Acceleration in an inertial worldline?

[runner108]

I think I'm beginning to understand general relativity. After doing some dabbling on the history of inertial frames vs accelerated frames from the time of Aristotle I've gotten to Einstein. Finally wrapped my mind around how an object in free fall is considered to be an inertial frame as measured by an accelerometer of a sphere inside another sphere with a gap i-between, whereas the same accelerometer would register acceleration when standing on the Earth's surface.

Despite this one "accelerates" in terms of velocity as one gets closer to the center of the earth. Am I correct in understanding the discrepancy is because at a small enough region of space even in curved space, the space is considered 'locally' flat, therefore free-fall = no acceleration.. where as as one moves over a region of curved space-time the curvature gives rise to an acceleration?

Thanks

 

[A.T.]

Despite this one "accelerates" in terms of velocity as one gets closer to the center of the earth.

The center is an arbitary point of reference, and so the velocity & acceleration measured in its frame are completely arbitrary. This relative(frame dependent) acceleration is called "coordinate acceleration", as opposed to the absolute(frame invariant) "proper acceleration" measured by an accelerometer.

Am I correct in understanding the discrepancy is because at a small enough region of space even in curved space, the space is considered 'locally' flat, therefore free-fall = no acceleration.. where as as one moves over a region of curved space-time the curvature gives rise to an acceleration?

There is no discrepancy just different ways to define acceleration. Coordinate acceleration depends on the frame of reference. Proper acceleration is always zero for free falling objects, in every frame of reference.

Proper acceleration of zero means no forces are acting, and a locally straight path in spacetime (geodesic).

Chapter 2.6 of this:
http://www.relativitet.se/Webtheses/tes.pdf
has nice visualizations of acceleration & non-acceleration in curved spacetime.

 

[runner108]

A.T. Thanks, keen answer and drives the point home. Appreciate it.

 

[runner108]

When Statics says that the net forces equalling zero makes the state a state of equilibrium, would this be measured by an accelerometer? Meaning would an apple on the ground be considered at equilibrium despite being accelerated as far as G.R. is concerned? Or would it be at a state of equilibrium despite measuring a proper acceleration via a accelerometer?

 

[pervect]

In statics, one usually introduces some particular coordinate system in which the geometry appears to be unchanging with time, i.e. static. This static geometry is only possible only if the problem has the right sort of symmetry to start with.

If a body has proper acceleration, as measured by an accelerometer, in this particular coordinate system, one introduces "fictitious forces". Gravity is an example of such a fictitous force - in the context of GR, anyway. In Newtonian mechanics, we pretend it's a real force, then scratch our heads about why the gravitational mass always equals inertial mass.

In a different problem, the problem of a merry-go-round, centrifugal force is another well known example of a "fictitous force".

Statics then says that the sum of all the forces (including the fictitious force) is zero.

 

[runner108]

So if someone is in free fall their accelerometer measures '0' therefore they are in a state of equilibrium. If they are on the face of the Earth according to Newtonian physics, gravity is pulling the person down and the ground is pushing back, therefore they are in a state of equilibrium. In G.R. the ground pushes against the person so we introduce a fictitious force gravity to counteract the ground pushing against us. Thus in all three situations we are at equilibrium despite the accelerometer measuring differently in the two principle different situations (one free fall the other standing)?

Thanks

 

[A.T.]

Thus in all three situations we are at equilibrium despite the accelerometer measuring differently in the two principle different situations (one free fall the other standing)?

Yes, because here "equilibrium" means zero coordinate acceleration, and the accelerometer measures proper acceleration. In a non-inertial system like the Earth's surface they are different.

 

[runner108]

Awesome, thanks a lot!

 

[runner108]

Just wanted to pursue one more thought here regarding proper acceleration vs coordinate acceleration. If you were falling into the Earth and the Earth was just a point mass, one would not feel a force of acceleration yet one would be speeding up based on coordinate acceleration. I can theoretically understand this. Just wanted to confirm that one would visually come to the conclusion that they were accelerating even though they wouldn't feel anything?

 

[A.T.]

If you were falling into the Earth and the Earth was just a point mass,

Falling into a point?

one would not feel a force of acceleration yet one would be speeding up based on coordinate acceleration.

Is "you" and "one" the same person? Speeding up based on coordinate acceleration depends on the frame of reference.

Just wanted to confirm that one would visually come to the conclusion that they were accelerating even though they wouldn't feel anything?

If "one" is in on the surface of the Earth and observes "them" falling, yes.