3 years, 78 days

## The matrix E and defining vectors...

Thank you. I am trying to incorporate your suggestions but am stuck trying to do a simple thing. Can you please tell me how I may define the matrix E[a,~mu] (i.e. $E_a^\mu$) whose rows are given by the vectors $V,S,Theta,Phi$ mentioned in the opening question or in the worksheet attached below?

I have a  confusions about the steps in my worksheet given below. It is highlighted in yellow.

 > restart
 >
 (1)
 >
 (2)
 > Setup(g_=-(c(t,r)^2 - v(t,r)^2)*dt^2 + 2*v(t,r)*dt*dr + dr^2 + r^2*dtheta^2 + r^2*sin(theta)^2*dphi^2)
 (3)
 >
 (4)
 > Define(beta(t,r));
 (5)
 > PDETools:-declare(beta(t,r))
 (6)
 > clear(V[~mu])
 (7)

First, I know this was a wrong command.Sorry!

 >
 (8)
 > Define(redo,S[~mu]=[-beta/(c*sqrt(1-beta^2)),(c+v*beta)/(c*sqrt(1-beta^2)),0,0]);
 (9)
 >
 (10)
 > Define(redo,V[~mu]=[1/(c*sqrt(1-beta^2)),-(v+c*beta)/(c*sqrt(1-beta^2)),0,0]);
 (11)
 >
 (12)

In both equations (11) and (12), it can be seen that there is a common factor of $\frac{1}{c \sqrt{1-\beta^2}}$ multiplying each entry of the vector fields $V^\mu$  and $S^\mu$. How can I define a vector field by taking the common factor outside the matrix, as in say $A^\mu = a(t,r) (1, 0,1,1)$? This is just easier to input.

 > Define(redo,Theta[~mu]=[0,0,1,0]);
 (13)
 > Define(redo,Phi[~mu]=[0,0,0,1]);
 (14)
 > with(LinearAlgebra):
 > Define()
 (15)
 >
 (16)
 >
 (17)
 > Define(E[a,~mu])
 >

I now want to construct the tensor $E_a^\mu$ whose rows are the vectors $V,S,\Theta,\Phi$ respectively i.e. $E_1^\mu = V^\mu$ and so on. I am not able to do this.

Finally, if I had defined the vector field V using the command

V := <1/(c*sqrt(1-beta^2)),-(v+c*beta)/(c*sqrt(1-beta^2)),0,0>

in the way matrices are defined within LinearAlgenra, would there be a difference in the way it is interpreted by the Physics package?

I am sorry if these are very naive questions. Any help would be greatly appreciated. Thank you.

## @tomleslie So, it is a good idea to put ...

@tomleslie So, it is a good idea to put a `.' between functions always!

## @ecterrab This is great. Thanks a lot...

@ecterrab This is great. Thanks a lot

## So, functional dependence should always ...

Thank you. It is a beautiful experience to work with the Physics package with your patient help. Thanks a lot. I have understood where I was going wrong. It is also nice to see how we can simply define the directional derivative in the direction of some vector field.

However, I have two further queries:

1. How do i type the nabla symbol for covariant derivative that you have used?

2. This is a curiosity: I notice that the tetrads are given by the notation l_,n_ etc. The metric is also given by g_. Do predefined tensors always have an underscore in the Physics package?

Thanks again.

## Thank you...

Thanks a lot for the explanation. I did not know that on a computer, lines and columns can be referenced by positive integers only.

## changing values taken by spacetime indic...

I discovered that Christoffel[0,0,0] and Christoffel[1,1,1] are the same. Both of them stand for \Gamma_{ttt}. Is there a way to change the values the spacetime indices take to (0,1,2,3) instead of (1,2,3,4)?

## Using 0 as an index...

Thanks a lot for the elaborate example.

However, I did not understand your warning, "Just be careful with using 0 as an index, since it always points to the timelike position,which now is 1". Suppose, I want to find $g_{tt}$. In Maple, should I write g_[00] or g_[tt] for that.?

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