Q & A: relativisitic falling

Q:
WE are told that in G.R. gravity is no longer a force but the curvature of space-time. I doubt that anybody can ever fully visualize intuitively a curved space-time and its implications and I am fully cognizant that, past a certain level of abstraction, our mind is unable to see the “truth” of certain propositions without making recourse to the mathematical language, which is beyond me. . I wonder, though, whether more of the G.R. constructs couldn’t be made conceptually understandable for the layman without the use of maths than many scientists commonly believe. Marcus Tullius Cicero, famous roman orator, used to say that if one grasps the subject matter, then words will follow..True, he didn’t have to give speeches on G.R. but one wonders if, sometimes at least, the layman’s failure to understand is not due to his teacher’s inability to have a clear conceptual grasp himself.. Scientists often fear that any attempt to explain scientific concepts without the use of maths will necessarily result in approximation, if not outright false statements..While nobody would like to learn what is not true, approximation is often the only alternative to total darkness and many of us, who have a poor grasp of mathematics, are willing to pay this price. Regarding G.R. gravity, I am struggling with two questions and I’d like to have a clearer grasp: 1. If bodies fall not due to a “ force” of gravity pulling on them but simply because they naturally follow the geodesics of curved space and are purportedly in uniform motion, how come that they still fall at an accelerating rate? 2. If one looks at the common diagrams showing how a body curves the fabric of space-time , he can clearly see the geodesics stretching around and under a half sunken body, not across it . How come then that bodies fall not only at an accelerating rate, as queried above, but also towards the earth and its center, rather than around it? Falling bodies do smash, after all, on earth. Does perhaps curved space and its geodesics also cut across the Earth itself ? I’ am sure my question may sound naïve to some, but I had to ask it. I’d also be grateful if somebody can recommend a book about G.R. in non-mathematical language, as much as possible. Thank you Ittiandro
- Franco Vivona (age 70)
A:
Those are nice questions.

1. The gravity causes only the "acceleration", as described in a Newtonian picture. Whether the ball happens at some particular time to be "falling", i.e. moving toward the earth, is just an accident of its history.

2. I think I know the type of picture you're referring to. They don't capture what's going on with the time coordinate but do show how the space coordinate differs from Euclidean geometry. The circumference of a circle around the object shows the circumference that would be measured by a collection of standard meter sticks used locally. The same goes for the diameter. The bulge means that the diameter measured that way is more than the circumference over 2 pi.

Drawing geodesics requires some way to represent the time coordinate too. It's certainly true that in theory as well as in life there are lots of geodesics that bump into the earth, as well as others that zip by it and others that represent orbits.

I've just been told that Thomas Moore is coming out with a good introduction to GR, perhaps along the lines you're looking for, but I haven't seen it yet.

Mike W.

(published on 10/16/2012)

Follow-Up #1: how to explain gravitational acceleration

Q:
Thank you for your prompt reply. I’m glad to see that you agree with my hypothesis that geodesics can cut across the earth, which explains why bodies moving along some of the geodesics can be perceived as falling on the Earth. Regarding the 2nd part of my question, i.e. why bodies still seem to fall (towards the earth) at an accelerating rate, I am still not clear. I agree with your statement that if the ball happens at some particular time to be "falling", i.e. moving toward the earth, it is just an accident of its history. Indeed, objects do not fall in absolute terms. They do no more than moving along a geodesic. They appear to be falling only when they follow a geodesic which happens to be oriented towards the Earth. However, if I am not mistaken, G.R. has done away with the notion of Gravity as a force and made acceleration redundant ( at least the newtonian gravity- engendered acceleration.), so that bodies inj G.R. always move in uniform motion. Yet, when a body “falls” ( in newtonian terms) towards the Earth, it does accelerate and we can even measure the acceleration rate as 9.68 mss. What causes this acceleration or how can it be explained in relativistic terms? Thank you again Franco
- Franco Vivona (age 70)
A:
When we're near the earth, we instinctively use an earth-based reference frame. So when we say we accelerate, we're referring to the change our velocity with respect to the earth.

Let's try to visualize a spacetime picture. That's not too hard here because we only need one space coordinate, the one along the line between the center of the earth and the center of you. Let's make the other, vertical, coordinate in a 2-D plot be time.

Say you start at rest with respect to the earth. In our plot, that means that you and the earth start out with parallel world lines. For convenience, we pick the earth frame so these are just vertical lines. The earth's warping of spacetime causes your world line to bend toward it, until they intersect.

Here's a start on how to think about this effect. If you make two identical clocks, and put one near the earth and one farther away, the near one will run slower. This effect can be seen and agreed upon by observers at either clock, although they may choose to call their own clock "correct". This effect applies to all clocks. Now the rate at which things tick, the frequency, is identical to the energy, except for a factor of Planck's constant. So the position-dependent frequency is really the same effect as the classical position-dependent energy. Take any quantum wave packet representing a particle. Let the quantum phase advance faster (higher frequency) on one side than the other. That shifts the interference pattens, causing the average velocity of the particle to change.  That gives the acceleration.

Here we've just ignored the spatial distortions, which become equally important for figuring out the trajectories of particles moving at the speed of light with respect to the earth. You can search around here for "light bending gravity" to find some discussions of that.

Mike W.

(published on 10/17/2012)