7
◼
We calcula e
Rμν Rμν
In[

]:=
Ricci enso squa ed =
Simpli y[Sum[in e seme ic[[m, i]]×in e seme ic[[l, j]]× icci[[i, j]]×
icci[[m, l]],{i, 1, n},{j, 1, n},{l, 1, n},{m, 1, n}]] // FullSimpli y
Ou [

]=
1
a6a2+ 24
4 a2a8+15 a6G M +189 a2G2M2 4+81 G2M2 6+9 a4G M 2(13 G M + )+9GMa2+ 22
A cTana
 -aa4+12 a2GM +18GM 3+3GMa4+3 a2 2+3 4A cTana

◼
We calcula e
Rμν Rμν
a = 0
In[

]:=
Ricci enso squa eda ze o =
Ricci enso squa ed /. A cTana
 → π
2-A cTan
a /. →0// FullSimpli y
Ou [

]=
4 a2-18aGMπ+27 G2M2π2
a6
◼
We calcula e he o m o
Rμν Rμν
a → ∞
In[

]:=
Se ies[Ricci enso squa ed, { , In ini y, 10}]
Ou [

]=
4 a4
8-96 a4G M
5 9-16 25 a6-39 a4G2M2
25 10 +O1
11
◼
We calcula e he K e schmann scala
Rμνρσ Rμνρσ
4
DBI_ ile2.nb
◼
The Riemann enso componen s e alua ed abo e a e o he o m
Rα
μνσ
◼
Hence, we lowe he i s index o he Riemann Tenso o ha e he componen s in he o m R
αμνσ
o calcula e u he
In[

]:=
iemanndown =Table[Sum[me ic[[l, m]]× iemann[[m, b, c, d]],{m, 1, 4}],
{l, 1, 4},{b, 1, 4},{c, 1, 4},{d, 1, 4}] // Simpli y;
In[

]:=
K=Sum[in e seme ic[[p, l]]×in e seme ic[[q, b]]×in e seme ic[[s, c]]×
in e seme ic[[ , d]]* iemanndown[[p, q, s, ]]× iemanndown[[l, b, c, d]],
{p, 1, 4},{l, 1, 4},{s, 1, 4},{c, 1, 4},{ , 1, 4},
{d, 1, 4},{q, 1, 4},{b, 1, 4}] // Simpli y
Ou [

]=
1
a6a2+ 24
12 a2a8+8 a6G M +42 a2G2M2 4+18 G2M2 6+a4G M 2(33 G M +2 )-
2aGMa2+ 2 2 a6+30 a2G M 3+18 G M 5+a4 (15 G M + ) A cTana
+
9 G2M2a2+ 22a4+2 a2 2+2 4A cTana
2
◼
We calcula e K e schmannscala a =0
In[

]:=
K e schmanna ze o =K/. A cTana
 → π
2-A cTan
a /. →0// FullSimpli y
Ou [

]=
34 a2-8aGMπ+9 G2M2π2
a6
◼
We calcula e he o m o K e schmannscala a → ∞
In[

]:=
Se ies[K, { , In ini y, 8}]
Ou [

]=
48 G2M2
6+32 a2G M
7+12 a4-16 a2G2M2
8+O1
9
De ining he DBI pa :
In[

]:=
ϕ[ ] = 2*a
B*(κ^2)
π
2-A cTana
* 3*(G*M)*
a^2 +1
Ou [

]=
2 a π
2-1+3GM
a2A cTana

Bκ2
◼
We calcula e ϕ a = 0
In[

]:=
ϕa ze o = ϕ[ ] /. A cTana
 → π
2-A cTan
a /. →0// FullSimpli y
Ou [

]=
0
◼
We calcula e he o m o ϕ a → ∞
DBI_ ile2.nb
5

In[

]:=
Se ies[ϕ[ ],{ , In ini y, 3}]
Ou [

]=
-6 G M +aπ
Bκ2-2 a2
Bκ2 +2 a2G M
Bκ2 2+2 a4
3 B κ2 3+O1
4
We conside he BH The modynamics in he (2GM>>a) limi
◼
We expand ( ) and conside e ms up o second o de in a
In[

