How quickly does the Earth feel the effects of a halved Sun's mass?

[PeterDonis]

By 'specific problem' I meant whether or not the gravitational effects from the Sun to the Earth are instantaneous

I answered this in post #26. Note, though, that another caveat is that if the system is stationary (i.e., the gravitational field does not change with time), then there is nothing to "propagate", so it makes no sense to ask how fast gravity "propagates" under such circumstances.

 

[pervect]

I have a doubt about gravitation. Suppose the mass of the Sun halves in an instant, after how long does the Earth ''notice'' it?
That is, does the gravitational force also decrease instantaneously?
Instinctively I would say yes, but I don't understand why it should be so. If, for example, we attach something to a spring and give a quick tug, the object doesn't immediately feel the force; so why should it be any different for gravitation?
Can you clarify this doubt? Thank you for any intervention.

P.S. You are very free to move this thread to another Forum if you feel that I have not posted it in the more appropriate one...

As others have pointed out, it's not possible for the sun to lose half it's mass in an instant. Thus, asking what the equations of GR say in this case is like asking what Maxwell's equations say about the electric field of a disappearing charge. The answer is similar - Maxwell's equations aren't consistent with charges just vanishing, and the equations of GR aren't consistent with mass vanishing.

To get around this, it's been suggested that you rephrase the question, though if someone has recommended exactly how, I haven't seen it.

I'll fill in this lack by suggesting how you can rephrase the question. While you can't make matter magically disappear, you can re-arrange it. Specifically, you can (in theory) blow up or explode the Sun.

The exact answer to your question will then depend on "how did I blow up the Sun"?

The easiest case to answer is if the explosion is spherically symmetrical. Note that to keep energy conserved, you'll need to include the source energy of the explosion in your calcuations. The "biggest boom" woud occur if the mass of the sun were totally converted to radiation in an instant, via a distributed, idealizazed, matter/anti-matter reaction.

There is an interesting question here - I was planning to invoke Birkhoff theorem, but when I looked at the fine print, this may not quite do the job, as it's not static. Which hapens I think because it's not a vacuum solution, either. :(.

However, I believe we can say that a spherical solution won't generate any gravitational waves, though I don't have a specific reference handy to shore up my recollection. And what I expect to happen in this case is that until the expanding wavefront of the exploding matter of the explosion reaches the observer, therre won't be any impact on the gravitation. If we imagine the "total conversion" explosion, this means there is no effect until the radiation from the explosoin reaches the observer.

IIRC - and again I don't have a specific reference - a non-spherical explosion does have the possibility of converting some of the energy of the explosion into gravitatioanl waves, so it becomes a harder problem.

A technical note - I'm assuming an asymptotically flat space-time, which is more-or-less required to talk about energy conservation in General Relativity.

 

[PeterDonis]

I believe we can say that a spherical solution won't generate any gravitational waves

That is correct. Any such waves would have to be monopole, and the lowest non-vanishing order for gravitational waves is quadrupole. This is well known and substantiated in the literature. (The basic reason is that gravity is spin-2; for spin-1, such as electromagnetism, the lowest non-vanishing order for waves is dipole; and you need spin-0 to get non-vanishing monopole waves.)

 

[renormalize]

the lowest non-vanishing order for gravitational waves is dipole.

Quadrupole.

 

[PeterDonis]

Quadrupole.

Oops, yes, thanks! Fixed now.

 

[berkeman]

Quadrupole.

So I have a quick question which is hopefully still on-topic for the OP's thread...

In my simplified scenario below, would the splitting hemispheres be considered a dipole, or somehow a quadrupole? If only a dipole, how would the change in gravitation propagate away from the explosive splitting event? Thanks.

To try to give you a better physical scenario and also make it easier to visualize, suppose that the Sun exploded all of a sudden and broke into two hemispheres that flew apart at high velocity, with the 2 halves moving perpendicular to the Solar System's orbital (ecliptic) plane.

The change in the Sun's gravitational field would propagate outward at the same velocity as the light from the Sun, so the changes in the gravitational field from the Sun would be "noticed" on Earth at the same time that we "saw" the two hemispheres separating and flying apart.

 

[PeterDonis]

would the splitting hemispheres be considered a dipole

As you've defined it, I believe so, yes. So no gravitational waves would be emitted. There would be a "change in the gravitational field" (more precisely, a change in spacetime curvature), but it would not be describable as gravitational waves.

 

[Vanadium 50]

Since the hemispheres are the same "charge" its a quadrupole. It's magnitude should be ~mz²/2. (Assumes separation is large compared to the sphere radius).

It will radiate.

However, there is also the near-field effect. Consider you are on a line in the same direction that the two haves separate in. Now half is closer and half is farther. But since this is an inverse square force, the part that moves towards you has a greater increase in attraction than is compensated for by the smaller attraction from the half that moves away.

A complication - it takes energy to move things around like this, and that energy gravitates too. That needs to be considered as well.

 

[PeterDonis]

Since the hemispheres are the same "charge" its a quadrupole.

Oops, yes, you're right.

A complication - it takes energy to move things around like this, and that energy gravitates too. That needs to be considered as well.

Yes, this is a good point.