<p>I make two different substitutions (m(r)=m1(r), m2(r)) to the modified Schwarzschild metrics:</p>
<p><span class="math-container">$${\it m1} \left( r \right) =1/2\,r{c}^{2} \left( 1+\sqrt {{\frac {r{c}^
{2}-2\,MG}{r{c}^{2}}}} \right) {G}^{-1}$$</span></p>
<p><span class="math-container">$${\it m2} \left( r \right) =1/2\,r{c}^{2} \left( 1+\sqrt {-{\frac {-r{c
}^{2}+2\,MG}{r{c}^{2}}}} \right) {\frac {1}{\sqrt {-{\frac {-r{c}^{2}+
2\,MG}{r{c}^{2}}}}}}{G}^{-1}$$</span></p>
<p><span class="math-container">$${\it coord}=[t,r,\theta,\phi]$$</span></p>
<p><span class="math-container">$$g={\it TABLE} \left( [{\it index\_char}=[-1,-1],{\it compts}= \left[ 
\begin {array}{cccc}  \left( 1-2\,{\frac {m \left( r \right) G}{r{c}^{
2}}} \right) {c}^{2}&amp;0&amp;0&amp;0\\\noalign{\medskip}0&amp;- \left( 1-2\,{\frac {
m \left( r \right) G}{r{c}^{2}}} \right) ^{-1}&amp;0&amp;0\\\noalign{\medskip}0
&amp;0&amp;-{r}^{2}&amp;0\\\noalign{\medskip}0&amp;0&amp;0&amp;-{r}^{2} \left( \sin \left( 
\theta \right)  \right) ^{2}\end {array} \right] ] \right)$$</span></p>
<p>I have now signatures (-,+,-,-) to these both metrics.</p>
<p>How do you call these signatures, if they are not Lorentz signatures ?</p>
<p>I refer here to my previous two postings (questions) in this general-relativity tag,
which were closed and I hope that they were not deleted due I refer here to them.</p>
<p>(Draft Neutron Star Model Hannu Poropudas 14052024)</p>
<p>Draft Neutron Star Models Hannu Poropudas 14052024</p>
<p>Testing two m(r) solutions for rho(r) and p1(r) given metrics H.P. 18.6.2024 / Maple 9 program</p>
<p>with(plots): with(plottools):</p>
<p>SI-units,
G = 6.67388<em>10^(-11), c= 2.99792458</em>10^8,
M = 1.5M_Sun neutron star,
M = 2.982705*10^30,
<span class="math-container">$$2\,{\frac {MG}{{c}^{2}}}= 4429.729416$$</span>
A. rho(r) case I and II Einstein eqs and solutions rho(r), p1(r)
<span class="math-container">$${\it m1} \left( r \right) =1/2\,r{c}^{2} \left( 1+\sqrt {{\frac {r{c}^
{2}-2\,MG}{r{c}^{2}}}} \right) {G}^{-1}$$</span>
B. p1(r) case I and II Einstein eqs and solutions rho2(r), p12(r)
<span class="math-container">$${\it m2} \left( r \right) =1/2\,r{c}^{2} \left( 1+\sqrt {-{\frac {-r{c
}^{2}+2\,MG}{r{c}^{2}}}} \right) {\frac {1}{\sqrt {-{\frac {-r{c}^{2}+
2\,MG}{r{c}^{2}}}}}}{G}^{-1}$$</span>
A. rho(r) case
<span class="math-container">$$g={\it TABLE} \left( [{\it compts}= \left[ \begin {array}{cccc} -
\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}&amp;0&amp;0&amp;0
\\\noalign{\medskip}0&amp;{\frac {1}{\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{
2}}}}}}&amp;0&amp;0\\\noalign{\medskip}0&amp;0&amp;-{r}^{2}&amp;0\\\noalign{\medskip}0&amp;0&amp;0
&amp;-{r}^{2} \left( \sin \left( \theta \right)  \right) ^{2}\end {array}
 \right] ,{\it index\_char}=[-1,-1]] \right)$$</span>
I
<span class="math-container">$$\left( r{c}^{2}+\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r-MG
 \right) {r}^{-3}=8\,\pi \,G\rho \left( r \right)$$</span>
<span class="math-container">$$\rho \left( r \right) =1/8\, \left( r{c}^{2}+\sqrt {{\frac {r{c}^{2}-2
\,MG}{r{c}^{2}}}}{c}^{2}r-MG \right) {\pi }^{-1}{G}^{-1}{r}^{-3}$$</span>
II
<span class="math-container">$$- \left( r{c}^{2}+\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r-M
G \right) {r}^{-2} \left( r{c}^{2}-2\,MG \right) ^{-1}=-8\,{\frac {
\pi \,G{\it p1} \left( r \right) }{{c}^{4}}}$$</span>
<span class="math-container">$${\it p1} \left( r \right) =1/8\,{c}^{4} \left( r{c}^{2}+\sqrt {{\frac 
{r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r-MG \right) {\pi }^{-1}{G}^{-1}{r}
^{-2} \left( r{c}^{2}-2\,MG \right) ^{-1}$$</span>
B. p1(r) case
<span class="math-container">$$g={\it TABLE} \left( [{\it index\_char}=[-1,-1],{\it compts}= \left[ 
\begin {array}{cccc} -{c}^{2}{\frac {1}{\sqrt {1-2\,{\frac {MG}{r{c}^{
2}}}}}}&amp;0&amp;0&amp;0\\\noalign{\medskip}0&amp;\sqrt {1-2\,{\frac {MG}{r{c}^{2}}}}
&amp;0&amp;0\\\noalign{\medskip}0&amp;0&amp;-{r}^{2}&amp;0\\\noalign{\medskip}0&amp;0&amp;0&amp;-{r}^{
2} \left( \sin \left( \theta \right)  \right) ^{2}\end {array}
 \right] ] \right)$$</span>
I
<span class="math-container">$$ \left( \sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r+r{c}^{2}-2
\,\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}MG-3\,MG \right) {c}^{4}{r
}^{-1} \left( r{c}^{2}-2\,MG \right) ^{-2}=8\,\pi \,G{\it rho2}
 \left( r \right)$$</span>
<span class="math-container">$${\it rho2} \left( r \right) =1/8\, \left( \sqrt {{\frac {r{c}^{2}-2\,M
G}{r{c}^{2}}}}{c}^{2}r+r{c}^{2}-2\,\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}
^{2}}}}MG-3\,MG \right) {c}^{4}{\pi }^{-1}{G}^{-1}{r}^{-1} \left( r{c}
^{2}-2\,MG \right) ^{-2}$$</span>
II
<span class="math-container">$$- \left( \sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r+r{c}^{2}-2
\,\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}MG-3\,MG \right) {r}^{-2}
 \left( r{c}^{2}-2\,MG \right) ^{-1}=-8\,{\frac {\pi \,G{\it p12}
 \left( r \right) }{{c}^{4}}}$$</span>
<span class="math-container">$${\it p12} \left( r \right) =1/8\, \left( \sqrt {{\frac {r{c}^{2}-2\,MG
}{r{c}^{2}}}}{c}^{2}r+r{c}^{2}-2\,\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^
{2}}}}MG-3\,MG \right) {c}^{4}{\pi }^{-1}{G}^{-1}{r}^{-2} \left( r{c}^
{2}-2\,MG \right) ^{-1}$$</span></p>
<p>plot(log10(m1(r)),r=4429.729416..8859.458832),
plot(log10(m2(r)),r=4429.729416..8859.458832),</p>
<p>plot(log10(rho(r)),r=4429.729416..8859.458832),
plot(log10(p1(r)),r=4429.729416..8859.458832),</p>
<p>plot([log10(rho(r)),log10(p1(r)),r=4429.729416..8859.458832]),
plot([log10(p1(r)),log10(rho(r)),r=4429.729416..8859.458832]),</p>
<p>plot(log10(rho2(r)),r=4429.729416..8859.458832),
plot(log10(abs(rho2(r))),r=4429.729416..8859.458832),
plot(log10(abs(rho2(r))),r=4429.729416..88594.58832),</p>
<p>plot(log10(p12(r)),r=4429.729416..8859.458832),
plot(log10(abs(p12(r))),r=4429.729416..8859.458832),
plot(log10(abs(p12(r))),r=4429.729416..88594.58832),</p>
<p>plot([log10(p12(r)),log10(rho2(r)),r=4429.729416..8859.458832]),
plot([log10(p12(r)),log10(rho2(r)),r=4429.729416..88594.58832]),
plot([log10(rho2(r)),log10(p12(r)),r=4429.729416..8859.458832]),
plot([log10(rho2(r)),log10(p12(r)),r=4429.729416..88594.58832]),</p>
<p>plot([log10(abs(p12(r))),log10(abs(rho2(r))),r=4429.729416..8859.458832]),
plot([log10(abs(p12(r))),log10(abs(rho2(r))),r=4429.729416..88594.58832]),
plot([log10(abs(rho2(r))),log10(abs(p12(r))),r=4429.729416..8859.458832]),
plot([log10(abs(rho2(r))),log10(abs(p12(r))),r=4429.729416..88594.58832]).</p>
<p>Best Regards,</p>
<p>Hannu Poropudas</p>

