What's all this about the curvature of spacetime?
It is unknown to me, and probably everybody else alive today, exactly what is going on when a
massive object curves spacetime. What truly causes the geometry of spacetime to be altered?
Someday we'll all know. Perhaps when the universe has been broken down to its smallest
components - whether that turns out to be these so-called superstrings or something else - then
we'll fully understand it.
For now, however, the effects of curved spacetime are easy to visualize and measure.
An unforced object traverses through relatively strongly curved spacetime (e.g. close to a large star
or planet) in the same way that an unforced object traverses through relatively flat spacetime (e.g.
out in deep space). So what are these observable effects?
First, let's talk about objects that have no way of having an external force applied to them. That is,
objects that are freely traveling through spacetime (i.e. following their geodesics).
Let's start way out in deep space:
Two good friends, Cheryl and Don, are floating together light-years away from any large mass.
Neither of them feels any external force, and because of this they're both a little nauseous (a
common effect of being "weightless"). Cheryl starts to feel better and gives Don a firm but loving
shove. During this push they both feel a force. Don feels a force on his stomach (in a direction from
his stomach towards his back) and Cheryl feels the same amount of force but in a direction from
her palm towards the back of her hand. The only time they feel any force is during this shove.
After the push, neither feel any force on their bodies and they now notice that they're moving apart
at a constant speed (they've got their frictionless tape measures and beeping watches to verify
this). Soon they're out of sight of each other, and that's the end of Cheryl and Don. Spaceflight can
be risky, but they knew the job was dangerous when they took it.
Now let's restart the trip of Cheryl and Don, but this time they're free falling five miles above the
Mojave Desert and headed towards the dirt. For this trip we're going to remove the Earth's
atmosphere so that no external forces will be able to be applied to either of them. They are now in
relatively strongly curved spacetime (compared with where they were in deep space) because
they're close to a planet.
What do Cheryl and Don experience while they're free falling above the Mojave Desert? What do
they feel, what speeds do they measure from one to the other, and what do they see when they look
at each other? Exactly the same as they did when they were out in deep space.
Let's discuss another effect of being unforced in spacetime.
We place a third person (Aaron) out in deep space, and we'll put Cheryl and Don back in their
position five miles above the Mojave Desert. Everyone now has really good binoculars, really long
frictionless tape measures, and watches that beep every second.
Aaron looks at Cheryl and Don and sees (and measures) that they're moving away from him. He is
able to measure that their relative speed is increasing as time goes on. Cheryl and Don notice and
measure the same thing about Aaron. During this whole experience, however, neither of the three of
them can feel any external forces.
So, there are two important effects that can be observed when unforced (i.e. following their
respective geodesics) in spacetime:
- Objects that are in regions of spacetime that have the same curvature magnitude and direction
experience the same effects as objects that are in a region of spacetime with a different degree of
curvature, and if no external forces are applied they do not measure any changes in speed from
any other unforced object in that same region of spacetime.
- Objects that are in regions of spacetime that have different magnitudes (and/or direction) of
curvature will observe a change in speed with respect to each other - even though no external force
is applied to either object.
This last effect is one that causes some controversy. How can there be a change of speed
between objects with no forces applied to either one of them? Until we figure out the fundamental
mechanism behind curved spacetime, the only thing that can be said is that it's just the way our
universe works, or it's just an effect of two unforced objects being in differently-curved regions of
Now, let's talk about objects that do have external means having a force applied. That is, objects
that are not freely traveling through spacetime.
Let's start with Lance and Melinda. They're both on the surface of the Earth. Lance is in his
supercharged hot rod and Melinda is standing on the street keeping an eye on Lance. They both
have their frictionless tape measures and watches ready. Lance takes off and keeps his car at a
leisurely 10mph. They both look at their tape measures and agree on this speed. Now, Lance
punches it and keeps the pedal to the metal. They both agree that their relative speed is now
changing at a rate of about 22mph per second. Lance slows down and turns around so they can
discuss what has happened. Since both of them know that speed must be measured with respect to
two objects, they try to figure out which one of them was causing the change in relative speed.
Melinda knows better, but she tries to fool Lance by saying, "I was the one that caused the change
in speed." Lance is a sly dog, however, and says, "Melinda, I felt my bucket seat pushing hard
against my back. Therefore, it was I that was responsible for the changing of our relative speed."
Now, we put Lance five miles above the ground and we remove the atmosphere around him (this is
commonly referred to as "adding insult to injury"). Melinda is gazing skywards as Lance's speed
begins to increase with respect to her. Lance observes the same increase in speed between the
two of them. Once Lance smacks into the ground and wipes himself clean, they begin to discuss
what is what. Lance says, "I was the one that caused the changing speed with respect to you. I
could see you closing in on me very quickly - especially just before I hit the Earth." However,
Melinda knows Einstein's great granddaughter and won't have any of it. Melinda asks Lance if he felt
any forces as he was approaching the Earth's surface. He answers, "No, I felt no forces until I met
up with that soft patch of crabgrass." Melinda ends the discussion by asserting, "Lance, as you
were falling, I felt a constant force on the bottom of my feet. Therefore, I was the one who gets the
credit for changing speed with respect to you."
- Objects that are in the same region of spacetime - regardless of the degree of curvature in that
spacetime - will only experience changes of speed with respect to each other if one or both of the
objects has an external force applied.
This particular aspect of our universe is perhaps the most difficult for us humans to accept. We are
all born here on Earth, and we spend most if not all of our lives on this planet. We naturally become
very Earth-centric. Our instincts tell us that our speed is usually zero and that everything else is
moving and changing with respect to us. Trying to imagine that everyone on the surface of the Earth
is being pushed outwards is hard to believe, as everyone is fairly stationary with respect to each
other. However, those who study General Relativity soon get accustomed to the notion that free
falling objects are unforced and are in what is called "inertial frames of reference". They also feel
comfortable with the fact that those stuck to the Earth do have a net upwards force on them and are
in what are called "accelerated frames of reference", and that this acceleration is with respect to
each of our local spacetime geometry which is curved inward toward the center of the Earth.