# Lorentz invariance and General invariance

#### mtak0114

Hi
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 $$p^\mu p_\mu = m^2$$, 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 $$V^\mu W_\mu = C$$ 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 $$u^\mu a_\mu = 0$$.

How does this change for general invariance

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

$$V^\mu W_\mu = C$$

is this still a constant C or is it a spacetime dependant quantity $$C(x^\mu)$$?

What about $$u^\mu a_\mu = 0$$
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

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