Confusion with Pseudo Scalars, Vectors, Tensors, and Tensor Densities

In Geometry the true mathematical objects (scalars, vectors, and tensors), while being covariant with respect to arbitrary transformation of coordinates, are, at the same time, identity-invariant. By this term we mean that any contraction between them to a scalar gives us invariant. We call them absolute objects or primary objects to distinguish them from so called “pseudo” mathematical objects and tensor densities. If we are making an arbitrary transformation of coordinates then the only way to find the description of a primary object in a new coordinate system is to use its transformation law. A “Pseudo” mathematical object actually is not original (primary) object at all. A “pseudo” object is usually constructed from the components of the primary objects (like determinant of a second rank tensor aik is constructed from its components). There is a straight forward way to obtain this determinant in a new coordinates by obtaining transformed tensor aik' first and then calculating determinant. After that is done one can see that determinant in a new coordinates is equal to determinant in old coordinates multiplied by the Jacobian with the weight w=2. It seems that we found a new transformation law and call determinant by the name pseudo-scalar. But it is only illusion. It seems that it has its own transformation law but this law is not original. The determinant is not a primary object. This flaw is very serious and it renders the whole chapter of tensor algebra (concerning pseudo quantities and tensor densities) as invalid. The mathematical objects that we use in physics to describe physical reality have to be identity-invariant. This requirement is obvious because the physical objects are independent of coordinate system that we use to describe them (we mean here strictly numerical description and not symmetry operations).

In this article we will prove that the very important in geometry (and physics) so-called determinant tensor (also called fully antisymmetric tensor, permutation tensor, or tensor Levi-Civita) does not need to be considered as pseudo tensor. It is an absolute tensor and therefore its use in physics is completely legal.

The determinant tensor is defined: ik...l e Dik...l = e g e(ik...l); D = e(ik...l) e g (1) where g is determinant of the covariant metric tensor, ε is the sign of this determinant, and e(ik...l) is the fully antisymmetric symbol (Levi-Civita symbol). The word 'symbol' underlines that it is deemed to be not a tensor. For this reason we include its indexes in brackets – not subscripts and not superscripts. The rank of this tensor is equal to the number of dimensions of space (n). The full contraction of this tensor with the same is:

1 Dik...l D =1 n! ik...l (1a)

We can raise or lower the indexes of this tensor in agreement with the expressions (1). We can make arbitrary transformation of coordinates and we return to the same expressions (1) in a new coordinates. Also all covariant derivatives of this tensor are zero (The same as all covariant derivatives of the metric tensor). This allows us to carry this tensor past the sign of covariant derivative. The determinant tensor as well as the metrics tensor are the very special tools in n-d geometry. Let us check the transformation properties of the determinant tensor (1):

¶x i ¶x k ¶x l ¶x i ¶x k ¶x l e Dik...l = ... Dab...c = ... e(ab...c) ¶xa ¶xb ¶xc ¶xa ¶xb ¶xc e g ed é¶x ù = e(ik...l) d = det ê ú Jacobian e g ë ¶x û (2)

We used here the algebraic definition of determinant. We notice that the square root from determinant of covariant metric tensor is the scalar density of weight w=-1 (see under “Tensor Density” in Wikipedia). That means that its transformation law (by the conventional treatment) is:

-1 e g = d e g (3)

By substituting this in (2) we find full agreement with the definition (1). That means that the discriminant tensor is an absolute tensor as defined in (1) (not a 'pseudo-tensor'), but this is after we accepted (3). Let us see what actually happens here. The transformation (3) is not an independent law. Actually we calculate the determinant of metric tensor independently in old and new coordinates, and by comparing them we see that:

-2 g = d g (3a)

After that we are taking the square root. The (3) is not unique consequence of (3a). By accepting (3) we have to realize that the sign of the square root will change under improper transformation of coordinates (note that the sign of Jacobian can not change anywhere in space because of the requirement on any coordinate transformation that d can not turn to zero or infinity in any point of space – it stays either positive or negative in all the space). From now on we have to keep track of the sign in front of the square root from determinant of metric tensor. If the sign is positive then the chirality of the coordinate system is positive. If the sign is negative then the chirality of the coordinate system is negative. With that improvement tensor Levi-Civita becomes a normal identity conserving (absolute) tensor.

First example: Volume element and action integral in 4-d space.

The action integral in physics is always written this way:

S = LdV dV = -gdW dW = dx0dx1dx2dx3 ò 4 4 V4 (4)

Let us introduce 4 small vectors:

ak (dx0 ,0,0,0) bk (0,dx1,0,0) ck (0,0,dx2 ,0) d k (0,0,0, dx3 )

i k l m then : dV4 = Diklma b c d = -gdW

Today's physics considers the contraction dV4 as a pseudo-scalar (because the general opinion that the determinant tensor is a pseudo tensor), Since we can integrate only true scalars, this leads us to the requirement that Λ also has to be pseudo-scalar (in order to compensate the product ΛdV4 into a true scalar). But if we keep the chirality of our coordinate system posted then dV4 is the true scalar and Λ does not have to be any pseudo quantity (as it supposed to be). A volume element of integration is not a pseudo-scalar any more: it is a true scalar. Second example: Vector product as it is defined in 3-d space.

A vector product as it conventionally defined is not identity-invariant (it is pseudo vector). The closest identity-invariant true vector in 3-d space would be:

a b 2 3 3 2 Ck = Dkab A B ; C1 = g (A B - A B ) C = g (A3B1 - A1B3 ); C = g (A1B2 - A2 B1) 2 3 (5)

With Euclidean metric and positive chirality (the square root is = +1) this coincides with the usual definition of vector product. But now C is not pseudo-vector any more. In general the rank of C (contraction of discriminant tensor with two vectors) is equal to the number of dimensions minus two: n-2. Only for 3-d space it is a vector. In 4-d and higher spaces C will be antisymmetric tensor of rank 2 or higher. This operation just a consequence of general tensor algebra.

From now on: the sign in front of the square root from the determinant of covariant metric tensor indicates the chirality of the coordinate system. With this convention the discriminant tensor (tensor Levi-Civita) becomes a normal identity conserving absolute tensor.

Another important result of this article is the requirement of Identity Invariance in theoretical physics (which actually stems from Special Relativity).

Also the use of any kind of determinants that would be “pseudo scalars” as a PRIMARY objects is prohibited in physics. It should be replaced by a wider use of determinant tensor (only the determinant of metric tensor is an exception from this rule).