CC BY-SA 4.0
2024-06-24T14:22:28.520Z
I've used some editing in order to introduce MathJax in your question, but I doubt if I did it right. Can you check and correct?
Dominique
CC BY-SA 4.0
2024-06-25T10:39:52.647Z
Can you explain why it's legit to replace a metric with 2 other metrics on the same manifold, each with different signatures? I'm sure it's valid and you don't have to teach a class, just provide a link, pls. I'm a mere GR padawan.
Miss Understands
CC BY-SA 4.0
2024-06-25T15:32:55.043Z
m1 and m2 are from solutions of Einstein equations which posting of mine was closed and deleted. date was near 14.5.2024 Hannu
Hannu Poropudas
CC BY-SA 4.0
2024-06-26T05:32:59.747Z
I have used following definitions in my calculations of m1(r) and m2(r) in Einstein equations I and II: rho(r)=dm(r)/dV(r), dV(r)=4*Pi*r^2/sqrt(1-2*M*G/(r*c^2)) and p1(r)=dm(r)*c^2/dV(r). In my, Draft Neutron Star Model Hannu Poropudas 14052024, this volume element dV(r) is Schwarzschild black hole's volume element. Hannu
Hannu Poropudas
<general-relativity>
What is called following two metric's signature if they are not Lorentz signatures?
Hannu Poropudas
# What is called following two metric's signature if they are not Lorentz signatures?

