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I am confused about these two related but different terms

Lorentz invariance/covariance and General invariance/covariance

As I understand it a Lorentz invariant is a scalar which is the same in all inertial reference frames i.e. it acts trivially under a Lorentz transformation

an example would be rest mass [tex]p^\mu p_\mu = m^2[/tex], all observers would agree on the value of the mass. But is this true for all scalars say for example the inner product between two arbitrary 4-vectors [tex]V^\mu W_\mu = C[/tex] would all inertial observers agree on the value of C? A good example may be the inner product between the 4-velocity and the 4-acceleration [tex]u^\mu a_\mu = 0[/tex].

How does this change for general invariance

[tex]p^\mu p_\mu = m[/tex] it is true that all observers would agree on the mass but how about for:

[tex]V^\mu W_\mu = C[/tex]

is this still a constant C or is it a spacetime dependant quantity [tex]C(x^\mu)[/tex]?

What about [tex]u^\mu a_\mu = 0[/tex]

I understand that this is still true in general relativity but is this a special scalar?

still very confused hope you can help

thanks

Mark