Understanding Gravity & Orbits in Light of Relativity

[PhysicsBob]

I've been re-reading Hawking's Brief book (for the fourth time :-) and trying to understand a bit more.

He discusses how General Relativity tell us that the planets don't orbit due to gravity, but due to the bending of space-time by mass/energy. That in fact, they are actually following a straight path (geodesic) in curved space-time. I was able to wrap my mind around all that after repeated attempts. (Feel free to correct me if I missed the boat). So, gravity doesn't seem to really apply in the Newtonian way.

I also understand roughly how Relativity has issues at the sub-atomic level and the want for a unified theory.

With that in hand, what I am now confused about is Newton's Theory of Gravity and objects that are in between. For example, dropping a ball and having it head towards the earth. Is Newton's Theory still valid for such observations? Or is there some other part of General Relativity that applies here to explain this attraction of the ball to the larger mass? Is the Theory of Gravity still valid at some level in light of General Relativity?

 

[russ_watters]

Welcome to PF!

He discusses how General Relativity tell us that the planets don't orbit due to gravity, but due to the bending of space-time by mass/energy.

That is gravity.

With that in hand, what I am now confused about is Newton's Theory of Gravity and objects that are in between. For example, dropping a ball and having it head towards the earth. Is Newton's Theory still valid for such observations?

General relativity reduces to Newton's law in situations where there speeds are low and/or distances short (may be other constraints too...). It works fine for most purposes, even space travel at current speeds/distances.

 

[PhysicsBob]

Welcome to PF!

That is gravity.

General relativity reduces to Newton's law in situations where there speeds are low and/or distances short (may be other constraints too...). It works fine for most purposes, even space travel at current speeds/distances.

So are you saying that Newton's Theory is known to be just an approximation... but close enough for the limited situations you mentioned? In other words, at this point we know he was actually wrong (no space/time consideration), but his numbers are good for reasonable calculations in simple situations?

Does this then say that in the case of the dropped ball, that the Earth's mass does warp space/time to such a major extent that the ball not only doesn't orbit, but that is actually pulled towards the center of the Earth's mass?

 

[Nugatory]

So are you saying that Newton's Theory is known to be just an approximation... but close enough for the limited situations you mentioned? In other words, at this point we know he was actually wrong (no space/time consideration), but his numbers are good for reasonable calculations in simple situations?

yes, Newton's theory is an approximation. That doesn't make it wrong, just incomplete.

Does this then say that in the case of the dropped ball, that the Earth's mass does warp space/time to such a major extent that the ball not only doesn't orbit, but that is actually pulled towards the center of the Earth's mass?

The Newtonian prediction for a dropped ball on Earth is identical, to a level of precision far greater than our best instruments can measure, to the general relativistic prediction. Both theories predict that the dropped ball will follow the same trajectory, one that intersects the surface of the Earth ("falls to the ground").

 

[PhysicsBob]

The Newtonian prediction for a dropped ball on Earth is identical, to a level of precision far greater than our best instruments can measure, to the general relativistic prediction. Both theories predict that the dropped ball will follow the same trajectory, one that intersects the surface of the Earth ("falls to the ground").

But why does the ball "drop"? In the case of a planet orbiting, if I understand it correctly, the planet is moving in a straight line, but the warping of space-time causes it to appear to us in 3D as if it's orbiting. Is the gravitational force on the surface of the Earth so strong that it warps space-time massively - to the point where the ball is pulled to the surface via that dimension ? Or is space-time not really a consideration on the Earth's surface?

 

[Nugatory]

But why does the ball "drop"? In the case of a planet orbiting, if I understand it correctly, the planet is moving in a straight line, but the warping of space-time causes it to appear to us in 3D as if it's orbiting. Is the gravitational force on the surface of the Earth so strong that it warps space-time massively - to the point where the ball is pulled to the surface via that dimension ? Or is space-time not really a consideration on the Earth's surface?

