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Straight lines
- Thread starter Resha Caner
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But doesn't the linear translation assume some underlying form of space? For example, is gravity's proposed curving of spacetime implying that gravitational force is applied to an object, or is it implying that the "linear" motion of an unforced object follows a curved path? Or is it an arbitrary choice?
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I understand inertia (and I can spell it). Maybe you don't understand what I'm asking. But what are you implying? What do you think I'm trying to do?
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Oafman
Try telling that to these bog brained murphys
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It's the latter. GR implies that there is no force of gravity. Instead, things follow their natural geodesic (straight line on a sphere etc) through a curved space time.
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Good to know. Since I work with classical mechanics I'm never quite sure what relativistic mechanics is claiming. I'm used to modelling gravity as a force in my "good enough" engineering world.
So, then, the object following the "natural geodesic" isn't accelerating?
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Oafman
Try telling that to these bog brained murphys
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In our frame of reference, gravity appears to be a force, so from an engineering POV, I guess it makes sense to treat it as such. As long as you're not changing the frame of reference (e.g. going into space, away from the earth's mass, where GR has to be accounted for, such as with GPS satellites)
Slightly out of my depth, so I had to look this up. Because obviously, things do appear to accelerate under gravity, but without a force, how can this be? Some of our resident physicists might be able to explain this succinctly, but in the meantime this article explains it well:
If gravity isn't a force, how does it accelerate objects?
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Thanks. That's a good starting point for me. So, it's the "inertial reference frame" thing.
In the end, my interest in this question ties back to engineering. The question would be, can the idea of an inertial reference frame be applied to any force, not just gravity. Let's take a simple example - a linear spring, which is represented by f = kx, where f = force, k = spring constant, x = displacement.
Can I create a reference frame where the mass feels no force from the spring? i.e. can I derive a "space" where some "straight line" considers the motion of the mass unforced?
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essentialsaltes
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Since it's in 'freefall' it doesn't feel any acceleration. But since the geodesics are no longer straight lines, there can be changes in distance and velocity over time.
Imagine two planes starting around the world trips, but on different great circle routes. Each is following a geodesic, so is travelling 'in a straight line'. But as they look at each other, they start to diverge, they're moving away from each other. But then when they both start approaching the antipodal point on their great circle, they notice that they're coming closer together again and meeting. Each one knows they're just moving in a straight line, so they might be inclined to think there was some force pulling the other plane toward them. But instead it's just a result of different geodesics in a curved space.
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Thanks. If you could also comment on my question at the end of post #30, that would be appreciated.
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Is the answer a circle? I'm probably outrunning my headlights, but I'm having one of those discovery rushes. If correct, this would be so very, very, cool. By generalizing on the idea of a "straight line" with geodesics, I can simply define away any force I choose. That's not meant to be flippant, but rather could be a very powerful answer to something I've been searching for a long time.
In my example with the linear spring, the descriptive equation is: mx" + kx = 0, where m = mass and (") indicates the second derivative. The solution is that x = x0*sin(sqrt(k/m)t), x0 = initial displacement and t= time.
This can be plotted in phase space as a circle. So, what I've been calling the phase space for all my models would equate to the "spacetime" of an unforced straight line. So cool ... if I'm right.
Why is it cool? Because I've long wondered if some non-Newtonian calculus could provide me a transform from nonlinear to linear models. My problem, however, was finding a commensurate transform for things like force. I was afraid I would "lose" certain acceleration terms during the transform. This may be the way to do it.
In my example with the linear spring, the descriptive equation is: mx" + kx = 0, where m = mass and (") indicates the second derivative. The solution is that x = x0*sin(sqrt(k/m)t), x0 = initial displacement and t= time.
This can be plotted in phase space as a circle. So, what I've been calling the phase space for all my models would equate to the "spacetime" of an unforced straight line. So cool ... if I'm right.
Why is it cool? Because I've long wondered if some non-Newtonian calculus could provide me a transform from nonlinear to linear models. My problem, however, was finding a commensurate transform for things like force. I was afraid I would "lose" certain acceleration terms during the transform. This may be the way to do it.
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essentialsaltes
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You can certainly create a coordinate system in which any particular object is not moving. But that's not the same as eliminating the force. Being in a freefalling elevator is not usual for us, but that's what it feels like to have no force acting on you -- it feels the same as if you were floating out in space far from anything. But if you're a mass on a spring, bungee-jumping, say. You experience accelerations. You feel them. They will register on an accelerometer that you are carrying. This is a message that some force is acting on you.
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I feel gravity. Accelerometers register the effects. So I don't see the difference.
I didn't mean to imply a coordinate system where the object isn't moving. In a phase plane the object is moving at constant velocity around the circle, as measured by the change in the phase angle.
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https://en.wikipedia.org/wiki/Inertial_frame_of_reference:
Looks like a no, at least if you're talking about an inertial reference frame.
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I think I did (post #33). However, I could have made a mistake. If someone wants to help me find a mistake, that would be great.
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No. The proper frame of reference would be the frame in which the spring and person are accelerating. You can't have a frame of reference where something starts accelerating spontaneously without any force.
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Post #33. There is no acceleration in the phase plane - just constant velocity ... no different than a plane flying along a geodesic.
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