Sun Disappearance: Impact on Earth's Path

[Will Learn]

TL;DR Summary

I've heard the following comment during lectures on General relativity:
Under Newtonian gravity, if the sun suddenly disappeared then Earth would instantly change it's trajectory (flying off in a straight line). Under General relativity things are different and the Earth would continue in its usual orbit for 8 minutes before any change. (8 mins = approx. light travel time from sun to earth).

Hi. I'm new and hoping for some discussion. I've been looking at some physics at home on my own while Covid-19 restrictions keep me off work. I'm not registered as a student anywhere and so don't have any chance to chat with other students or university staff. Hence, I'd be grateful for some discussion here.
The summary box explains the situation but I'll flesh out some of the details here:

1. Newtonian Gravity should be familiar from school level physics. It is assumed that this gravitational force acts over an arbitrary distance instantly. Planet Earth's orbit is explained with Newtonian mechanics (inlcuding Newtonian gravity), as usual.

2. General Relativity explains gravity differently. There is a metric defined on all of spacetime and planet Earth follows a geodesic path as usual.

3. This is where a comment has often been thrown into the You Tube lectures and other short videos I have been watching. The comment is something like the following (This is NOT an exact quote):
Under Newtonian gravity, if the sun suddenly disappeared then Earth would instantly change it's trajectory (flying off in a straight line). Under General relativity things are different and the Earth would continue in its usual orbit for a while* without any change.
(* Sometimes, instead of saying "a while" they say 8 minutes or a similar approximation of the time it takes light to travel from the sun to the earth).What I think: That comment is at best partially true. It's certainly not obvious and there are so many assumptions and caveats required to make it true that it isn't really fit to be anything more than a throw-away statement.
1. How many other people have heard something like this? I get the impression it's a commonly used statement but maybe I'm just listening to the worst lectures and YT videos.
2. What are your thoughts?
I can write more about why I think the statement is rubbish but I'm aware I've already taken a lot of your time and I'm only too happy to hear from others.

Thanks.

 

[Will Learn]

---> It has just been brought to my attention that there is a similar thread from "Willowz" in 2009 entitled "If the sun suddenly dissapeared". I'm going to read that now.

 

[PAllen]

In general relativity it is impossible to formulate the disappearance of the sun. Thus the question is meaningless. As if asking what the square root of 2 is if 2+2=5.

 

[Will Learn]

Thanks PAllen.

I've also skim read through the other, older thread I mentioned earlier.
It seems that I have an answer to one of my first questions --> By the look of all the similar threads that have been posted here, many other people HAVE heard a similar statement while learning G.R.

I also agree with you (PAllen) about the impossibility. I would cite the local conservation of energy equation: The sun just can't "dissappear suddenly" because the stress-energy tensor can't be that discontinuous, it has to be differentiable with respect to every co-ordinate including time for the conservation of energy equation to even make any sense. I'll consider this good reason No.1 why the statement is terrible and should not be used in lectures or texts that aim to teach GR.

Thanks again. I've put a like on your comment, I'm new here and I don't know if there's anything else I'm supposed to do.

 

[Will Learn]

One thing that no-one seems to have mentioned directly is that the 8 minute delay all rests on the assumption that the sudden change propagates just like a gravitational wave.
There is no evidence for this as far as I know and no way you could glean the information from the Einstein field equations.

Gravitational waves do NOT arise as a result of sudden and total dissapearances of mass. Instead gravitational waves are just the result of a smooth, continuous, if rapid re-positioning of mass. For example, two Neutron stars spiral around each other and then merge. A smooth time change in the stress-energy tensor is something we can work with and get information from the Einstein field equations.

By comparison, the sudden dissapearance of mass would seem to create an abrupt (discontinuous) change in the stress-energy tensor at that point. We may assume a corresponding discontinuous change in the metric occurs at that place and time. If this is anything like a gravitational wavefront then it is more like a vertical shock wave instead of some smooth oscillation. It is then, as far as I can see, wild speculation that this should propagate just like the real gravitational waves we have observed.

