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 dependent 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

 

[atyy]

A scalar g(u,v) is a generally invariant quantity, and hence a Lorentz invariant quantity. However, it can be a spacetime dependent quantity - ie. has a different value at different points in spacetime (spacetime dependence), but at any particular point in spacetime it has the same value under arbitrary coordinate changes (generally invariant).

However, a different use of Lorentz invariant is the "form" of the laws of physics. In special relativity, Lorentz covariance means the laws are supposed to maintain the "same form" under a Lorentz transformation, where the "same form" means that we are NOT allowed to use Christoffel symbols (covariant derivatives disallowed) to write the laws of physics.

If we are allowed to use Christoffel symbols (covariant derivatives allowed), then any law is generally covariant, and hence also Lorentz covariant.