def g_n2(y, n, RA, RB, rh, l, pm1, deltaphi, isP):
sigma = sigma_btz_n(y, n, RA, RB, rh, l, pm1, deltaphi, isP) + 2
if sigma == 0:
return 0
elif sigma > 0:
return 1 / (4 * fp.pi * fp.sqrt(2) * l) * 1 / fp.sqrt(sigma)
else:
return -fp.j / (4 * fp.pi * fp.sqrt(2) *
l) * 1 / fp.sqrt(-sigma) * fp.sign(y)
def integrand_Pdotp_n(s, tau, n, R, rh, l, pm1, Om, lam, sig, deltaphi, eps):
K = lam**2 * rh / (2 * fp.pi * fp.sqrt(2) * l *
fp.sqrt(R**2 - rh**2)) * fp.exp(-tau**2 / 2 / sig**2)
# Zp = mp.mpf(rh**2/(R**2-rh**2)*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*n)) + 1))
# if Zp == fp.mpf(mp.cosh(rh/l/mp.sqrt(R**2-rh**2) * s)):
# return 0
# else:
return K * fp.exp(-(tau-s)**2/2/sig**2) * fp.exp(-fp.j*Om*s) \
/ fp.sqrt(gammap_btz_n(s,n,R,rh,l,deltaphi,eps))
def sigma_geon_n(y, n, RA, RB, rh, l, pm1, deltaphi, isP):
if isP:
return RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) - 1\
+ fp.sqrt(RA**2-rh**2)*fp.sqrt(RB**2-rh**2)/rh**2 \
* fp.mpf(mp.cosh(rh/l/mp.sqrt(RA*RB-rh**2) * y) )
else:
return RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) - 1\
+ fp.sqrt(RA**2-rh**2)*fp.sqrt(RB**2-rh**2)/rh**2 \
* fp.mpf(mp.cosh(rh/l**2 * y) )
def integrand_deltaPdot_n(s, tau, n, R, rh, l, pm1, Om, lam, sig, deltaphi):
Zm = mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*(n+1/2))) - 1))
Zp = mp.mpf(rh**2/(R**2-rh**2)\
*(R**2/rh**2 * fp.cosh(rh/l * (deltaphi - 2*fp.pi*(n+1/2))) + 1))
K = lam**2 * rh / (2 * fp.pi * fp.sqrt(2) * l *
fp.sqrt(R**2 - rh**2)) * fp.exp(-tau**2 / 2 / sig**2)
return K * fp.exp(-(tau-s)**2/2/sig**2) * fp.exp(-fp.j*Om*s)\
* ( 1/fp.sqrt(Zm + mp.cosh(rh/l/mp.sqrt(R**2-rh**2) * s))\
- pm1*1/fp.sqrt(Zp + mp.cosh(rh/l/mp.sqrt(R**2-rh**2) * s)) )
def XGEON_integrand_nBA(y, n, RA, RB, rh, l, pm1, Om, lam, sig, deltaphi):
bA = mp.sqrt(RA**2 - rh**2) / l
bB = mp.sqrt(RB**2 - rh**2) / l
K = 1
alp2 = bA**2 * bB**2 / 2 / (bA**2 + bB**2) / sig**2
bet2 = (bA + bB) * bA * bB / (bA**2 + bB**2)
E = (bB - bA) / fp.sqrt(2) / fp.sqrt(bB**2 + bA**2) * (
(bB + bA) * y / 2 / sig + fp.j * sig * Om)
return K*mp.exp(-alp2*y**2)*mp.exp(-fp.j*bet2*Om*y) *fp.erfc(E) \
* XGEON_denoms_n(y,n,RA,RB,rh,l,pm1,Om,lam,sig,deltaphi)
def g_n1(y, n, RA, RB, rh, l, pm1, deltaphi, isP):
sigma = sigma_btz_n(y, n, RA, RB, rh, l, pm1, deltaphi, isP)
