How Does Mass Influence Spacetime Curvature?

[bkelly13]

TL;DR Summary

Does a large mass pull spacetime in closer to it?

I have read widely about spacetime but not deeply. I am not good with high mathematics. I am aware of and comfortable with the concept: “matter tells spacetime how to curve, and curved spacetime tells matter how to move.” But not much deeper. This is a thought experiment.

Is there an equation that calculates how mass curves spacetime? Specifically, the question is how much a black hole, or any large mass, will draw in space time. I suspect it does, but am not positive this is a valid concept.

To be explicit, presume we stand off from a solar system in space and that we can see the lines of space and that we are not affected by changes. Presume we can see the lines where planets A, B, and C orbit the sun S. We replace sun S with one that is ten or a hundred times as massive. How would those circles for the planet orbits move? From our distant and unaffected location, the presumption is we would see the line for each planet would move in closer to the sun and the closer ones would be affected more than those further away. Would those orbit lines move? If so how much would they move? (Let’s ignorine the sudden change on the planets themselves for several reasons.)

But how would we measure it? From our location use Pythagorean’s theorem. Set the adjacent as the line to the sun, the hypotenuse as the line to each of the planets, and the opposite is the distance between the sun and the planet. And since this is a thought experiment, presume our visual lines of sight to the planets and the sun are not changed by the changes in mass and that we can see the changes instantly.

Is this a valid thought experiment? Your thoughts please.

 

[Ibix]

Is there an equation that calculates how mass curves spacetime?

Einstein's field equations.

Specifically, the question is how much a black hole, or any large mass, will draw in space time. I suspect it does, but am not positive this is a valid concept.

It does not. "Pulling in spacetime" doesn't make sense. Outside of the "river" model of a part of Schwarzschild spacetime (which I think is a poor description that sets people up for failure in less trivial circumstances) I don't think "pulling in space" makes sense either.

How would those circles for the planet orbits move?

I doubt there's a sense in which you can do the thought experiment you're intending.

If you mean instantaneously substitute a larger mass for the Sun, you can't do that - attempting to force it in to the maths will lead you to a 1=0 contradiction.

If you mean smoothly increase the mass of the Sun you could do that, but the result would depend on how you did it and how fast, and the maths may be rather challenging. The answer will probably be highly elliptical orbits once the mass settles down.

If you mean "consider the case where the planets are in orbit around a different star" then the question is what orbits do you mean. You can certainly solve this one, but it's not going to tell you anything except "orbits are circular or elliptical".

 

[Ibix]

we can see the lines of space

I don't know what you mean by this, but I suspect you've been mislead by pictures with warped gridlines around the Earth. Unfortunately they're little more than artist's impressions of... something. Maybe something real, maybe not. Either way "lines of space" is not a concept in relativity.

 

[phinds]

I have read widely about spacetime but not deeply.

Based on your question, I'd say it's clear that the reading you have done has been in popular science presentations rather than peer reviewed papers or text books.

Those are entertainment, intended primarily to make someone money and somewhat secondarily to interest people in science. They are NOT science education and will generally lead you astray much, or even most, of the time.

 

[bkelly13]

I am a bit surprised to the resistance of the concept of a thought experiment. Seems to me that it is a valid concept to imagine doing something something imposible to get to some condition then thinking about the new condition and making comparisions.

Specifically, the question is how much a black hole, or any large mass, will draw in space time. I suspect it does, but am not positive this is a valid concept.

It does not. "Pulling in spacetime" doesn't make sense. Outside of the "river" model of a part of Schwarzschild spacetime (which I think is a poor description that sets people up for failure in less trivial circumstances) I don't think "pulling in space" makes sense either.

Consider something like an asteroid moving through space. It tends to go in a straight line. Then it approaches a star. From the perspective of the asteroid it continues in a straight line. From the perspective of an observer the path now curves toward the star. My interpretation is that the star has curved space time. Since the asteroid curves toward the star the star has, to some amount, drawn in spacetime.

 

[Ibix]

Seems to me that it is a valid concept to imagine doing something something imposible

Depends what you mean by "impossible". If you mean "in some way beyond our technological capability", sure that's fine. If you mean "in a way that the laws of physics specifically forbid" then no, you can't get anything useful. It's like asking "what would happen if I let go of a ball and it went down and up at the same time". The premise is contradictory and you can't do anything with it.