]:=
Se ies[ [ ],{a, 0, 3}]
Ou [

]=
1-2 G M
+2 G M a2
5 3+O[a]4
◼
We loca e he ho izon o he abo e space ime me ic
In[

]:=
Sol e[(1-(2GM)/ )+(2 G M a^2)/(5 ^3) ⩵ 0, ]// Simpli y
Ou [

]=
 →1
3
2GM+4×51/3G2M2
-27 a2G M +40 G3M3+3 81 a4G2M2-240 a2G4M41/3+
-27 a2G M +40 G3M3+3 81 a4G2M2-240 a2G4M41/3
51/3,
 →2 G M
3-2ⅈ51/3-ⅈ+ 3G2M2
3-27 a2G M +40 G3M3+3 81 a4G2M2-240 a2G4M41/3+
ⅈ ⅈ+ 3 -27 a2G M +40 G3M3+3 81 a4G2M2-240 a2G4M41/3
6×51/3,
 →2 G M
3+2ⅈ51/3ⅈ+ 3G2M2
3-27 a2G M +40 G3M3+3 81 a4G2M2-240 a2G4M41/3-
ⅈ -ⅈ+ 3 -27 a2G M +40 G3M3+3 81 a4G2M2-240 a2G4M41/3
6×51/3
◼
We expand he eal solu ion up o second o de in a
In[

]:=
Ho izon =Se ies1
32GM+4×51/3G2M2
-27 a2G M +40 G3M3+3 81 a4G2M2-240 a2G4M41/3+
-27 a2G M +40 G3M3+3 81 a4G2M2-240 a2G4M41/3
51/3,
{a, 0, 2} // FullSimpli y
Ou [

]=
2 G M -a2
10 (G M)+O[a]3
6
DBI_ ile2.nb
◼
We ob ain he loca ion o he ho izon a = 2GM -
a2
/10GM
◼
Su ace g a i y is de ined as
1
2
*
'( h
) whe e
h
is he ho izon and ( ) is he nega i e o he
g
componen o he me ic
◼
We calcula e he su ace g a i y up o second o de in a
In[

]:=
Su aceG a i y =1
2*D[(1-(2GM)/ )+(2 G M a^2)/(5 ^3), ] /.
-> 2 G M -a^2/(10 (G M)) // FullSimpli y
Ou [

]=
100 G3M3a4-100 a2G2M2+400 G4M4
a2-20 G2M24
◼
We w i e he inal exp ession o su ace g a i y which is co ec up o second o de in a
In[

]:=
inalsu aceg a i y =Se ies[Su aceG a i y, {a, 0, 2}]
Ou [

]=
1
4 G M -a2
80 G3M3+O[a]3
◼
Hawking Tempe a u e is de ined as
TH
=
1
2π
* (Su ace g a i y)
◼
We w i e he exp ession o he co esponding Hawking Tempe a u e co ec up o second o de
in a
In[

]:=
HawkingTempe a u e =1
2*π * inalsu aceg a i y
Ou [

]=
1
8 G M π-a2
160 G3M3π +O[a]3
◼
We w i e he exp ession o he a ea o he ho izon
In[

]:=
Ho izona ea =4*π*((No mal[Ho izon])^2 +a^2)
Ou [

]=
4 a2+ - a2
10 G M +2 G M
2
π
◼
We w i e he exp ession o he e apo a ion a e P co ec up o second o de in a
In[

]:=
P= (HawkingTempe a u e^4)*(Ho izona ea)// FullSimpli y
Ou [

]=
1
256 G2M2π3-a2
5120 G4M4π3+O[a]3
◼
The i s e m in he abo e exp ession is he co esponding e apo a ion a e o he Schwa zchild
me ic and he second e m is he co ec ion.
DBI_ ile2.nb
7