# Curvature of space question

I’m hoping I get the wording of this right, but here goes: when I throw a ball and see it move forward and then drop to the surface of the earth, amI seeing before my very eyes the way in which earth is curving space due to its gravitational field? What I think I mean is: is the ball essentially traveling in a straight line, but that line is curved by the earth’s gravitational field?

What I think I mean (II) is: one of the proofs of general relativity was the deflection of the path of starlight around the sun that was demonstrated during a full eclipse. IIRC, the starlight was considered to be traveling a straight line, but space was curved around the sun, so it appeared to be deflection. Is the deflection of the ball toward the earth a similar relativistic effect, or is it a much more mundane phenomenon?

Can’t give you a good answer, because I don’t understand relativity well enough, but as I understand it the main difference between the scenarios is that photons are massless, so they can’t be affected by gravity (the force).

The short answer to your question is yes: the deflection of the path of a ball as it moves near the surface of the Earth is equivalent to the deflection of light. Both particles (balls & photons) are moving on what are called geodesics in spacetime; the only difference is how fast these paths are moving. In the presence of a massive body, the laws of general relativity cause the surrounding spacetime to become curved, and so the paths of objects deviate from what they would be in the absence of this massive body.

And yes, photons are affected by gravity even though they have no mass; the pithy way of explaining why this is so is to note that they have energy, and energy is equivalent to mass.

The ball moves in a purely Newtonian fashion.

The straight lines that are referred to in relativity are the shortest possible paths between two distances, technically known as a geodesic. Each type of space has a different geodesic. For a flat piece of paper it’s a line. For the surface of a globe it is a diameter. For three-dimensional space it’s whatever the configuration of that space defines it as (and there are many types of three-dimensional spaces). Obviously a ball rolled along the surface of a sphere travels a shorter path than one thrown through the air. A light ray from a distant source gets noticed if the path observed varies from the one that is calculated. But that’s because the underlying space itself has been altered.

The closer analogy would be measuring a ball rolled along the earth. If there were bumps or depressions on the ground, the ball would move and arrive in a way different from a ball rolled along a perfectly smooth surface. By measuring the difference in timing you could tell whether the sphere varied from perfection. That’s essentially what measuring a light ray bent by the sun did to space.

But a space ship sent around the sun wouldn’t behave like a light ray. It moves on a path that is never least distance. Though relativity may play a part in the calculation, you’re mostly using Newtonian physics just like the thrown ball.

Space is flat. Einstein was curved.

Does it ever make sense to consider the gravitational attraction between two objects, and the resultant acceleration of them toward each other, as being due to the space between them getting smaller? Not the distance between them, but the actual ‘amount’ of space, itself. In other words, is the acceleration of the two objects toward each other actually the space between them shrinking in an accelerating manner? We see that as their separation distance decreasing.

Probably all this is just silly, but I’ll ask about it anyway - I’ve embarrassed myself too many times already to be worried about it happening one more time.

No, it’s still a least-distance path. You’ve just got to realize that we’re measuring distance in spacetime between two events, not just distance in space between two locations. For instance, consider the location where the Earth is right now, and the location on the opposite side of the Sun from that. The shortest distance in space between those two points is a straight line that goes through the center of the Sun. But now, consider the point where the Earth is right now at a time that’s right now, and the point on the opposite side of the Sun six months from now: Now, you’ll find that the shortest path between those two spacetime points is along the Earth’s orbit, and the Earth follows that shortest path.

Your answer confuses me because it seems to mix up three separate motions.

A neutrino can move from earth now to the location of earth plus six months right through the sun in a straight line in sixteen minutes.

The earth follows a least resistance path in a stable orbit around the sun in spacetime and gets to the opposite side in six months.

A space craft sent from earth now to the position of earth plus six months would take a different path than earth’s orbit, in a time that depends on initial thrust plus on-board thrust.

I was trying to distinguish those three different effects. Don’t you need to do so to answer the OP?

You missed one important word in Chronos’ reply: spacetime. These three separate trajectories start at the same event in spacetime (i.e., the same (t,x,y,z) coordinates) but end at three different events; and the neutrino and Earth both follow geodesic trajectories to get from here-and-now to the same point (x,y,z) in space, though at different times. (The spacecraft may or may not follow a geodesic, depending on whether it is thrusting.) As MikeS said, any freely-falling trajectory is a geodesic in spacetime, although it looks curved in three dimensions.

Your reply that the ball is moving in a “purely Newtonian fashion,” while correct, is I think not responding to the OP. It’s true that Newtonian mechanics is adequate for predicting its trajectory, but the OP is asking whether you can think of its path as bent simply by the curvature of spacetime; and that is also true. The geodesic calculation in general relativity simply calculates a trajectory in spacetime which is unaffected by external (nongravitational) forces, and it works just the same for a massive body as for a massless body, except that for a massive body the trajectory is timelike and for a massless body the trajectory is lightlike.

Absolutely, demonstrably incorrect. Photons are indeed affected by gravity, otherwise black holes wouldn’t be black. Also, we can measure the mass of stuff by how light bends around it due to gravity.

And, in fact, a photon bends twice as much as a slow moving object, because it’s moving fast enough to be affected by the curvature of space as well as well as the curvature of time.

Not quite: The faster something’s moving, the less it bends. What you meant to say is that a photon bends twice as much as Newton’s theory would have predicted for a photon.

Just to clarify: some relativity-haters early last century tried to resurrect Newtonian dynamics after star location measurement during eclipses drove the stake through its heart in favor of GR. They postulated that massless particles were accelerated in a gravitational field by the same amount that a particle with mass would be, just without applying an opposing force to the massive body that created the field. (Otherwise, you’d lose conservation of momentum, at which point all hell breaks loose.) But the math didn’t work out–it turns out that you’d have to apply an arbitrary factor of two to make this work, which was too much even for the most die-hard Newton fan. Plus, this doesn’t help explain the precession of Mercury’s orbit; eventually, the evidence just forced them to surrender.

Sorry to muddy the waters; I didn’t want someone leaving the thread with the impression that Newtonian physics included photons, which were first integrated into any successful modern theory of light in the twentieth century by the developers of quantum theory–particularly by Einstein himself, in his Nobel-winning paper on the photoelectric effect. In Newton’s time, the question of whether light consisted of particles or waves was an open question, until Newton’s own research into refraction pretty much conclusively proved … that light consisted of waves, and not particles. :smack: So Albert pretty much demolished all of Isaac’s greatest accomplishments. Makes you wonder if he had something against the poor guy. Had he only invented a new version of the calculus, his victory would have been complete …

No, refraction doesn’t really say anything one way or the other: One can certainly construct particle-based models of refraction. And IIRC, Newton himself fell on the particle side of the debate. You need diffraction experiments to prove the wave model, and those didn’t come about until the early 1800s.