(Asked by: username_0 on 2024-06-24)

I make two different substitutions (m(r)=m1(r), m2(r)) to the modified Schwarzschild metrics:

$${\it m1} \left( r \right) =1/2\,r{c}^{2} \left( 1+\sqrt {{\frac {r{c}^
{2}-2\,MG}{r{c}^{2}}}} \right) {G}^{-1}$$

$${\it m2} \left( r \right) =1/2\,r{c}^{2} \left( 1+\sqrt {-{\frac {-r{c
}^{2}+2\,MG}{r{c}^{2}}}} \right) {\frac {1}{\sqrt {-{\frac {-r{c}^{2}+
2\,MG}{r{c}^{2}}}}}}{G}^{-1}$$

$${\it coord}=[t,r,\theta,\phi]$$

$$g={\it TABLE} \left( [{\it index\\_char}=[-1,-1],{\it compts}= \left[
\begin {array}{cccc} \left( 1-2\,{\frac {m \left( r \right) G}{r{c}^{
2}}} \right) {c}^{2}&0&0&0\\\noalign{\medskip}0&- \left( 1-2\,{\frac {
m \left( r \right) G}{r{c}^{2}}} \right) ^{-1}&0&0\\\noalign{\medskip}0
&0&-{r}^{2}&0\\\noalign{\medskip}0&0&0&-{r}^{2} \left( \sin \left(
\theta \right) \right) ^{2}\end {array} \right] ] \right)$$

I have now signatures (-,+,-,-) to these both metrics.

