This worksheet is designed to determine the connections, and curvatures for the flat Minkowski metric

          expressed in oblate spheroidal coordinates

>	S:=sqrt(sigma^2+a^2*(cos(theta))^2);

>	Delta:=sqrt(sigma^2+a^2);

>

grtw();

>	makeg(OblSpher);

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]?

makeg>

3;

Enter coordinates as a LIST (eg. [t,r,theta,phi]):

makeg>

[sigma,theta,phi,t];

Would you like to enter 1) covariant components,

                        2) contravariant components, or

                        3) both.

makeg>

1;

Enter the covariant components of basis vector '1' as a LIST (eg. [1,0,0,0]) or differential (eg. d[x] + d[y]):

makeg>

(S/Delta)*d[sigma];

Enter the covariant components of basis vector '2' as a LIST (eg. [1,0,0,0]) or differential (eg. d[x] + d[y]):

makeg>

S*d[theta];

Enter the covariant components of basis vector '3' as a LIST (eg. [1,0,0,0]) or differential (eg. d[x] + d[y]):

makeg>

Delta*sin(theta)*d[phi];

Enter the covariant components of basis vector '4' as a LIST (eg. [1,0,0,0]) or differential (eg. d[x] + d[y]):

makeg>

d[t];

Is the basis inner product  1) Diagonal, or

                            2) Symmetric?

makeg>

1;

Enter eta[1,1]:

makeg>

1;

Enter eta[2,2]:

makeg>

1;

Enter eta[3,3]:

makeg>

1;

Enter eta[4,4]:

makeg>

-1;

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={}]: 

makeg>

;

{}

You may choose to 0) Use the metric WITHOUT saving it,

                  1) Save the metric as it is,

                  2) Re-enter a basis vector,

                  3) Re-enter the inner product,

                  4) Add/change constraints,

                   5) Add a text description, or

                  6) Abandon this metric and return to Maple.

makeg>

1;

Information written to: `C:/Grtii(6)/Metrics/OblSpher.mpl`

Do you wish to use this spacetime in the current session?

(1=yes [default], other=no): 

makeg>

1;

Initializing: OblSpher

makeg() completed.

>	grcalc(rot(bdn,bdn,bdn));

Created a definition for e(bdn,dn,pdn)

>	grdisplay(rot(bdn,bdn,bdn));

>	grcalc(R(bdn,bdn,bdn,bdn));

Created definition for rot(bdn,bup,bdn) 

>	grdisplay(R(bdn,bdn,bdn,bdn));

>	gralter(R(bdn,bdn,bdn,bdn),simplify,trig);

Component simplification of a GRTensorII object:

Applying routine simplify to object R(bdn,bdn,bdn,bdn)

Applying routine `simplify[trig]` to object R(bdn,bdn,bdn,bdn)

>	grdisplay(R(bdn,bdn,bdn,bdn));

The plots just below first show graphs of curves, in the y,z-plane, for instance, of rho versus z (in cylindrical coordinates), for various fixed values of sigma:  a, 2a, 3a, 4a, and 5a, where everything is measured in units of a.  

              The 2-dimensional ones are ellipses, with foci at +1 and -1.  

If we had also plotted the situation for sigma = 0, we would have just the horizontal axis between  +1 and -1.

Then they show a 3-dimensional plot for a given, single value of sigma; in this case sigma=1.  It is of course an oblate spheroid.

>	Rho:=sqrt(sigma^2+a^2)*sin(theta); Z:=sigma*cos(theta);

>	plot([seq([subs(a=1,sigma=n,Rho),subs(sigma=n,Z),theta=0..Pi],n=1..5),seq([subs(a=1,sigma=n,-Rho),subs(sigma=n,Z),theta=0..Pi],n=1..5)],scaling=`constrained`,color=[red,green,yellow,blue,cyan,red,green,yellow,blue,cyan], title=`y,z-plane; constant sigma`);

>	plot3d([subs(a=1,sigma=1,Rho*cos(phi)),subs(a=1,sigma=1,Rho*sin(phi)),subs(a=1,sigma=1,Z)],theta=0..Pi,phi=0..2*Pi,scaling=`constrained`,title=`constant sigma = 1; oblate spheroid`);

The graphs below show graphs for constant values of theta.

 You can see, on the 2-dimensional one below, in the y,z-plane, that they are pairs of hyperbolae.

>	plot([seq([subs(a=1,theta=2*n,Rho),subs(theta=2*n,Z),sigma=-5..5],n=1..5),seq([subs(a=1,theta=2*n,-Rho),subs(theta=2*n,Z),sigma=-5..5],n=1..5)],scaling=`constrained`,color=[red,green,yellow,blue,cyan,red,green,yellow,blue,cyan],title=`constant theta; y,z-plane`);

Below I also give two different perspectives of the surfaces of constant theta, which are hyperboloids of one sheet; the generalization of

    the pair of cones that one would have gotten for such surfaces if theta had been a member of a set of spherical coordinates.

>	plot3d([subs(a=1,theta=5,Rho*cos(phi)),subs(a=1,theta=5,Rho*sin(phi)),subs(a=1,theta=1.5*Pi,Z)],sigma=-5..5,phi=0..2*Pi,title=`constant theta; hyperboloids of one sheet`);

>	plot3d([subs(a=1,theta=1.5*Pi,Rho*cos(phi)),subs(a=1,theta=1.5*Pi,Rho*sin(phi)),subs(a=1,theta=1.5*Pi,Z)],sigma=-5..5,phi=0..2*Pi,title=`constant theta; hyperboloids of one sheet`);

>