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Strees-energy-momentum tensor

 nothing changes in general. I wonder if this is a correct view of it.
I found that confusing !! really....

Let's consider an object moving in free fall relative to an observer following a particular path. If an identical particle at a different velocity, or an identical particle with the same velocity but also angular momentum then passes the observer, the particles will in general follow three different paths.

That's the physics ,I think we can agree.

In the view I gave, the second particle with angular momentum changes gravitational spacetime curvature relative to the first; the third particle with only additional velocity has a different 'visible curvature'....but it is not part of 'the amount of gravity'.

That is just a convention, but one that seems to make sense to me regarding fast moving particles not becoming black holes.

 Quote by dextercioby http://en.wikipedia.org/wiki/Tangent_space Both definitions here (first and second) ascribe an algebraic (i.e. according to the axioms here http://en.wikipedia.org/wiki/Vector_space) character to the tangent space in a point x of a general manifold. Where does the norm come from which would induce the topology on T_x (M) ? Your first sentence induces the set inclusion {vector spaces} $\subset$ {topological spaces} which is not not correct (the axioms of a vector space don't mention a norm, so there wouldn't be any norm-induced topology ). The right set connection is: {topological vector spaces} = {vector spaces} $\cap$ {topological spaces}
Oh, yes, you're absolutely right, I was wrong in admitting a norm as an axiom (or a direct consequence of an axiom) of vector space definition. I guess it comes from my time spent studying functional analysis!
Anyway, thanks for clearing that up for me.
But that's troubling, as in the wikipedia article of the exponential map it is said explicitly
 The radius of the largest ball about the origin in TpM that can be mapped diffeomorphically...
So a notion of distance is used... I wonder how.
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