Is the stress energy tensor continuous at the interface between an object and vacuum? The stress energy tensor has (as I understand) zero value in the region where there is no matter and non-zero value where there is matter. Suppose there is one matter object in space, now how does the value of the stress energy tensor change as we go from inside of the object to outside, is it a continuous change or does it jump to zero?
 A: The stress-energy tensor is a local property. Its value at a point is determined by the properties at that point and not by the surroundings of the point.
Consider a hypothetical object with a sharp edge i.e. the density changes discontinuously from a finite value to zero at the edge. In that case the stress-energy tensor will also change discontinuously. If the object is static and we can ignore any pressure within it then (as usually written) the only non-zero component of the stress-energy tensor will be $T_{00}$, and this is just the density. So at the edge of the object $T_{00}$ will change discontinuously from a non-zero value to a zero value.
I don't know whether this will help, but I think the best way to understand the stress-energy is to look at the stress-energy tensor for a point particle:
$$ T^{\alpha\beta}({\bf x},t) = \gamma m v^\alpha v^\beta $$
at the position of the particle and zero everywhere else. The variable $v$ is the velocity vector $(c, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt})$ i.e. it is the derivative of the position with respect to coordinate time (not proper time).
So if your object is a stationary point particle then at the position of the particle $T^{00} = mc^2$ and all other components are zero. Everywhere apart from the position of the particle all components of $T$ are zero. Any extended object can be (conceptually) built up as a collection of point particles.
A: Is it a continuous change or does it jump to zero?
It jumps from zero. Think about a photon. It has an non-zero "active gravitational mass", and it's just an E=hc/λ wave in space. It doesn't have an outside, there is no discontinuity. But put this photon through pair production to create an electron and a positron. and each exists as a standing wave. We can diffract electrons. The wave nature of matter is not in doubt. So we can reason that the electron is just a 511keV wave in a closed spin-½ path rather than a linear path. And then we can reason that because it's in a closed path, the field variation looks like a standing field. Standing wave, standing field. From this we can reason that this "object" isn't some billiard-ball thing that has an inside and an outside. It's more like a cyclone. The eye of the storm is the bit in the middle, but that's where there is no wind. That's where there is no hurricane, because wind is what it is. It's the same for an electron. It isn't some billiard-ball particle that has a field, because field is what it is. And that field doesn't shut off at some place "outside" the particle, just as the storm doesn't end when you go outside the eye. Instead, that's where it starts.
