# Einstein's notion of "covariant"

In his The Meaning of Relativity, pg. $$11-12$$, Einstein explains the notion of "covariant" along the following lines:
Consider a point $$\mathbf x$$ on a straight line $$\mathbf x -\mathbf A=\lambda\mathbf B$$ in $$\mathbb R^3$$. Without a loss of generality we can assume that $$\mathbf B^T\mathbf B =1$$. Now, consider an orthogonal linear transformation (OLT) from the $$K_{\mathbf x}$$ frame to $$K'_{\mathbf x}$$ defined by: $$\mathbf x'=\mathbf a +b\mathbf x$$ where $$b=[b_{\mu\nu}]$$ is the transformation matrix satisfying $$b^Tb=I$$. Then the straight line in our transformed frame becomes $$\mathbf{x'-A'=}\lambda\mathbf B'$$ with $$\mathbf A'=b^T\mathbf A$$ and $$\mathbf B'=b^T\mathbf B$$. It therefore implies that straight lines have an underlying property which is independent of the system of coordinates. Formally, this depends upon the fact that the (vector) quantity $$\vec q :=(\mathbf x -\mathbf A)-\lambda\mathbf B$$ is transformed as components of an interval, $$\Delta\mathbf x$$. If $$\vec q=\vec 0$$ for one system of Cartesian coordinates, then $$\vec {q'}=\vec 0$$ for all systems. Thus we can say that the equation of a straight line is "covariant" with respect to OLT's.

What I think of the assumption $$\mathbf B^T\mathbf B =1$$ is that here Einstein is referring to set of homothetic transformations centered at the point $$\mathbf A,$$ that an OLT maps a homothety centered at $$\mathbf A$$ to one centered at $$\mathbf A'$$, preserving their scaling factor $$\lambda.$$ So, straight lines are mapped to straight lines by an OLT, the points on which are therefore called "covariant" vectors. Is my interpretation correct?

### Edit

After further reading I find that he is talking about the covariance of laws of physics expressed as equations: a law $$F=0$$ where $$F$$ may be a vector or scalar or tensor expression in general, if this law preserves its nature under a transformation, that is if it is transformed to $$F'=0$$ under the said transformation where $$F'$$ has similar mathematical form as $$F$$, then the law is said to be covariant with respect to that transformation.