#print('y',y,' sigma',sigma)
if n == 0 and deltaphi == 0 and RA == RB: #Deal with this case separately
return 0
elif sigma == 0:
return 0
elif sigma > 0:
return 1 / (4 * fp.pi * fp.sqrt(2) * l) * 1 / fp.sqrt(sigma)
else:
return -fp.j / (4 * fp.pi * fp.sqrt(2) *
l) * 1 / fp.sqrt(-sigma) * fp.sign(y)
def Xn_plus(n, RA, RB, rh, l, pm1, Om, lam, tau0, width, deltaphi, isP):
intlim2 = l**2/rh * mp.acosh( (RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) + 1)\
/ fp.sqrt(RA**2-rh**2)/fp.sqrt(RB**2-rh**2)*rh**2)
#print('plus, %.4f ;'%intlim2)
f = lambda y: integrandof_Xn_plus(y, n, RA, RB, rh, l, pm1, Om, lam, tau0,
width, deltaphi, isP)
if intlim2 < width:
return fp.quad(f, [0, intlim2]) + fp.quad(f, [intlim2, width])
else:
return fp.quad(f, [0, width])
def f02(y, n, R, rh, l, pm1, Om, lam, sig):
K = lam**2 * sig / 2 / fp.sqrt(2 * fp.pi)
a = (R**2 - rh**2) * l**2 / 4 / sig**2 / rh**2
b = fp.sqrt(R**2 - rh**2) * Om * l / rh
Zp = mp.mpf((R**2 + rh**2) / (R**2 - rh**2))
if Zp - mp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zp - mp.cosh(y)))
elif Zp - mp.cosh(y) < 0:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zp))
else:
return 0
def f02(y,n,R,rh,l,pm1,Om,lam,sig):
K = lam**2*sig/2/fp.sqrt(2*fp.pi)
a = (R**2-rh**2)*l**2/4/sig**2/rh**2
b = fp.sqrt(R**2-rh**2)*Om*l/rh
Zp = mp.mpf((R**2+rh**2)/(R**2-rh**2))
if Zp == mp.cosh(y):
print("RIP MOM PLSSS")
#print(Zp, y, fp.cosh(y))
if Zp - mp.cosh(y) > 0:
return K * fp.exp(-a*y**2) * fp.cos(b*y) / fp.mpf(mp.sqrt(Zp - mp.cosh(y)))
elif Zp - mp.cosh(y) < 0:
return -K * fp.exp(-a*y**2) * fp.sin(b*y) / fp.mpf(mp.sqrt(mp.cosh(y) - Zp))
else:
return 0
def g(x, RA, RB):
# bA = mp.sqrt(RA**2-rh**2)/l
# bB = mp.sqrt(RB**2-rh**2)/l
# return fp.j*4 * fp.exp(-fp.j*Om*(bB+bA)*(tau0+width/2))/(Om*(bB+bA))\
# * fp.cos( Om/2*(bB+bA)*(x-width) ) * fp.cos(Om/2*(bB-bA)*x)\
# * g_n2(x,0,RA,RB,rh,l,pm1,deltaphi)
sigma = sigma_btz_n(x, n, RA, RB, rh, l, pm1, deltaphi) + 2
if sigma == 0:
return 0
elif sigma > 0:
return 1 / (4 * fp.pi * fp.sqrt(2) * l) * 1 / fp.sqrt(sigma)
else:
return -fp.j / (4 * fp.pi * fp.sqrt(2) *
l) * 1 / fp.sqrt(-sigma) * fp.sign(x)
def fn1(y, n, R, rh, l, pm1, Om, lam, sig):
K = lam**2 * sig / 2 / fp.sqrt(2 * fp.pi)
a = (R**2 - rh**2) * l**2 / 4 / sig**2 / rh**2
