Any spherically symmetric exact vacuum solution to Einstein's field equations should be nothing more than a coordinate transformation of the Schwarzschild solution. I've attached a grtensorII spreadsheet anyone can download the program for and follow to verify that the line element I entered is an exact vacuum solution.
It is close to the isotropic coordinate expression for the Schwarzschild solution except the stuff multiplied by k which is any constant. I didn't find this solution by doing a transformation and just did so this morning so at the moment I have no idea what coordinate transformation would take this to the isotropic coordinate expression without the k. So here's the challenge. Find a coordinate transformation that demonstrates that this is equivelent to the Schwarzschild solution.
Now heres the real stumper. Some of the Riemann tensor terms diverge at the event horizon at r=R/4 for nonzero k.
> grtw();
GRTensorII Version 1.79 (R6)
6 February 2001
Developed by Peter Musgrave, Denis Pollney and Kayll Lake
Copyright 1994-2001 by the authors.
Latest version available from: http://grtensor.phy.queensu.ca/
C:/Grtii(6)/Metrics
> makeg(isotropic);
Makeg 2.0: GRTensor metric/basis entry utility
To quit makeg, type 'exit' at any prompt.
Do you wish to enter a 1) metric [g(dn,dn)],
2) line element [ds],
3) non-holonomic basis [e(1)...e(n)], or
4) NP tetrad [l,n,m,mbar]?
> 2;
Enter coordinates as a LIST (eg. [t,r,theta,phi]):
> [ct,r,theta,phi];
Enter the line element using d[coord] to indicate differentials.
(for example, r^2*(d[theta]^2 + sin(theta)^2*d[phi]^2)
[Type 'exit' to quit makeg]
ds^2 =
> ((((1-((R/4)/r))/(1+((R/4)/r)))^2)*((d[ct])^2))-(((1+((R/4)/r))^4)*(((d[r])^2)+((r^2)*((d[theta])^2))+((r^2)*((sin(theta))^2)*((d[phi])^2))))-(((k)*(R/4)/(r*((1-((R/4)/r))^2)))*(((((1+((R/4)/r))^2)*(d[r]))-(((1-((R/4)/r))/(1+((R/4)/r)))*(d[ct])))^2));
If there are any complex valued coordinates, constants or functions
for this spacetime, please enter them as a SET ( eg. { z, psi } ).
Complex quantities [default={}]:
> {};
The values you have entered are:
Coordinates = [ct, r, theta, phi]
Metric:
1 k R
g[ct] [ct] = ------------ - 1/4 -----------------------------
/ 1/4 R\2 2 / 1/4 R\2
|1 + -----| r (1 - 1/4 R/r) |1 + -----|
\ r / \ r /
3
k R
- 1/64 ------------------------------
3 2 / 1/4 R\2
r (1 - 1/4 R/r) |1 + -----|
\ r /
2
1/8 k R R
+ ------------------------------ - 1/2 --------------
2 2 / 1/4 R\2 / 1/4 R\2
r (1 - 1/4 R/r) |1 + -----| |1 + -----| r
\ r / \ r /
2
1/16 R
+ ---------------
/ 1/4 R\2 2
|1 + -----| r
\ r /
2
k R
g[ct] [r] = 1/16 -----------------------------
2 2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
4
k R
- 1/256 -----------------------------
4 2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
1/4 k R
+ ----------------------------
2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
3
k R
- 1/64 -----------------------------
3 2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
g[ct] [theta] = 0
g[ct] [phi] = 0
2
k R
g[r] [ct] = 1/16 -----------------------------
2 2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
4
k R
- 1/256 -----------------------------
4 2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
1/4 k R
+ ----------------------------
2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
3
k R
- 1/64 -----------------------------
3 2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
4 3
R R k R
g[r] [r] = - 1/256 ---- - 1/16 ---- - 1/4 ----------------
4 3 2
r r r (1 - 1/4 R/r)
5 4
k R k R
- 1/1024 ----------------- - 1/64 -----------------
5 2 4 2
r (1 - 1/4 R/r) r (1 - 1/4 R/r)
3 2
k R k R
- 3/32 ----------------- - 1/4 ----------------- - 1 - R/r
3 2 2 2
r (1 - 1/4 R/r) r (1 - 1/4 R/r)
2
R
- 3/8 ----
2
r
g[r] [theta] = 0
g[r] [phi] = 0
g[theta] [ct] = 0
g[theta] [r] = 0
4 3
R R 2 2
g[theta] [theta] = - 1/256 ---- - 1/16 ---- - r - R r - 3/8 R
2 r
r
g[theta] [phi] = 0
g[phi] [ct] = 0
g[phi] [r] = 0
g[phi] [theta] = 0
4 2 3 2
R sin(theta) R sin(theta)
g[phi] [phi] = - 1/256 -------------- - 1/16 --------------
2 r
r
2 2 2 2 2
- 3/8 R sin(theta) - R r sin(theta) - r sin(theta)
You may choose to 0) Use the metric WITHOUT saving it,
1) Save the metric as it is,
2) Correct an element of the metric,
3) Re-enter the metric,
4) Add/change constraint equations,
5) Add a text description, or
6) Abandon this metric and return to Maple.
