Private theorem

The private theorem is the statement that if the result of multiplying a vector by an amount with an arbitrary number of upper and lower indices is a tensor for any vector, then the value with upper and lower indices is a tensor.

Content

Formulation

Let the value $P_{\lambda \mu \nu }$ ${\ displaystyle P _ {\ lambda \ mu \ nu}}$ is such that for any vector $A^{\nu }$ ${\ displaystyle A ^ {\ nu}}$ magnitude $A^{\lambda }P_{\lambda \mu \nu }$ ${\ displaystyle A ^ {\ lambda} P _ {\ lambda \ mu \ nu}}$ is a tensor. In this case, the value $P_{\lambda \mu \nu }$ ${\ displaystyle P _ {\ lambda \ mu \ nu}}$ is a tensor.

Proof

Consider the transformation from the old curvilinear coordinate system, where the vector has coordinates $x^{\mu }$ ${\ displaystyle x ^ {\ mu}}$ to a new coordinate system, where the same vector has coordinates $x^{\mu '}$ ${\ displaystyle x ^ {\ mu '}}$ . We agree to designate ${\frac {\partial x^{\mu '}}{\partial x^{\nu }}}=x_{,\nu }^{\mu '}$ ${\ displaystyle {\ frac {\ partial x ^ {\ mu '}} {\ partial x ^ {\ nu}}} = x _ {, \ nu} ^ {\ mu'}}$ . Denote the value $Q_{\mu \nu }=A^{\lambda }P_{\lambda \mu \nu }$ ${\ displaystyle Q _ {\ mu \ nu} = A ^ {\ lambda} P _ {\ lambda \ mu \ nu}}$ . By condition, $Q_{\mu \nu }$ ${\ displaystyle Q _ {\ mu \ nu}}$ there is a tensor, therefore $Q_{\beta \gamma }=Q_{\mu '\nu '}x_{,\beta }^{\mu '}x_{,\gamma }^{\nu '}$ ${\ displaystyle Q _ {\ beta \ gamma} = Q _ {\ mu '\ nu'} x _ {, \ beta} ^ {\ mu '} x _ {, \ gamma} ^ {\ nu'}}$ . Then $A^{\alpha }P_{\alpha \beta \gamma }=A^{\lambda '}P_{\lambda '\mu '\gamma '}x_{,\beta }^{\mu '}x_{,\gamma }^{\nu '}$ ${\ displaystyle A ^ {\ alpha} P _ {\ alpha \ beta \ gamma} = A ^ {\ lambda '} P _ {\ lambda' \ mu '\ gamma'} x _ {, \ beta} ^ {\ mu '} x _ {, \ gamma} ^ {\ nu '}}$ . Because $A^{\lambda }$ ${\ displaystyle A ^ {\ lambda}}$ is a vector, according to the rules of vector transformation we have: $A^{\lambda '}=A^{\alpha }x_{,\alpha }^{\lambda '}$ ${\ displaystyle A ^ {\ lambda '} = A ^ {\ alpha} x _ {, \ alpha} ^ {\ lambda'}}$ . In this way: $A^{\alpha }P_{\alpha \beta \gamma }=A^{\alpha }x_{,\alpha }^{\lambda '}P_{\lambda '\mu '\nu '}x_{,\beta }^{\mu '}x_{,\gamma }^{\nu '}$ ${\ displaystyle A ^ {\ alpha} P _ {\ alpha \ beta \ gamma} = A ^ {\ alpha} x _ {, \ alpha} ^ {\ lambda '} P _ {\ lambda' \ mu '\ nu'} x_ {, \ beta} ^ {\ mu '} x _ {, \ gamma} ^ {\ nu'}}$ This equality must be true for all. $A^{\alpha }$ ${\ displaystyle A ^ {\ alpha}}$ , Consequently $P_{\alpha \beta \gamma }=P_{\lambda '\mu '\nu '}x_{,\alpha }^{\lambda '}x_{,\beta }^{\mu '}x_{,\gamma }^{\nu '}$ ${\ displaystyle P _ {\ alpha \ beta \ gamma} = P _ {\ lambda '\ mu' \ nu '} x _ {, \ alpha} ^ {\ lambda'} x _ {, \ beta} ^ {\ mu '} x_ {, \ gamma} ^ {\ nu '}}$ . Magnitude $P_{\alpha \beta \gamma }$ ${\ displaystyle P _ {\ alpha \ beta \ gamma}}$ is a tensor. The proof can be easily generalized to any number of superscripts and superscripts ^[1] .

Notes

↑ Dirac, 1978 , p. 14.

Literature

Dirac P.A.M. General theory of relativity. - M .: Atomizdat, 1978. - 64 p.

[_1e6f8761a02e6fc2-1] Dirac, 1978 , p. 14.