b = fp.sqrt(R**2 - rh**2) * Om * l / rh
Zm = mp.mpf(rh**2 / (R**2 - rh**2) *
(R**2 / rh**2 * fp.cosh(2 * fp.pi * rh / l * n) - 1))
if Zm == mp.cosh(y):
return 0
elif Zm - fp.cosh(y) > 0:
return fp.mpf(K * mp.exp(-a * y**2) * mp.cos(b * y) /
mp.sqrt(Zm - mp.cosh(y)))
else:
return fp.mpf(-K * mp.exp(-a * y**2) * mp.sin(b * y) /
mp.sqrt(mp.cosh(y) - Zm))
def T_P_int_geon(x, tau, tau0, n, R, rh, l, pm1, Om, lam, sig, deltaphi=0):
K = lam**2 / (np.sqrt(2) * np.pi * l * Om) * rh / np.sqrt(R**2 - rh**2)
Gm = rh**2 / (R**2 - rh**2) * (R**2 / rh**2 * fp.mpf(
mp.cosh(rh / l * (deltaphi - 2 * np.pi * (n + 0.5)))) - 1)
Gp = rh**2 / (R**2 - rh**2) * (R**2 / rh**2 * fp.mpf(
mp.cosh(rh / l * (deltaphi - 2 * np.pi * (n + 0.5)))) + 1)
if tau < tau0:
return 0
elif x < (tau + tau0):
return K * fp.sin(Om*(2*tau0+x))\
* (1/fp.sqrt(Gm + fp.mpf(mp.cosh(rh/l**2 * x))) - 1/fp.sqrt(Gp + fp.mpf(mp.cosh(rh/l**2 * x))))
else:
return K * fp.sin(Om*(2*tau-x))\
* (1/fp.sqrt(Gm + fp.mpf(mp.cosh(rh/l**2 * x))) - 1/fp.sqrt(Gp + fp.mpf(mp.cosh(rh/l**2 * x))))
def P_BTZn(n,R,rh,l,pm1,Om,lam,sig):
b = fp.sqrt(R**2-rh**2)/l
lim = 20*sig*rh/b/l**2
print('limit: ',lim)
Zm = fp.mpf(1) #rh**2/(R**2-rh**2)*(R**2/rh**2 * fp.cosh(2*fp.pi*rh/l * n) - 1)
Zp = rh**2/(R**2-rh**2)*(R**2/rh**2 * fp.cosh(2*fp.pi*rh/l * n) + 1)
print('Zm: ',Zm, 'Zp: ',Zp, 'acosh(Zp): ', fp.mpf(mp.acosh(Zp)))
#print(Zm,fp.acosh(Zm))
# plt.figure()
# xaxis = fp.linspace(0,lim/5,50)
# yaxis = [f02(x,n,R,rh,l,pm1,Om,lam,sig) for x in xaxis]
# plt.plot(xaxis,yaxis)
# plt.show()
if pm1==-1 or pm1==1 or pm1==0:
if n==0:
return 0\
- fp.quad(lambda x: f02(x,n,R,rh,l,pm1,Om,lam,sig),[0,fp.mpf(mp.acosh(Zp))])\
- fp.quad(lambda x: f02(x,n,R,rh,l,pm1,Om,lam,sig),[fp.mpf(mp.acosh(Zp)),lim])
#fp.quad(lambda x: f01(x,n,R,rh,l,pm1,Om,lam,sig),[-fp.inf,fp.inf])
# else:
# if fp.cosh(lim) < Zm or Zm < 1:
# return fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
# - fn2(x,n,R,rh,l,pm1,Om,lam,sig),[0,lim])
# else:
# return fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
# - fn2(x,n,R,rh,l,pm1,Om,lam,sig),[0,fp.mpf(mp.acosh(Zm))])\
# - fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
# - fn2(x,n,R,rh,l,pm1,Om,lam,sig),[fp.mpf(mp.acosh(Zm)),lim])
else:
print("Please enter a value of 1, -1, or 0 for pm1.")
