Spin Time Questions

[Secular Sanity]

It is vaguely possible that the coordinate system of the surface of the Earth is rushing up to hit the 'falling'  chalk (like the back of the rocket) - unless anyone has any sensible ideas I'll see if I can find any cats with an opinion about this.

I think that's the general idea.

[confused2]

It is vaguely possible that the coordinate system of the surface of the Earth is rushing up to hit the 'falling'  chalk (like the back of the rocket) - unless anyone has any sensible ideas I'll see if I can find any cats with an opinion about this.

I think that's the general idea.

Wouldn't that suggest the Earth is getting bigger at an ever increasing rate?

[Secular Sanity]

It is vaguely possible that the coordinate system of the surface of the Earth is rushing up to hit the 'falling'  chalk (like the back of the rocket) - unless anyone has any sensible ideas I'll see if I can find any cats with an opinion about this.

I think that's the general idea.

Wouldn't that suggest the Earth is getting bigger at an ever increasing rate?

You know the answer, but I’ll play along.

If I’m accelerating up and so is everyone else around the world, and presumably the whole surface of the earth, then shouldn’t the earth be expanding?

No. It’s possible for you to be accelerating, even though your spatial coordinates are not changing. In curved spacetime, you need to accelerate to just standstill. In GR, a freefalling object is not accelerating. →[straight path] [zero gravity]

But there’s no such thing as a free lunch. You’re an inertial observer during freefall, but you’ll pay the price when you encounter another object, such as the earth. During the impact, you’ll experience an upwards acceleration, and for a split second, you’ll weigh more. You might even break a leg…but gravity is not a force in the traditional sense.

[confused2]

No. It’s possible for you to be accelerating, even though your spatial coordinates are not changing. In curved spacetime, you need to accelerate to just standstill. In GR, a freefalling object is not accelerating. →[straight path] [zero gravity]

The Earth's surface and my chalk are both (obviously) close to the mass that causes the effect. How would you feel about the falling chalk getting the full benefit of the effect but the Earth's surface resisting it by virtue of being held in place by all the stuff underneath. This is an alternative to the Earth's surface doing the work (like the accelerating rocket) and the chalk being a victim of the circumstance it finds itself in.

So how can the equivalence principle be true?

Maybe there's a trick. As you drop the chalk above the Earth it stays in the same frame (SR wise) as it started off in . only when it hits the surface and 'stops' does it shift to the GR frame as dictated by gravitational potential. The 'bump' is the shift from the SR frame at the top to the new frame at the bottom. The frame at the bottom isn't accelerating with respect to the frame at the top - it has a fixed 'velocity' which can be worked out from the difference in gravitational potential. Thus (I claim) the acceleration is an illusion caused by the greater the height you start from the greater the difference in gravitational potential and apparent velocity between starting and finishing points. You might spot a problem with this .. I'm ready for that!

[Secular Sanity]

No. It’s possible for you to be accelerating, even though your spatial coordinates are not changing. In curved spacetime, you need to accelerate to just standstill. In GR, a freefalling object is not accelerating. →[straight path] [zero gravity]

The Earth's surface and my chalk are both (obviously) close to the mass that causes the effect. How would you feel about the falling chalk getting the full benefit of the effect but the Earth's surface resisting it by virtue of being held in place by all the stuff underneath. This is an alternative to the Earth's surface doing the work (like the accelerating rocket) and the chalk being a victim of the circumstance it finds itself in.

So how can the equivalence principle be true?

Maybe there's a trick. As you drop the chalk above the Earth it stays in the same frame (SR wise) as it started off in . only when it hits the surface and 'stops' does it shift to the GR frame as dictated by gravitational potential. The 'bump' is the shift from the SR frame at the top to the new frame at the bottom. The frame at the bottom isn't accelerating with respect to the frame at the top - it has a fixed 'velocity' which can be worked out from the difference in gravitational potential. Thus (I claim) the acceleration is an illusion caused by the greater the height you start from the greater the difference in gravitational potential and apparent velocity between starting and finishing points. You might spot a problem with this .. I'm ready for that!

I'm not following. Can you reword it? Will be busy all weekend.

