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t and g don't relate in Newtonian physics, so I doubt your old text will talk about it. I'll try to give the first steps towards what's involved. I don't expect people without a technical background to understand this.
The metric in Euclidean geometry is given by the pythagorean theorem, dL^2=dx^2+dy^2+dz^2. {x,y,z} are not intrinsically meaningful. There's nothing preventing you from going to any number of different coordinate systems. L does have meaning though. The distance between any two points is the same for everyone.
Special relativity modified this relation by introducing a fourth coordinate -- time:
dL^2=-(cdt)^2+dx^2+dy^2+dz^2
Just like {x,y,z}, t is not fundamental. You can do coordinate transformations on it. L is again the same to everybody.
This is used to define what it means for something to be moving at the speed of light independent of coordinate choices. If dL=0 along an objects path (in 4D spacetime), then it is moving at light speed. In the above coordinates, this means e.g. dx/dt=c, as expected.
Objects moving at less than the speed of light have dL^2<0 (its ok for dL to be imaginary). One can define what's called the proper time as dT^2=-dL^2. This is uniquely defined, and has many nice properties.
If dL^2>0, then this represents something moving faster than light. Its possible to show with the above equation that an object moving on this type of path will violate causality.
Now as stated before, there are many possible forms for dL depending on the coordinates chosen. If there's no gravity, though, it is always possible to transform back to the metric given above. Gravity modifies the equation for L such that it can no longer be brought into that simple form by any choice of coordinates. Its probably better to say that gravity IS the (invariant) difference in the true metric and the special relativistic version.
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