def wightman_btz_n(s, n, R, rh, l, pm1, Om, lam, sig, deltaphi=0, eps=1e-4):
if n == 0:
Bm = (R**2 - rh**2) / rh**2
Bp = (R**2 + rh**2) / rh**2
else:
Bm = R**2 / rh**2 * fp.cosh(rh / l * (deltaphi - 2 * fp.pi * n)) - 1
Bp = R**2 / rh**2 * fp.cosh(rh / l * (deltaphi - 2 * fp.pi * n)) + 1
Scosh = (R**2 - rh**2) / rh**2 * mp.cosh(rh / l**2 * (s - fp.j * eps))
# if s == 0:
# return 0
# elif Bm == Scosh or Bp == Scosh:
# return 0
# else:
return 1/(4*fp.sqrt(2)*fp.pi*l) * (-fp.j/fp.sqrt(2*Bm)/\
mp.sinh(rh/2/l**2 * (s - fp.j*eps)) - pm1/fp.sqrt(Bp - Scosh))
def PGEON_n(n, R, rh, l, pm1, Om, lam, sig):
"""
Om = energy difference
lam = coupling constant
"""
if pm1 == -1 or pm1 == 1 or pm1 == 0:
return lam**2*sig*fp.sqrt(fp.pi) * fp.mpf(mp.exp(-sig**2 * Om**2)) *\
fp.quad(lambda x: h_n(x,n,R,rh,l,pm1) * PGEON_gaussian(x,sig), [-fp.inf, fp.inf])
else:
print("Please enter a value of 1, -1, or 0 for pm1.")
def P_BTZn(n, R, rh, l, pm1, Om, lam, sig):
b = fp.sqrt(R**2 - rh**2) / l
lim = 20 * sig * rh / b / l**2
#Zp = mp.mpf(rh**2/(R**2-rh**2)*(R**2/rh**2 * fp.cosh(2*fp.pi*rh/l * n) + 1))
#print('Zm: ',Zm, 'Zp: ',Zp)
#print(Zm,fp.acosh(Zm))
if pm1 == -1 or pm1 == 1 or pm1 == 0:
if n == 0:
return fp.quad(lambda x: f01(x, n, R, rh, l, pm1, Om, lam, sig),
[-fp.inf, fp.inf])
else:
Zm = rh**2 / (R**2 - rh**2) * (
R**2 / rh**2 * fp.cosh(2 * fp.pi * rh / l * n) - 1)
if fp.cosh(lim) < Zm or Zm < 1:
return fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
- fn2(x,n,R,rh,l,pm1,Om,lam,sig),[0,lim])
else:
return fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
- fn2(x,n,R,rh,l,pm1,Om,lam,sig),[0,fp.mpf(mp.acosh(Zm))])\
- fp.quad(lambda x: fn1(x,n,R,rh,l,pm1,Om,lam,sig)\
- fn2(x,n,R,rh,l,pm1,Om,lam,sig),[fp.mpf(mp.acosh(Zm)),lim])
else:
print("Please enter a value of 1, -1, or 0 for pm1.")
def P_BTZn(n,
RA,
RB,
rh,
l,
pm1,
Om,
lam,
tau0,
width,
deltaphi=0,
eps=1e-8,
isP=True):
f = lambda y: integrandof_P_BTZn(y, n, RA, RB, rh, l, pm1, Om, lam, tau0,
width, deltaphi, isP)
if n == 0 and deltaphi == 0 and RA == RB:
# P0m
P0m = 2 * fp.quad(
lambda x: integrandPBTZ_0m(x, RA, rh, l, pm1, Om, lam, tau0, width,
eps), [0, width])
intlim = l**2/rh * mp.acosh( (RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) + 1)\
/ fp.sqrt(RA**2-rh**2)/fp.sqrt(RB**2-rh**2)*rh**2)
if intlim < width:
return fp.quad(f, [0, intlim]) + fp.quad(f, [intlim, width]) + P0m
else:
return fp.quad(f, [0, width]) + P0m
else:
intlim1 = l**2/rh * mp.acosh( (RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) - 1)\
/ fp.sqrt(RA**2-rh**2)/fp.sqrt(RB**2-rh**2)*rh**2)
