Would objects really be at rest relative to each other in orbit?

Question

I am currently reading Gravitation by Misner, Thorne, and Wheeler (MTW). In the very first chapter on weightlessness they make the following claim:

“Contemplate the interior of a spaceship and a key, penny, nut and pea by accident or design set free inside. Shielded from all view of the world outside by the walls of the vesse, each object stays at rest relative to the vessel. Or it moves through the room in a straight line with uniform velocity. That is the lesson which experience shouts out. Forego talk of acceleration! That, paradoxically is the lesson of the circumstance that ‘all objects fall with the same acceleration’. Whose fault were those accelerations after all? They came from allowing a ground based observer into the act: the push of the ground under his feet was driving him away from a natural world line.”

It seemed reproducing the offending quote would highlight if my question rests on misunderstanding what it says.

My question is this: if this is supposed to be saying that a spaceship in orbit feels no acceleration, being in free fall, then does that mean such a ship and its contents would feel no apparent rotational forces?

I would have thought that, given from a Newtonian perspective it is moving in a circle, objects inside would not act as if they were in an inertial frame. There would be a slight apparent centrifugal force. There would appear to be Coriolis forces. Am I wrong? Would no such forces be apparent if one were actually in orbit?

They clearly are talking about orbit - there wouldn’t be any point in the quote if it were just a ship floating in empty space.

Answer 1

From a Newtonian perspective the objects free-floating inside the spaceship are subject to (almost) exactly the same gravitational acceleration as the spaceship and travel on similar orbits. The walls of the spaceship do not "shield" them from gravitational attraction. Hence there is no apparent centrifugal or coriolis force on those objects in the accelerated frame of reference of the spaceship. There will be tidal forces, since the gravitational acceleration will not be identical in all parts of the spaceship.

Answer 2

The referenced passage is comparing a Newtonian description of three objects orbiting the earth with a relativistic one. Newton described bodies as being accelerated by gravity toward the center of the earth which changes the direction of their motion from a straight line to an ellipse. The author is pointing out that if these objects were encapsulated in a box they would freely float around the box as though the earth were not there at all - think astronauts in the space station. From this perspective the objects are not accelerating at all they are just following their natural path through curved spacetime.

Of course Newton would have reached the same conclusion that the objects would move together in unison, and that none of them would “feel” an acceleration the way you feel an accelerating car or bicycle. Einstein’s realization is that the reason that you don’t feel an acceleration is that there is none. And that subtle difference is the foundation of general relativity.

Answer 3

The objects floating in the spaceship all have orbits that are very similar to each other, and to the orbit of the spaceship, so their relative velocities are almost zero.

To keep things simple, let's assume that the spaceship is in an ideal circular equatorial orbit, with no atmospheric drag, orbiting an ideal Earth with a gravitational field that has perfect spherical symmetry.

If the floating objects and the centre of mass of the spaceship are all at the same radial distance from the centre of the Earth they all have the same constant orbital speed, so they appear at rest relative to each other. We can use simple calculus to estimate the drift speed of objects that have a different radial distance.

In a circular orbit of radius $r$, the orbital period $T$ is $$T = 2\pi r / v$$

The orbital speed is given by $$v^2 = \mu/r$$ where $\mu = GM$ is the gravitational parameter. Differentiating, $$2v \frac{dv}{dr} = -\mu/r^2 = -v^2/r$$ So $$\frac{dv}{dr} = -\frac12 v/r = -\pi/T$$ Now $$\frac{dv}{dr} \approx \frac{\Delta v}{\Delta r}$$ So if we change the orbital radius by $\Delta r$, the orbital speed changes by $$\Delta v \approx -(\pi/T)\Delta r$$

Eg, if $\Delta r=1$ unit, the drift speed is $-\pi/T$, and we expect the object to drift a distance of $\pi$ units over one orbital period.

For Earth, $\mu \approx 398600.435507 \, \rm{km^3/s^2}$. Its equatorial radius is $R = 6378.137 \, \rm{km}$.

For a satellite with period $93$ minutes = $5580$ seconds (which is very close to the period of the ISS), the orbit radius is ~$6795.681 \, \rm{km}$, and its altitude is ~$417.544 \, \rm{km}$. Its orbital speed is ~$7658.653 \, \rm{m/s}$.

If $\Delta r=1 \, \rm{m}$ then $\Delta v= 0.0005635 \, \rm{m/s}$, just over half a millimetre per second.

---

In an elliptical orbit, things are a little more complicated because the orbital velocity isn't constant. It varies with the orbital radius, in accordance with the vis-viva equation:

$$v^2 = \mu\left(\frac2r - \frac1a\right)$$ where $a$ is the semi-major axis of the ellipse (which is the mean of the minumum and maximum orbital radii). I have a derivation of the vis-viva equation here.

Kepler discovered that an orbiting body "sweeps out" equal areas in equal times. Newton showed that's because angular momentum is conserved in a central potential field. Here's a diagram for an orbit of eccentricity $e=3/5$. The dots on the ellipse are plotted at equal time steps of $T/24$, so each sector has equal area.

Near perigee the points on the orbit are spread out, and near apogee they are bunched closer together.

Answer 4

The idea is a spaceship free in a flat universe. Galilei's idea a bit elevated: Galilei considered the man on the top of the mast, letting a stone fall freely to a point on deck (captains head, e.g.). He successfully hits the goal independent of the ships constant velocity.

If the spaceship orbits the earth on a circle, the parallelly freely moving objects, orbit on slightly different Kepler ellipses, so they change relative orientation and their periods may be different.

What MTW want to demonstrate, is the fact, that everyday gravitational forces are the opposite of the obvious:

Fixing an object to the surface of the earth by its gravitational pull is really the permanent force of the ground acting on the object against the acceleration on the free motion, starting on the ground, and falling freely as if the gravitating mass of the earth was concentrated in a smaller ball around the center, as it happens on a cliff.

By Newtons actio=reactio at rest, the accelerating force is equal to the exerted force onto the balance on the floor.

Answer 5

The point he is trying to make is that a spaceship in orbit is the same to one drifting in deep space. They're both moving in a straight line, the difference being that the space is curved around a massive body. Ignoring small tidal forces, you should view them as equivalent from the point of view of the spaceship and everything inside it.

Answer 6

There are tidal forces at play but they are very small in this situation. The tidal forces are a slight bias in the near-zero apparent gravitation within the ship so that at one end of the cabin it has a tiny positive force and at the other end it is slightly negative. The rotational "forces" you refer to would be manifested inside the ship as a rotation of this very small tidal bias.

Answer 7

Free rotation is different from free fall. While linear velocity is relative, angular velocity is absolute. A pair of gyroscopes inside a spacecraft can determine its angular velocity without reference to the outside (although for practical reasons we usually use more than two). A few years ago, an ops blunder spun the Hitomi spacecraft so fast it broke into 11 pieces from the stress.

Greg Egan, in his sci-fi novel Incandescence, describes the trajectories that free objects take inside an orbiting, rotating body. The context is an alien scientist, living in that environment, working out the physics of inertia and gravity.