> 1;
Information written to: `C:/Grtii(6)/Metrics/isotropic.mpl`
Do you wish to use this spacetime in the current session?
(1=yes [default], other=no):
> 1;
Initializing: isotropic
Default spacetime = isotropic
For the isotropic spacetime:
Coordinates
x(up)
a
x = [ct, r, theta, phi]
Line element
/
2 | 1 k R
ds = |------------ - 1/4 -----------------------------
|/ 1/4 R\2 2 / 1/4 R\2
||1 + -----| r (1 - 1/4 R/r) |1 + -----|
\\ r / \ r /
3
k R
- 1/64 ------------------------------
3 2 / 1/4 R\2
r (1 - 1/4 R/r) |1 + -----|
\ r /
2
1/8 k R R
+ ------------------------------ - 1/2 --------------
2 2 / 1/4 R\2 / 1/4 R\2
r (1 - 1/4 R/r) |1 + -----| |1 + -----| r
\ r / \ r /
2 \ /
1/16 R | 2 |
+ ---------------| d ct + 2 |
/ 1/4 R\2 2| |
|1 + -----| r | |
\ r / / \
2
k R
1/16 -----------------------------
2 2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
4
k R
- 1/256 -----------------------------
4 2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
1/4 k R
+ ----------------------------
2 / 1/4 R\
r (1 - 1/4 R/r) |1 + -----|
\ r /
3 \ /
k R | |
- 1/64 -----------------------------| d ct d r + |
3 2 / 1/4 R\| |
r (1 - 1/4 R/r) |1 + -----|| \
\ r //
4 3
R R k R
- 1/256 ---- - 1/16 ---- - 1/4 ----------------
4 3 2
r r r (1 - 1/4 R/r)
5 4
k R k R
- 1/1024 ----------------- - 1/64 -----------------
5 2 4 2
r (1 - 1/4 R/r) r (1 - 1/4 R/r)
3 2
k R k R
- 3/32 ----------------- - 1/4 ----------------- - 1 - R/r
3 2 2 2
r (1 - 1/4 R/r) r (1 - 1/4 R/r)
2 \
R | 2
- 3/8 ----| d r
2 |
r /
/ 4 3 \
| R R 2 2| 2
+ |- 1/256 ---- - 1/16 ---- - r - R r - 3/8 R | d theta
| 2 r |
\ r /
/ 4 2 3 2
| R sin(theta) R sin(theta)
+ |- 1/256 -------------- - 1/16 --------------
| 2 r
\ r
\
2 2 2 2 2|
- 3/8 R sin(theta) - R r sin(theta) - r sin(theta) | d
|
/
2
phi
makeg() completed.
> grcalc(G(up,up));
Created definition for G(up,up)
CPU Time = .141
> grdisplay(G(up,up));
For the isotropic spacetime:
G(up,up)
G(up, up)
a b
G = All components are zero
>