 

[Will Learn]

Just read through some more of the older threads similar to this one: A few people did mention that we must just assume the change in the metric propagates like a gravitational wave.

Even if we do make that assumption I think there is still another problem. It is the metric field that is experiencing this change. New values for the metric are being assigned to the metric field at all the points in space where this gravitational wave is assumed to have reached. So 1 second after the sun disappears, all points in space within a sphere of radius 1 light-second have been given the new metric values. (I should say they WERE previously thought to be within 1 light-second). The problem is that the metric value has changed here, so the way we measure distance and time here has changed. As a result that sphere probably doesn't measure up as 1 light-second anymore.

To analyse the trajectory of planet Earth you have to measure FROM somewhere. It would seem (to me) most sensible to check if that orbit is changing by measuring the radius of that orbit. Measuring from the Earth to where the sun used to be is a problem because (as outlined above) some of the space between Earth and the sun has already expanded. As the gravitational wave continues to progress toward earth, more of that space has expanded. As a result it would seem that the radius of orbit HAS been changing before the gravitational wave actually reached planet earth.

To get the impression that Earth's trajectory was entirely unchanged for 8 minutes you would have to avoid measuring over any part of space that has already been touched by that gravitational wave. This could be done if you were only looking at space locally to the planet Earth for example.

Please correct me if any of this seems wrong. I'm fairly sure that the statement is terrible and should not be used in any teaching to introduce G.R. but maybe I've just completely mis-understood the situation.

 

[PAllen]

This may or may not be above your head, but let me describe a little more precisely what the problem is with "sudden disappearance of the sun" in GR (general relativity). In GR, a solution is the complete history of the universe, represented mathematically as a manifold with pseudo-Riemannian metric tensor , typically called g. From the metric tensor, you can construct another tensor called the Einstein tensor often written G. By construction, as forced as 2+2=4, this tensor's covariant divergence is zero. The physics (rather than math) of GR is contained in the Einstein field equations which state that the stress-energy tensor is proportional to G. This means that the divergence of the stress energy tensor is also 0. And this means that the sun can't just disappear.

Somewhat similar, is that in Maxwell's equations for EM, there is no way to represent that some electric charge vanishes (as opposed to flows from one place to another).

 

[FactChecker]

You should be careful here. You want to disallow instantaneous changes. There are changes that are very fast and where the time required for the change is much smaller than the time that the gravitational waves get to us. So they can be thought of as practically instantaneous. The recent measurements of gravitational waves at LIGO took only 0.2 seconds for the "chirp signal" but took about 1.4 billion years to reach us. In the context of this question, one can think of that as a practically instantaneous change that did not affect us gravitationally for 1.4 billion years.

 

[PAllen]

You should be careful here. You want to disallow instantaneous changes. There are changes that are very fast and where the time required for the change is much smaller than the time that the gravitational waves get to us. So they can be thought of as practically instantaneous. The recent measurements of gravitational waves at LIGO took only 0.2 seconds for the "chirp signal" but took about 1.4 billion years to reach us. One can think of that as a practically instantaneous change that did not affect us gravitationally for 1.4 billion years.

Note, though, that this says, for example, that the sun can explode in very brief period compared to light travel time to some observer. However, the effects of an explosion are not even qualitatively similar to a disappearance. The latter simply cannot be represented in GR.

 

[phinds]

How would the sudden dissapearance [sic] of the sun affect Earth's path?

check this out; it specifically mentions your scenario

https://www.physicsforums.com/insights/how-to-avoid-breaking-physics-with-your-what-if-question/

 

[FactChecker]

Note, though, that this says, for example, that the sun can explode in very brief period compared to light travel time to some observer. However, the effects of an explosion are not even qualitatively similar to a disappearance. The latter simply cannot be represented in GR.