How do you call these signatures, if they are not Lorentz signatures ?

I refer here to my previous two postings (questions) in this general-relativity tag,
which were closed and I hope that they were not deleted due I refer here to them.

(Draft Neutron Star Model username_0 username_0 14052024)

Draft Neutron Star Models username_0 username_0 14052024

Testing two m(r) solutions for rho(r) and p1(r) given metrics H.P. 18.6.2024 / Maple 9 program

with(plots): with(plottools):

SI-units,
G = 6.67388*10^(-11), c= 2.99792458*10^8,
M = 1.5M\_Sun neutron star,
M = 2.982705\*10^30,
$$2\,{\frac {MG}{{c}^{2}}}= 4429.729416$$
A. rho(r) case I and II Einstein eqs and solutions rho(r), p1(r)
$${\it m1} \left( r \right) =1/2\,r{c}^{2} \left( 1+\sqrt {{\frac {r{c}^
{2}-2\,MG}{r{c}^{2}}}} \right) {G}^{-1}$$
B. p1(r) case I and II Einstein eqs and solutions rho2(r), p12(r)
$${\it m2} \left( r \right) =1/2\,r{c}^{2} \left( 1+\sqrt {-{\frac {-r{c
}^{2}+2\,MG}{r{c}^{2}}}} \right) {\frac {1}{\sqrt {-{\frac {-r{c}^{2}+
2\,MG}{r{c}^{2}}}}}}{G}^{-1}$$
A. rho(r) case
$$g={\it TABLE} \left( [{\it compts}= \left[ \begin {array}{cccc} -
\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}&0&0&0
\\\noalign{\medskip}0&{\frac {1}{\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{
2}}}}}}&0&0\\\noalign{\medskip}0&0&-{r}^{2}&0\\\noalign{\medskip}0&0&0
&-{r}^{2} \left( \sin \left( \theta \right) \right) ^{2}\end {array}
\right] ,{\it index\\_char}=[-1,-1]] \right)$$
I
$$\left( r{c}^{2}+\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r-MG
\right) {r}^{-3}=8\,\pi \,G\rho \left( r \right)$$
$$\rho \left( r \right) =1/8\, \left( r{c}^{2}+\sqrt {{\frac {r{c}^{2}-2
\,MG}{r{c}^{2}}}}{c}^{2}r-MG \right) {\pi }^{-1}{G}^{-1}{r}^{-3}$$
II
$$- \left( r{c}^{2}+\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r-M
G \right) {r}^{-2} \left( r{c}^{2}-2\,MG \right) ^{-1}=-8\,{\frac {
\pi \,G{\it p1} \left( r \right) }{{c}^{4}}}$$
$${\it p1} \left( r \right) =1/8\,{c}^{4} \left( r{c}^{2}+\sqrt {{\frac
{r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r-MG \right) {\pi }^{-1}{G}^{-1}{r}
^{-2} \left( r{c}^{2}-2\,MG \right) ^{-1}$$
B. p1(r) case
$$g={\it TABLE} \left( [{\it index\\_char}=[-1,-1],{\it compts}= \left[
\begin {array}{cccc} -{c}^{2}{\frac {1}{\sqrt {1-2\,{\frac {MG}{r{c}^{
2}}}}}}&0&0&0\\\noalign{\medskip}0&\sqrt {1-2\,{\frac {MG}{r{c}^{2}}}}
&0&0\\\noalign{\medskip}0&0&-{r}^{2}&0\\\noalign{\medskip}0&0&0&-{r}^{
2} \left( \sin \left( \theta \right) \right) ^{2}\end {array}
\right] ] \right)$$
I
$$ \left( \sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r+r{c}^{2}-2
\,\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}MG-3\,MG \right) {c}^{4}{r
}^{-1} \left( r{c}^{2}-2\,MG \right) ^{-2}=8\,\pi \,G{\it rho2}
\left( r \right)$$
$${\it rho2} \left( r \right) =1/8\, \left( \sqrt {{\frac {r{c}^{2}-2\,M
G}{r{c}^{2}}}}{c}^{2}r+r{c}^{2}-2\,\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}
^{2}}}}MG-3\,MG \right) {c}^{4}{\pi }^{-1}{G}^{-1}{r}^{-1} \left( r{c}
^{2}-2\,MG \right) ^{-2}$$
II
$$- \left( \sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}{c}^{2}r+r{c}^{2}-2
\,\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^{2}}}}MG-3\,MG \right) {r}^{-2}
\left( r{c}^{2}-2\,MG \right) ^{-1}=-8\,{\frac {\pi \,G{\it p12}
\left( r \right) }{{c}^{4}}}$$
$${\it p12} \left( r \right) =1/8\, \left( \sqrt {{\frac {r{c}^{2}-2\,MG
}{r{c}^{2}}}}{c}^{2}r+r{c}^{2}-2\,\sqrt {{\frac {r{c}^{2}-2\,MG}{r{c}^
{2}}}}MG-3\,MG \right) {c}^{4}{\pi }^{-1}{G}^{-1}{r}^{-2} \left( r{c}^
{2}-2\,MG \right) ^{-1}$$

