Does General Relativity allow sudden gravitational source changes?

With the Bianci identity and the Einstein field equation it is pretty easy to show from first principles that in general relativity the stress energy tensor always obeys $$\nabla_\nu T^{\mu\nu}=0$$

This means that if you draw a small box in spacetime (i.e. a box that has finite extension in time as well as in space) then any energy and momentum density that enters one face must exit another face of the box. This is a continuity condition, and indicates a locally conserved quantity, which in this case is the four-momentum.

Does General Relativity allow sudden gravitational source changes?

Yes, $T^{\mu\nu}$ can change suddenly, provided that those sudden changes obey the above continuity equation. This is a severe restriction, but it is not a complete elimination of all rapid changes. The example of the "exploding sun" is an example. The momentum density terms of $T^{\mu\nu}$ could change suddenly, but only in a fashion that conserves momentum, such as splitting into two chunks that go in opposite directions.

By analogy, you can think of this similar to how the electromagnetic four-current obeys the continuity condition $\nabla_\mu J^\mu = 0$. The difference being that in the latter it is a scalar (charge) that is conserved while in the former it is a vector (four-momentum) that is conserved. In that sense, the stress-energy tensor is the "flow" of the conserved four-momentum, just like the four-current is the "flow" of the conserved charge. Also, note that in both the gravitational and the EM case, the source of the fields is the "flow", not the conserved quantity itself. This makes the analogy pretty useful. Furthermore, in the weak field limit the Einstein field equations reduce to a set of equations with the same form as Maxwell’s equations. This makes the analogy mathematically accurate in the appropriate limit.

So, let's apply this analogy to understanding your question and arguments.

the effective source strength ... the total effective source strength ... the source strength ... that source strength

If we look at the EM case, the conserved quantity is the charge, a scalar, but the source is a vector, the four-current "flow" of the charge. So even in EM, there is not a single "strength" number that represents the source. There is a whole vector that represents the source.

The gravitational case is even stronger. In the gravitational case even the conserved quantity itself is not a scalar that could be represented as a single "strength". And when you look at the "flow" that is the source of the gravitational field, it makes even less sense to speak of a strength. You simply cannot represent $T^{\mu\nu}$ as a single-number "strength".

So, the source can undergo sudden changes, but there is no such thing as the source "strength", so there is no sense in which the non-existent source "strength" could be said to change suddenly.

Now, there is one situation where the equations can be simplified to the point that the source can be summarized as a single-number "strength". This is the special case of spherical symmetry. It is important to remember that this special case where a source "strength" is meaningful is due to the spherical symmetry and is not generalizable to non-spherically symmetric cases. In particular, it does not apply to a sun which splits into two halves, but it does apply to a sun which explodes spherically outward (or collapses spherically inward).

Consider the EM case. Suppose we have a spherical charge which suddenly explodes spherically outward. In this case the EM source $J^\mu=(\rho,\vec j)$ has an immediate change, we immediately go from $\vec j=0$ to $\vec j = j(r) \hat r$ and $\partial \rho /\partial t$ related to $\vec j$ by the continuity equation. So, the source, a vector quantity, has changed everywhere.

But what about the source "strength" which is a simplified quantity due to the spherical symmetry? It turns out that due to Newton's shell theorem, the source strength for the E field is unchanged, and because there is no monopole radiation for EM, the source strength for the B field is also unchanged. Essentially, the suddenly appearing currents produce B fields that curl around $\vec j$, and the curling B field from each point is canceled out by the B field from the neighboring points. So the spherical symmetry prevents the sudden change in the source vector from manifesting as a sudden change in the source "strength".

The gravitational case is similar. An exploding sun will immediately have source terms that change like $\vec j = j(r) \hat r$. Those will produce gravitomagnetic fields that will be canceled out by their neighbors. And the spherical symmetry and lack of monopole gravitational radiation will again prevent the sudden change in the source tensor from manifesting as a sudden change in the source "strength". The proof that gravity behaves in this fashion is called Birkhoff’s theorem. The theorem is not an approximation, so although I used the EM analogy to explain how it works, it is not limited to cases where the EM approximation is good.

So, the issue that you have is not with sources being unable to change due to the conservation laws (they can). Rather, the issue is with the source "strength" being unable to change due to the spherical symmetry that is required for the source tensor to be simplified as a single "strength" number. Spherical symmetry is highly restrictive and so speaking of source "strength" places you into a restrictive scenario. Your confusion comes from confounding the general source with the very restrictive concept of a source "strength", specifically in non-spherical scenarios where the source "strength" concept does not exist.

Consider a system of two objects of equal total energy, moving directly towards each other with the same speed $v$, which then collide and remain at rest (that is, the reverse of the "exploding sun" example). Just before the collision the sum of the stress-energy tensors for the two objects has diagonal terms only, because the momentum terms cancel, but the transfer of momentum terms relate to the square of the speed and have the same sign, so the total effective source strength is enhanced by the factor $(1+v^2/c^2)$. Just after the collision, the transfer of momentum terms are now zero and the only non-zero term is the energy, which is locally conserved so it should be unchanged. This means the source strength has abruptly lost the $v^2/c^2$ term.