intlim2 = l**2/rh * mp.acosh( (RA*RB/rh**2 * fp.mpf( mp.cosh(rh/l * (deltaphi-2*fp.pi*n)) ) + 1)\
/ fp.sqrt(RA**2-rh**2)/fp.sqrt(RB**2-rh**2)*rh**2)
#print('%.4f, %.4f'%(intlim1,intlim2))
if intlim2 < width:
return fp.quad(f, [0, intlim1]) + fp.quad(
f, [intlim1, intlim2]) + fp.quad(f, [intlim2, width])
elif intlim1 < width:
return fp.quad(f, [0, intlim1]) + fp.quad(f, [intlim1, width])
else:
return fp.quad(f, [0, width])
def h_n2(y, n, RA, RB, rh, l, pm1, deltaphi, isP):
return 1 / (4 * fp.pi * fp.sqrt(2) * l) * 1 / fp.sqrt(
sigma_geon_n(y, n, RA, RB, rh, l, pm1, deltaphi, isP) + 2)
# checkfcn.append( (g_n1(y[i],0,Ri,Ri,rh,l,pm1,0)*fp.exp(fp.j*Om*y[i])).real )
# checkfcn2.append( (g_n1(y[i],0,Ri,Ri,rh,l,pm1,0)*fp.exp(fp.j*Om*y[i])).imag )
#plt.plot(y,checkfcn,label='real of g')
##plt.plot(y,checkfcn2,label='imag of g')
#plt.legend()
P0_gauss = []
P_btz, P_geon = [], []
X_btz, X_geon = [], []
PB_btz, PB_geon = [], []
# Create y-axis array for BTZ and geon
for n in np.arange(0, nmax + 1):
print('n = ', n, '; i = ', end='')
for i in range(np.size(dR)):
print(i, end=' ', flush=True)
bA = fp.sqrt(RA[i]**2 - rh**2) / l
bB = fp.sqrt(RB[i]**2 - rh**2) / l
tau0B = tau0 * bB / bA
widthB = width * bB / bA
if n == 0:
P0_gauss.append(P0m_GAUSS(RA[i], rh, l, pm1, Om, lam, sig).real)
P_btz.append(
P_BTZn(0, RA[i], RA[i], rh, l, pm1, Om, lam, tau0, width))
P_geon.append(
2 * PGEON_n(0, RA[i], RA[i], rh, l, pm1, Om, lam, tau0, width))
PB_btz.append(
P_BTZn(0, RB[i], RB[i], rh, l, pm1, Om, lam, tau0B, widthB))
PB_geon.append(
2 *
PGEON_n(0, RB[i], RB[i], rh, l, pm1, Om, lam, tau0B, widthB))
X_btz.append(
def integrandPBTZ_0m(y, R, rh, l, pm1, Om, lam, tau0, width, eps):
#print( mp.sinh(rh/2/l**2 * (y - fp.j*eps)) )
return (- fp.j*rh*lam**2 / (8*fp.pi*l * fp.sqrt(R**2-rh**2))\
* fp.exp(-fp.j*Om*y)/mp.sinh(rh/2/l**2 * (y - fp.j*eps)) * (width - np.abs(y))).real
def T_P_int_geon(x, tau, n, R, rh, l, pm1, Om, lam, sig, deltaphi=0):
return lam**2*fp.exp(-sig**2*Om**2) * fp.sqrt(fp.pi)*sig * fp.erf(-(x-2*tau/2/sig)-fp.j*sig*Om)\
* fp.exp(-x**2/4/sig**2) * h_n(x,n,R,rh,l,pm1)
def h_n(x, n, R, rh, l, pm1):
return 1/(4*fp.sqrt(2)*fp.pi*l) * (1/fp.sqrt(sigma_geon(x,n,R,rh,l)) \
- pm1 * 1/fp.sqrt(sigma_geon(x,n,R,rh,l) + 2))
def integrandPBTZ_0m_GAUSS(y, R, rh, l, pm1, Om, lam, sig, eps):
#print(mp.sqrt(1 - mp.cosh( y -fp.j*eps )))
return fp.exp(-fp.j*fp.sqrt(R**2-rh**2)*Om*l/rh * y)\
/ mp.sqrt(1 - mp.cosh( y -fp.j*eps ))\
* lam**2*sig/(2*fp.sqrt(2*fp.pi))\
* fp.exp(-(R**2-rh**2)*l**2/(4*sig**2*rh**2) * y**2)
