Koobee Wublee

2007-01-15 07:36:01 UTC

I see that the discussion of whether the metric qualifies as a tensor

or not has splintered into many different localized groups which are

very confusing. Thus, I would like to put everything into a nutshell

of the pros and cons. Although the tensors can cover more ranks, the

discussion should only be limited what GR covers. That is rank 2

tensors or the 4-by-4 matrices.

The definition of a tensor as agreed by almost everyone says that any

matrix invariant under a coordinate transformation is a tensor while

the elements to the matrix can change under the same transformation.

**** Pros' argument:

The geometry described is invariant under any transformation. Thus, it

can be represented by an operator operating on the same vector twice.

ds^2 = f(dq,dq)

Where

** ds^2 = the invariant geometry

** f() = the operator

** dq = the coordinate vector

Of course, we all know that

ds^2 = g_ij dq^i dq^j = g'_mn dq'^m dq'^n

Where

** dq' = Different coordinate system

The question is the matrices ([g] = [g']).

**** Cons's argument:

According to the pros, although the elements of the matrices (g_ij =/=

g'_ij), the martrix ([g] = [g']). In doing so, they can never

describe what constitutes how these 2 matrices are identical. They

avoid it as if a plague in fact. They can only hand-wave it by saying

over and over again that they are indeed identical.

**** What is at stake?

Does it matter if the metric is a tensor or not for the sake of the

mathematics involved? Yes, it does. The interpretation to the

infinite number of solutions to the field equations is at stake. The

existence of the black holes is at stake.

Apparently, the pros have never followed through the derivation of the

solutions to the field equations. Each solution in terms of g_ij is

only valid to the choice of coordinate system where each solution very

different from the others must describe a different geometry using the

same coordinate system. This means the field equations do have an

infinite number of solutions in which the Schwarzschild metric,

Schwarzschild's original metric, or any other is just as valid as any

other where each describes a different geometry. This would shatter

the general theory of relativity.

The field equations in free space are

R_ij(q^0, q^1, q^2, q^3) = 0

Where

** R_ij(q) = Ricci tensor as a function of q

** g1_ij(q) = Solution as function of q

** g2_ij(q) = Solution as function of q

** g3_ij(q) = Solution as function of q

** ...

And all these different geometries described by each solution with the

same coordinate system.

** ds1^2 = g1_ij dq^i dq^j

** ds2^2 = g2_ij dq^i dq^j

** ds3^2 = g3_ij dq^i dq^j

** ...

The silly claim by the pros is that ([g1] = [g2] = [g3] = ...). Thus,

(ds1^2 = ds2^2 = ds^3 = ...). They are wrong. The simple mathematics

shows so. They cannot prove why ([g1] = [g2] = [g3] = ...). The

Schwarzschild metric is not unique. The existence of black holes is

based on a non-unique solution to the field equations. 100 years of

physics is totally BS based on this wrong concept of linear algebra.

or not has splintered into many different localized groups which are

very confusing. Thus, I would like to put everything into a nutshell

of the pros and cons. Although the tensors can cover more ranks, the

discussion should only be limited what GR covers. That is rank 2

tensors or the 4-by-4 matrices.

The definition of a tensor as agreed by almost everyone says that any

matrix invariant under a coordinate transformation is a tensor while

the elements to the matrix can change under the same transformation.

**** Pros' argument:

The geometry described is invariant under any transformation. Thus, it

can be represented by an operator operating on the same vector twice.

ds^2 = f(dq,dq)

Where

** ds^2 = the invariant geometry

** f() = the operator

** dq = the coordinate vector

Of course, we all know that

ds^2 = g_ij dq^i dq^j = g'_mn dq'^m dq'^n

Where

** dq' = Different coordinate system

The question is the matrices ([g] = [g']).

**** Cons's argument:

According to the pros, although the elements of the matrices (g_ij =/=

g'_ij), the martrix ([g] = [g']). In doing so, they can never

describe what constitutes how these 2 matrices are identical. They

avoid it as if a plague in fact. They can only hand-wave it by saying

over and over again that they are indeed identical.

**** What is at stake?

Does it matter if the metric is a tensor or not for the sake of the

mathematics involved? Yes, it does. The interpretation to the

infinite number of solutions to the field equations is at stake. The

existence of the black holes is at stake.

Apparently, the pros have never followed through the derivation of the

solutions to the field equations. Each solution in terms of g_ij is

only valid to the choice of coordinate system where each solution very

different from the others must describe a different geometry using the

same coordinate system. This means the field equations do have an

infinite number of solutions in which the Schwarzschild metric,

Schwarzschild's original metric, or any other is just as valid as any

other where each describes a different geometry. This would shatter

the general theory of relativity.

The field equations in free space are

R_ij(q^0, q^1, q^2, q^3) = 0

Where

** R_ij(q) = Ricci tensor as a function of q

** g1_ij(q) = Solution as function of q

** g2_ij(q) = Solution as function of q

** g3_ij(q) = Solution as function of q

** ...

And all these different geometries described by each solution with the

same coordinate system.

** ds1^2 = g1_ij dq^i dq^j

** ds2^2 = g2_ij dq^i dq^j

** ds3^2 = g3_ij dq^i dq^j

** ...

The silly claim by the pros is that ([g1] = [g2] = [g3] = ...). Thus,

(ds1^2 = ds2^2 = ds^3 = ...). They are wrong. The simple mathematics

shows so. They cannot prove why ([g1] = [g2] = [g3] = ...). The

Schwarzschild metric is not unique. The existence of black holes is

based on a non-unique solution to the field equations. 100 years of

physics is totally BS based on this wrong concept of linear algebra.