I think that is dodging the purpose of the question -- does a gravitational effect transmit instantaneously. Just because the scenario is flawed does not mean that the core question can not be answered simply and directly. IMHO, saying that the scenario can not happen is side-tracking the intended question.
And discussions about the smoothness of the equations are also sidetracking the question.

 

[Will Learn]

Thanks again PAllen and FactChecker.

To PAllen --> It's taken me a moment to find my textbook and follow what you've written.
Div G = 0 is a consequence of the twice contracted Binachi identiy. When you said this construction was forced - did you mean "forced" as in artificially made to have that property or "forced" as in it must be true. If it's the later then I agree - it MUST have that property.
Anyway the rest of it makes sense but seems to boil down to Div T = 0 which is (by my understanding) exactly the same as saying the local conservation of energy prevents the sudden dissapearance of the sun.
I have more of a background in Maths than Physics and that is why I see the time discontinuity as the biggest problem. I won't differentiate functions that aren't differentiable so the Div T = 0 equation doesn't even make sense if T is not differentiable with respect to time. It's not as if the Divergence of T is something other than 0, it isn't anything as far I'm concerned and that is what prevents the sudden dissapearance of the sun.

To FactChecker ---> Yes seems reasonable, thanks. Earlier threads had comments like "it still isn't known that gravitational waves travel at c ". However I think that is old or inaccurate now. I haven't looked much at the chapter on gravitational waves but I believe we know that grav. waves propagate at c. Is that right?

To Phinds ----> I'll check your link now. Thanks.

 

[Will Learn]

check this out; it specifically mentions your scenario

https://www.physicsforums.com/insights/how-to-avoid-breaking-physics-with-your-what-if-question/

Wow, almost exactly the question I asked. More evidence that this is a statement that is in common use. Thanks.

I should make it clear that I'm asking about this statement because it appeared in videos I have seen about G.R. It's not a statement I put together myself. Personally I think it's an awfull statement and should not be used in the teaching of GR. My main question now is - why is it used so often? By the look of the responses I have got here, many Physicists have taken the time to think about and realized it's many flaws.

 

[Will Learn]

Somewhat similar, is that in Maxwell's equations for EM, there is no way to represent that some electric charge vanishes (as opposed to flows from one place to another).

Thanks for the advice but here's something that will make you laugh:
The textbook I bought myself does exactly this kind of thing. It keeps talking about this Maxwell guy, as if that will help me understand. I'm a Mathematician, I preferred it when they were talking about Manifolds. I've had to stop reading my text on GR and switch to a book called Electrodynamics by Griffiths because apparently this Maxwell guy did something so important that everyone is mentioning it.

 

[phinds]

apparently this Maxwell guy did something so important that everyone is mentioning it.

That's a lot like saying "this old guy, Einstein, seems to have done something important". Yep. Sure did.

 

[PeterDonis]

One thing that no-one seems to have mentioned directly is that the 8 minute delay all rests on the assumption that the sudden change propagates just like a gravitational wave.
There is no evidence for this as far as I know and no way you could glean the information from the Einstein field equations.

This is correct, but the reason it is correct is that, as you have already agreed, the "sudden change" in question (a sudden disappearance of the Sun, or any other massive object) is impossible according to GR. So of course there will be no way to derive the propagation of this impossible thing in GR.

A few people did mention that we must just assume the change in the metric propagates like a gravitational wave.

This is a misleading way to put it. The metric doesn't "change". Spacetime is a 4-dimensional manifold with metric; the metric just is. When people say the metric "changes", it is a sloppy way of saying that the metric is different at different points of spacetime. When people say that changes in the metric "propagate" as gravitational waves, it is a sloppy way of saying that the metric at points of spacetime on the future light cone of some gravitational wave source (such as a black hole merger) has wavelike properties (or more precisely that the difference in the metric at those points from some average "background" value has wavelike properties).