When you release the ball, you're standing on the surface of the Earth 6400 kilometers away from the center that you're using to define the $\vec{r}$ in Newton's $\vec{F}=mGM_E/\vec{r}^2$. To really see what's going on, you have to imagine the Earth as a point mass instead of a sphere with a diameter 12,800 kilometers; and you're standing on a high-dive platform 6400 kilometer high.

The ball only drops straight towards the center of the Earth if it is released with no tangential velocity at all. If there is any tangential velocity no matter how small, the falling ball is also moving sideways so will not fall straight towards the center. Instead, you've put it in a very acute elliptical orbit (long axis 6400 kilometers, short axis only a few tens of meters) with the low point on the opposite side of the center of the Earth from you and very close to the center. (In practice, of course, we don't observe this because almost all of this trajectory is in the region $r<6400$ km which lies below the surface of the real non-point Earth - instead we just observe that the ball collides with the surface of the earth).

Newtonian mechanics gets this result by solving for the behavior of a ball moving under the influence of a force given by $\vec{F}=mGM_E/\vec{r}^2$. If there is zero transverse velocity the solution is a trajectory that moves in a straight line to the central point. If there is any initial transverse velocity at all, the solution is an elliptical orbit around the central point (and this is how we calculate the orbits of the panets around the sun).

General relativity gets the same result (aside from tiny corrections that are well and thoroughly insignificant when dealing with balls and the Earth's gravity) by considering the straight lines (geodesics) in spacetime through the point at which the ball is released. The geodesic that has no transverse spatial velocity (that is, points exactly towards the center of the earth) intersects the path that the center point of the Earth follows, so when the ball follows that geodesic it moves towards and eventually reaches the center of the earth. The geodesics that don't point directly at the center of the Earth miss that center and follow the curvature of spacetime on a path that corresponds to an elliptical orbit.

 

[PhysicsBob]

Thanks... excellent answer, and I appreciate your patience with me, but I think my meager understanding is causing me to ask the wrong question.

How about if I address it this way: Does the gravitational force due to mass do anything other than bend space-time? My schoolboy understanding (pre-reading Hawking) was Newtonian i.e. that big objects attract smaller ones through a force known as "gravity". Then Hawking showed me how the mass was actually bending space-time. But, are there two things at work here? Is my schoolboy understanding still correct to some extent, or is the only force gravity exerts the bending of space-time?

 

[Nugatory]

But, are there two things at work here? Is my schoolboy understanding still correct to some extent, or is the only force gravity exerts the bending of space-time?

There is one physical phenomenon here: If I place two objects in empty space, back off and watch them, they will accelerate towards one another (and the trajectories that they follow will depend on their initial velocities).

There are two ways of describing this phenomenon mathematically: Newton's theory which says that there is a force between them given by $\vec{F}=Gm_1m_2/\vec{r}^2$; and Einstein's theory which says that both bodies are following geodesics through curved spacetime, and the curvature is given by the Einstein field equations (which take, among other things, the masses of the two objects as inputs). The two theories usually agree to the limits of accuracy of our instruments, but observation has shown that when they disagree by enough to measure it is Einstein's theory and not Newton's that matches the observation.

I interpret these facts as saying that there is only one thing at work here, but that that we have two descriptions of it, one good (Newton's) and one better (Einstein's).

 

[PhysicsBob]

There is one physical phenomenon here: If I place two objects in empty space, back off and watch them, they will accelerate towards one another (and the trajectories that they follow will depend on their initial velocities).

There are two ways of describing this phenomenon mathematically: Newton's theory which says that there is a force between them given by $\vec{F}=Gm_1m_2/\vec{r}^2$; and Einstein's theory which says that both bodies are following geodesics through curved spacetime, and the curvature is given by the Einstein field equations (which take, among other things, the masses of the two objects as inputs). The two theories usually agree to the limits of accuracy of our instruments, but observation has shown that when they disagree by enough to measure it is Einstein's theory and not Newton's that matches the observation.

I interpret these facts as saying that there is only one thing at work here, but that that we have two descriptions of it, one good (Newton's) and one better (Einstein's).

Ah... OK, Now I get it. Thanks.