Defining Inertial Reference Frames in General Relativity

[hartlw]

How can you have an inertial reference frame in which a body can remain at rest or move with constant velocity unless you postulate the disappearance of the universe?

In the Michelson Morley experiment the Earth is not moving with constant velocity, it is accelerating. So the postulates of special relativity do not apply.

 

[DrGreg]

How can you have an inertial reference frame in which a body can remain at rest or move with constant velocity unless you postulate the disappearance of the universe?

In the Michelson Morley experiment the Earth is not moving with constant velocity, it is accelerating. So the postulates of special relativity do not apply.

Special relativity (SR) is only an approximation to the real universe, in which the effects of gravity are assumed to be negligible. You might say that SR postulates the negligibility of gravity. The approximation works well enough in many situations.

In the Michelson Morley experiment, the effects of the Earth's acceleration are assumed to be much less than the effect that is being looked for.

From the point of view of General Relativity, the centre of the Earth is not accelerating: it is in free fall. But the surface of the Earth is accelerating due to the Earth turning on its axis.

 

[shalayka]

As well, kinematic time dilation (that is, due to relative motion) is a key contributor to the relativistic precession of bodies orbiting a mass described by the Schwarzschild solution. The other 2/3rds of the relativistic precession is due to gravitational time dilation. So, even for accelerating bodies, the effects described by Special Relativity still apply in a very real sense. In fact, if you want to switch to Special Relativity, just take the instantaneous velocity of the body and simply ignore what happened/happens before and after that instant (thus removing acceleration from your considerations).

 

[hartlw]

Dr Greg wrote: "But the surface of the Earth is accelerating due to the Earth turning on its axis."

And about the sun.

Thanks for reply Dr Greg.

 

[hartlw]

Dr Greg,

Whoops, I see what you mean about free fall. Like a satellite orbiting the earth. With respect to a frame in a space station, a body would remain at rest, if you neglect masses other than the earth. But would an object move at constant velocity wrt to this frame? It would seem the object would move to a new orbit corresponding to its velocity.

Thanks again.

 

[hartlw]

Suppose you are in a space station in stationary orbit about the Earth (or sun), and there is no other gravitational influence. Looking out the window you see no movement wrt to the Earth (or sun). Would you not have to conclude, absent any evidence to the contrary, that you are at rest, and that if you performed the Michelson Morley experiment the results would be what you expected?

The results of the M&M on the space station should be the same whether or not the Earth (or sun) is spinning about an axis through its center.

 

[HallsofIvy]

Suppose you are in a space station in stationary orbit about the Earth (or sun), and there is no other gravitational influence. Looking out the window you see no movement wrt to the Earth (or sun).

Assuming that the space station is rotating so that a window is always facing toward the earth, you would see no movement with respect to the Earth but why in the world no motion with respect to the sun? You have to be orbiting the Earth once every 24 hours so you would certainly see motion with respect to the sun.

Would you not have to conclude, absent any evidence to the contrary, that you are at rest, and that if you performed the Michelson Morley experiment the results would be what you expected?

The results of the M&M on the space station should be the same whether or not the Earth (or sun) is spinning about an axis through its center.

 

[hartlw]

Assuming that the space station is rotating so that a window is always facing toward the earth, you would see no movement with respect to the Earth but why in the world no motion with respect to the sun? You have to be orbiting the Earth once every 24 hours so you would certainly see motion with respect to the sun.

Absoluteley correct.

I submit that it is impossible for a coordinate system fixed in an object moving under the influence of the sun to have constant velocity. Whether the object is free-falling or rotating in orbit it is accelerating, and hence postulates of special relativity do not apply, aka M&M.

 

[jtbell]

Quite true. The inertial reference frames of relativity and Newtonian mechanics are mathematical abstractions and idealizations which are nevertheless useful approximations to many real-world situations. We routinely use Newton's laws to analyze motion on the surface of the earth, and don't include Coriolis forces when they don't significantly affect our results.

An important part of physics is knowing when one can safely apply various approximations to simplify analysis. Remember that special relativity is itself an approximation to general relativity, useful when spacetime curvature is small enough that we can neglect it.

 

[MeJennifer]

Whether the object is free-falling or rotating in orbit it is accelerating, and hence postulates of special relativity do not apply, aka M&M.

A non rotating free falling test object does not accelerate in GR.

 

[hartlw]

You can attach a reference frame to that object, but any other object in that frame cannot remain stationary or move with constant velocity.

Oh, GR, General Relativity. Sorry, I can't address GR, know nothing about it.