Asking what happens if the Sun suddenly doubled in mass violates local conservation of stress-energy, which is directly baked in to the Einstein field equations. You can get from "the Sun doubles in mass out of nowhere" to 1=0 in two lines of maths. You cannot get anything useful from it.

So no, we don't resist a thought experiment. I am simply telling you that your thought experiment contains a subtle self-contradiction and you can get no sense from it. And all the ways I can think of to fix it give you results that depend on details of choices you make in fixing it.

From the perspective of the asteroid it continues in a straight line. From the perspective of an observer the path now curves toward the star.

This is something of a misstatement. The path is either straight or it is not - there is no "perspective" that can change it from one to the other.

The asteroid's path through spacetime is straight. Its path through space is curved. All observers will agree this.

My interpretation is that the star has curved space time

Yes.

the star has, to some amount, drawn in spacetime.

No. This statement makes no sense. I suppose you are thinking of the rubber sheet picture of spacetime. Unfortunately, again, that's vaguely illustrative at best and actually misses out the major component of curvature. Primarily it illustrates the difference between Newtonian and relativistic predictions, which are so tiny compared to the predictions themselves that it took us two centuries to notice.

 

[cianfa72]

As @Ibix said, the asteroid's path through spacetime is geodesic hence it has by definition zero path curvature (do not be fooled from spacetime curvature -- i.e. from the fact that there is in general non-zero geodesic deviation).

 

[bkelly13]

If I were the asteroid I would not feel any change in my trajectory. I could detect it but not feel it. I suspect, not certain, that accelerometers would not detect the accelerations of traveling in the curved line. From my perspective I would be continuing on a straight line.
The new trajectory curves in towards the star, not away, and not unaffected.
These two concepts produce the thoughts that spacetime is drawn in to the star.

 

[berkeman]

Thread closed for Moderation...

 

[berkeman]

We are trying to figure out if this question was asked by his other brother Darryl a couple years ago. We'll figure it out by tomorrow...

TL;DR Summary: Does mass draw space-time in to it?

Presume we look at a two-dimensional view of space time, with no local masses, and we draw a grid of equidistance spaced lines. The intent is to look at space but not time.
As we begin, we look in all directions and the grid lines are evenly spaced.
Begin adding mass to the center of the grid. To my understanding, so far, the mass will draw the lines of space in towards it. They become closer together in closer to the mass.
We might also draw those grid lines as circles about a point in space. As we add mass, space is distorted and the lines are closer to the mass. They are closer together.

Here is the question:
Looking some distance from the mass, and as we add mass, are the circles drawn in towards the mass?
That means that space time is compressed within some distance from the mass.
It also indicates that space, some greater distance from the mass, is stretched. The circles are further apart.
Are these valid conclusions?Thank you for your time.

 

[berkeman]

Okay, that original thread was closed due to personal speculation, which is off limits at PF. This new incarnation of that user is asking some of the same questions in this thread, but so far has not tried to introduce any of that previous speculation.

Thread is reopened provisionally.

 

[PeterDonis]

I am a bit surprised to the resistance of the concept of a thought experiment.

A thought experiment does not mean "just make something up" or "just throw some words together". It means picking some aspect of the laws of physics, drawing logical consequences, and seeing what they say.

When you talk about "mass pulling in spacetime", you are not picking some aspect of laws of physics. You are just throwing some words together. The laws of physics are not expressed in words. They are expressed in math. So doing a thought experiment in physics means doing the math.

My interpretation is that the star has curved space time.

Yes, but...

Since the asteroid curves toward the star the star has, to some amount, drawn in spacetime.

No. Curving spacetime is not the same thing as "drawing in spacetime".

Consider an analogy: two people at different points on the Earth's equator start moving due North. Their paths are initially parallel, but as they go their paths are drawn together--and eventually they meet at the North pole. This indicates that the Earth's surface is curved. Does it also indicate that the Earth's surface gets "drawn in" by something? Does the Earth shrink as the people move North?

 

[bkelly13]

Okay, that original thread was closed due to personal speculation, which is off limits at PF. This new incarnation of that user is asking some of the same questions in this thread, but so far has not tried to introduce any of that previous speculation.

Thread is reopened provisionally.

I did not realize that personal speculation is off limits. That is, in part, a specific goal of my series of questions and comments. I will stop now. I did not want to just abandon so am posting this.

 

[PeterDonis]

I did not realize that personal speculation is off limits. That is, in part, a specific goal of my series of questions and comments.

Given that, this thread is now closed.