When properly understood, it is true that "changes in the metric propagate like a gravitational wave"; that is one way of viewing what gravitational waves are. But the "changes in the metric", as you note, are not due to mass suddenly disappearing or appearing. They are due to particular continuous properties of certain sources (like black hole mergers).

 

[PeterDonis]

why is it used so often?

It might be used often in pop science presentations, like videos. I don't think you will find that it is used often in actual textbooks or peer-reviewed papers.

 

[Will Learn]

I think that is dodging the purpose of the question -- does a gravitational effect transmit instantaneously. Just because the scenario is flawed does not mean that the core question can not be answered simply and directly. IMHO, saying that the scenario can not happen is side-tracking the intended question.
And discussions about the smoothness of the equations are also sidetracking the question.

I've had a bit longer to think about it now and might go about offering an answer like this:

We can preserve the essence of the original statement by simplifying our universe. We need only consider a spacetime where there is just the sun and the earth. Furthermore, we are only interested in the effect on Earth's path, so we can treat the Earth as a "test mass" - a point particle that experiences the metric in its surroundings but does not contribute to the stress-energy tensor itself.
With this simplification, we can see that while the sun is there, the scenario meets the criteria for the Schwarzschild solution. The Sun plays the role of the gravitating mass.
The Schwarzschild solution has been well studied. Outside of the sun, our spacetime is entirely and uniquely described by the Schwarzschild metric and its corresponding natural co-ordinates (see Birkhoff's theorem - we are assuming the sun was perfectly spherical and symmetric). In particular the metric is not dynamic, it is unchanging with time.

AFTER the sun has disappeared our spacetime becomes empty space without any mass (the Earth is still only a test mass). We know the solution of Einstein's field equations for this situation equally well. It is just flat empty space. It has the usual metric Eta in the usual (cartesian) co-ordinates. In particular the metric is again completely static, unchanging with time.

The metric is therefore ALWAYS static at all times except possibly at the instant when the sun disappeared (and we have no way of knowing what really happens there - but it's only one instant). By the nature of the original statement - that the sun suddenly dissapears - we have to assume that no co-ordinate time elapses while the sun dissappears. So it would seem that the universe has instantly changed from being one way to being another way. This isn't JUST a co-ordinate change, although the Ricci curvatures may look the same, the full Riemann Curvature Tensors of the two spacetimes are intrinsically different. In particular the components of the Riemann curvature tensor are constant for flat space. There has been an intrinsic change in the nature of the universe at every point and that change was instant.

In many respects it's ike the instant change that was seen under Newtonian gravity. However, there is one caveat we have to add: The co-ordinates we used to describe our spacetime have changed before the sun disappeared and after the sun disappeared. It's not obvious how you could map one set of co-ordinates seemlessly to the other. Fortunately we don't need to worry too much: With the identification of the change in curvature we already know that the metric is intrinsically different at every point in space and that change occurred instantly. That change, at-least, did seem to transmit instantaneously.

(I can only apologise for my part in any earlier discussions of smoothness and differentiability).

 

[PeterDonis]

might go about offering an answer like this

No. This will not work. You can't just make the Sun disappear. This point has already been made repeatedly.

 

[PeterDonis]

The OP question has been addressed. Thread closed.

 

[PeterDonis]

One additional follow-up:

I think that is dodging the purpose of the question -- does a gravitational effect transmit instantaneously.

The answer to that question is simple: no, it doesn't. In spacetime terms, the "gravity" at any particular event (which means the metric, spacetime curvature, and any other quantity associated with the spacetime geometry) is entirely determined by what is in the past light cone of that event. This is the technically correct way of stating what is often stated as "gravity propagates at the speed of light in GR", or something to that effect. (The statement that electromagnetic effects propagate at the speed of light has to be restated in a similar way in GR if it is to be technically correct.)

A good discussion of how to consistently pose questions about "the speed of gravity" in GR can be found in this classic paper by Carlip:

https://arxiv.org/abs/gr-qc/9909087