plot(log10(m1(r)),r=4429.729416..8859.458832),
plot(log10(m2(r)),r=4429.729416..8859.458832),

plot(log10(rho(r)),r=4429.729416..8859.458832),
plot(log10(p1(r)),r=4429.729416..8859.458832),

plot([log10(rho(r)),log10(p1(r)),r=4429.729416..8859.458832]),
plot([log10(p1(r)),log10(rho(r)),r=4429.729416..8859.458832]),

plot(log10(rho2(r)),r=4429.729416..8859.458832),
plot(log10(abs(rho2(r))),r=4429.729416..8859.458832),
plot(log10(abs(rho2(r))),r=4429.729416..88594.58832),

plot(log10(p12(r)),r=4429.729416..8859.458832),
plot(log10(abs(p12(r))),r=4429.729416..8859.458832),
plot(log10(abs(p12(r))),r=4429.729416..88594.58832),

plot([log10(p12(r)),log10(rho2(r)),r=4429.729416..8859.458832]),
plot([log10(p12(r)),log10(rho2(r)),r=4429.729416..88594.58832]),
plot([log10(rho2(r)),log10(p12(r)),r=4429.729416..8859.458832]),
plot([log10(rho2(r)),log10(p12(r)),r=4429.729416..88594.58832]),

plot([log10(abs(p12(r))),log10(abs(rho2(r))),r=4429.729416..8859.458832]),
plot([log10(abs(p12(r))),log10(abs(rho2(r))),r=4429.729416..88594.58832]),
plot([log10(abs(rho2(r))),log10(abs(p12(r))),r=4429.729416..8859.458832]),
plot([log10(abs(rho2(r))),log10(abs(p12(r))),r=4429.729416..88594.58832]).

Best Regards,

username_0 username_0

**Tags:** general-relativity

> **Comments:**
> **username_1**: I've used some editing in order to introduce MathJax in your question, but I doubt if I did it right. Can you check and correct?  
> **username_2**: Can you explain why it's legit to replace a metric with 2 other metrics on the same manifold, each with different signatures? I'm sure it's valid and you don't have to teach a class, just provide a link, pls. I'm a mere GR padawan.  
> **username_0**: m1 and m2 are from solutions of Einstein equations which posting of mine was closed and deleted. date was near 14.5.2024 username_0  
> **username_0**: I have used following definitions in my calculations of m1(r) and m2(r) in Einstein equations I and II: rho(r)=dm(r)/dV(r), dV(r)=4*Pi*r^2/sqrt(1-2*M*G/(r*c^2)) and p1(r)=dm(r)*c^2/dV(r). In my, Draft Neutron Star Model username_0 username_0 14052024, this volume element dV(r) is Schwarzschild black hole's volume element. username_0  

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## 0 answers