 

[lbrits]

Dr Greg wrote: "But the surface of the Earth is accelerating due to the Earth turning on its axis."

And about the sun.

Thanks for reply Dr Greg.

I believe it is principally due to the rotation of the Earth that the Earth's surface is considered accelerating. The center of the Earth is not accelerating as it is moving around the sun. It is moving along a geodesic.

 

[hartlw]

As I understand it, you are saying that the center of the Earth is not accelerating. I learned that it was, centripetal acceleration.

I'll try to look at it from your point of view. A space station orbiting the sun. Is it moving at constant velocity or accelerating? WRT to what? OK, let's say we are testing the assumption that the station is
moving at constant velocity wrt to an aether (3d coordinate system) attached to the sun. The speed is constant but the velocity isn't. Furthermore, an object will not move at constant velocity, or speed, in the space station frame, so it is not an inertial reference frame;
a bullet fired from the space station will not have constant velocity wrt the space station frame.

I will agree that a body placed at the center of mass of the space station will remain at rest wrt a coordinate system fixed in the space station. If we are dealing in priciples, you can't ignore the mass of the space station when you try to define an inertial reference frame.

In a frame attached to an orbiting body, so that the body is at rest in that frame, is there a force acting on the body? If I don't look "out the window," there isn't. If I do , there obviously is. OK, my physical laws are based on what I measure (see) in the frame. But am I allowed to test for the presence of a gravitational field? In that case I would have to conclude there is a force acting on the body and it is not moving so it is not an inertial refeemce frame. Which brings us full circle.

There is no such thing as an inertial reference frame.

 

[lbrits]

Well, sure there is. In general relativity, an inertial reference frame is a frame upon which no forces act. It is the generalization of the special relativistic condition
$$\frac{d^2 x^\mu}{d\tau^2} = 0$$
and is written
$$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$$.
If you are concerned that $$\Gamma^\mu_{\alpha\beta}$$ is the "force", then I'll just say that you can pick local coordinates such that that term vanishes. This is the principle of equivalence stated mathematically.

Of course, a true "force" is something that you would put in place of zero on the right hand side... for example, if you're standing on the surface of the Earth, the ground is pushing back up on you, which prevents you from assuming a geodesic (an elliptic orbit). Thus the observer on the ground is not traveling in a geodesic because of the electromagnetic force + Pauli repulsion holding him up, but the center of the Earth is moving along a geodesic (for all practical purposes), because no force is acting on it.

 

[D H]

You can attach a reference frame to that object, but any other object in that frame cannot remain stationary or move with constant velocity.

Oh, GR, General Relativity. Sorry, I can't address GR, know nothing about it.

An inertial reference frame in GR is a bit different than an inertial reference frame in Newtonian mechanics. That constant velocity thang is how an inertial reference frame is defined according to Newton's first law. Per Newton's second law, an inertial reference frame can also be defined in terms of an object that is not subject to external forces.

Viewed in this light, the concept of an inertial reference frames is not all that different between GR and classical mechanics. The two concepts differ in what is considered to be a force. Gravitation is a force in classical mechanics but not in GR. It is instead a pseudo-force resulting from viewing things from the perspective of a non-inertial frame, just as centrifugal force and the Coriois effect are pseudo-forces resulting from viewing things from the perspective of a rotating frame. In classical mechanics, a distinguishing feature of a pseudo-force acting on some object is that the force is proportional to the mass of the object. The gravitational force is the only classical force that is proportional to the mass of the object. This suggests that gravity is not a real force.

How to define an inertial frame in GR? In a way, its even easier than in classical physics. Accelerometers measure all forces acting on them except gravity. (In GR, accelerometers measure all forces acting on them, period.) Colocate an ideal point-sized inertial measurement unit (accelerometer plus gyroscope) on an object. If the IMU measures zero acceleration and zero angular velocity, a reference frame with origin at the IMU and axes fixed with respect to the IMU is an inertial frame.

 

[hartlw]

How to define an inertial frame in GR? In a way, its even easier than in classical physics. Accelerometers measure all forces acting on them except gravity. (In GR, accelerometers measure all forces acting on them, period.) Colocate an ideal point-sized inertial measurement unit (accelerometer plus gyroscope) on an object. If the IMU measures zero acceleration and zero angular velocity, a reference frame with origin at the IMU and axes fixed with respect to the IMU is an inertial frame.

But no three particles in this (freely falling) frame can remain in a straight line, or, no other particle can remain fixed or move with constant velocity in this frame.

Or did I misunderstand? Is the above simply saying that any movement due to gravitational forces is discarded?