The source that you describe here has a changing quadrupole moment. Only the monopole term can be represented as a single "strength". The dipole term requires a vector, and the quadrupole term requires a tensor. So, the source in this scenario cannot be summarized with a single "strength". It requires a tensor representation of the source.

Because the quadrupole moment is changing, there will be a measurable change in the field seen by a distant observer. However, those measurable field changes will propagate outward at $c$ as gravitational radiation. So although the quadrupole moment can change suddenly, the field will not.

Again, consider the EM analogy. A pair of charges going towards each other would have a changing quadrupole moment and would produce a quadrupole field. But that quadrupole field could not be described by a single “strength”. The correct way to analyze it is through a multipole expansion, not just a single “strength” number.

This appears to make it possible that the gravitational source strength can be varied abruptly (at least by a small amount), for example when objects collide or a support breaks. This seems unexpected, but appears to me to be valid. Is this correct? If not, where does it go wrong?

It goes wrong in assuming that the source has a relevant single number “strength” at all. Yes, you can change the SET immediately subject to the continuity condition. No, such internal changes will not change the monopole term and hence will not change the “strength”. Such changes may suddenly change the quadrupole term, and produce gravitational waves. But the quadrupole terms are represented by a symmetric trace-free tensor of rank 2, not a single “strength” number.

Again, consider the EM analogy. You can vary the current density abruptly, subject to the continuity condition. You cannot change the “strength” which would be the monopole term or the net charge. You can suddenly change the dipole terms, but the dipole terms cannot be represented as a single number.

The Einstein field equation is a system of differential equations relating terms of the Riemann and the stress energy tensors. And this system of equations is valid for each point of the space-time.

In the example of two colliding objects, depending on the point of the space-time, we are relating the Riemann tensor of one or the other object with the stress-energy tensor in this point.

So, I can't understand

the sum of the stress-energy tensors for the two objects

If we want to know the gravity field in the vacuum, out of those bodies, (where the stress energy tensor is zero), it is necessary somehow to integrate the contributions of all that points. An analogy is applying Biot-Savart to find the magnetic field along the axis normal to a circular circuit with a constant current. The sources are the $\vec J$'s along the circuit. The sum of all $\vec J$ is zero, but of course there is a magnetic field out of the circuit. That vectorial sum has no meaning to calculate the field.

I have been reminded that if this question relates to a monopole situation, where the effective scalar gravitational potential changes abruptly, then it is already known that for a spherically symmetrical situation, Birkhoff's theorem shows that dynamic internal changes, such as radial oscillations, do not affect the external field, even though they affect the internal pressure and its overall integral, which in a static situation contributes to the gravitational source strength. Although the situation in this question is more general, it seems reasonable to assume that whatever mechanism compensates for the dynamic pressure changes as part of Birkhoff's theorem would also apply in this more general case.

I haven't managed to understand the mechanism of Birkhoff's theorem in sufficient detail to work this out for myself, but back in 2018 I emailed Kenneth Nordtvedt to ask about the spherically symmetric case with dynamic changes and wondered if he had any explanation as to what compensates for the pressure changes. He did not have a specific answer, but kindly provided the following response, showing that he clearly understood my question and had some useful ideas about possible answers:

In my paper Phys Rev 169 p1017 (1968) in which I calculated the MG/MI ratio for general metric theories of gravity, I purposely rejected using concepts like pressure, etc. I used a gas model for a star with neutral "atoms" moving at various speeds as the mechanism which resisted gravitational collapse. I indeed found a left over virial term which only vanishes if the massive body is in equilibrium. If body were oscillating about equilibrium the virial would have oscillating non-zero value.

In later work I found that when body oscillates about equilibrium it generates a non-zero value for d h(vector) /dt with h(vector) being the mixed space-time compoents of the metric tensor g_munu. Since the acceleration of a test particle near this massive body is given by grad g_00 + d h(vector) /dt, one finds that the time varying h(vector) term produces a contribution which cancels the non-zero virial contribution to MG of the massive body oscillating about equilibrium (at least in the cases where the body has some spherical symmetry properties).

Also it must be considered that the non-zero h(vector) is producing some oscillating variations in the proper coordinates of space and time in the vicinity of the massive body. So pseudo accelerations could very well be induced in the non-proper space and time coordinates of a test particle.

...

For a massive body exploding with spherical symmetry, like in case of super novae, which is then far from being in equilibrium, one could speculate whether there might be some anomalous acceleration of test particles in vicinity of this exploding body, and which might be observable by astronomers? Presence or absence of same might be considered a new test of gravity for the interesting values of g_00 and h (vector) which would be generated by this exploding body.

Kenneth Nordtvedt

I think this is about as good an answer as I could hope to get. It seems likely from this that even in situations which are not spherically symmetric, there is some effect relating to tensor derivatives which cancels or smooths any attempt to make a sudden dynamic change, even though the source tensor terms can change abruptly. I will try to follow up his suggestions to see if I can identify the relevant compensating effect. It's clearly not something which is widely known!