[confused2]

It’s possible for you to be accelerating, even though your spatial coordinates are not changing. In curved spacetime, you need to accelerate to just standstill. In GR, a freefalling object is not accelerating. →[straight path] [zero gravity]

But there’s no such thing as a free lunch. You’re an inertial observer during freefall,

My problem is reconciling (say) a clock being hit by an accelerating rocket and a clock in a gravitational field hitting the surface of the Earth. It results in a prediction which may or not be correct. I can find Scharzchild radiuses and escape velocities but nothing that confirms what comes next.

A clock falling off a cliff (height h) will aquire a velocity v^2=2gh by the time it reaches the bottom. So the time dilation relative to a 'stationary' clock at the bottom of the cliff will be √(1-v²/c²) = √(1-2gh/c²)

My prediction is the time dilation (due to the difference in gravitational potential) between a clock at the bottom of the cliff and a clock at the top of the cliff must also be √(1-2gh/c²)

Yaay... occaisionally .. even in Idiotville..

https://plus.maths.org/content/what-general-relativity

For small distance changes, this approximation works pretty well…
t' ≈ t
√(1 − 2g∆h/c2)
where…

t = duration of an event in the gravitational field of some object (a planet, a sun, a black hole)
t' = duration of the same event when viewed from slightly higher up
g = local gravitational field (local acceleration due to gravity)
∆h = height difference between the event and the observer
c = speed of light in a vacuum

[Secular Sanity]

You drop your chalk towards a massive object like earth. As the chalk falls, its gravitational potential energy decreases, but the total energy-momentum of the chalk-planet system remains constant, because the decrease in gravitational potential energy is accompanied by an increase in the kinetic energy of the chalk (as it accelerates), the individual components of the energy change, but the total energy-momentum of the system stays the same.

But are your thoughts about acceleration being an illusion leaning towards Mach’s principle, which is sort of related to the idea that acceleration is an illusion, because it suggests that it's not an intrinsic property of objects, but rather a result of the distribution of mass in the universe?

https://www.youtube-nocookie.com/embed/cPEwkMHRjZU

[confused2]

Couple of points first..
Ideally we'd read, mark and inwardly digest The GR Book - I think I'm right that a description of what a thing does isn't quite the same as how it does it. Personally I'm happy with a poor description than captures the mechanism .. which is what I think we're looking for here.

Always work with potentials if possible. In the last post I could have used M1 and M2 and the gravitational constant, maybe the Schwarzchild radius to get g®, integrated the field to find the potential difference and/or shown that treating the field as uniform is ok. Instead I wrote down gh .. which was a lot easier.

There seems to be a few cats that think gravity 'works' by time dilation so I'm kind of keeping my paw on that.

You drop your chalk towards a massive object like earth. As the chalk falls, its gravitational potential energy decreases, but the total energy-momentum of the chalk-planet system remains constant, because the decrease in gravitational potential energy is accompanied by an increase in the kinetic energy of the chalk (as it accelerates), the individual components of the energy change, but the total energy-momentum of the system stays the same.

Basically agree - I think.

If you just use gravitational potential (cf gravitational potential energy) then mass is just a distraction.

To look at momentum we need mass so I'd suggest placing two identical planets of mass M 'some distance' apart and seeing what happens. The total momentum is always zero so we don't have to keep calculating it as we go along. The gravitational potential is pretty much the same on the surface of both planets so this might kill off some of the time dilation cats. All I can do is stare at this and hope to see 'something'. Anyone got any ideas about why the planets will move towards each other?

But are your thoughts about acceleration being an illusion leaning towards Mach’s principle, which is sort of related to the idea that acceleration is an illusion, because it suggests that it's not an intrinsic property of objects, but rather a result of the distribution of mass in the universe?

"I know nothing" seems like the best option here. Somehow Mach’s principle seems to bounce off me .. I see the words but .. nothing.

Edit.. Hm. They're actually heading for a centre of mass so gravitational potential (and time dilation) comes back into the game.

[Secular Sanity]

Do you know if the length contraction has ever been directly measured? 

What’s the difference between the length contraction and the Lorentz transformation?

[confused2]

Length contraction measured .. so far as I know not exactly measured .. but this is darn close..

The muons of interest are travelling close to the speed of light .. their rapid decay gives a way to see how much time has passed in the stationary muon frame of reference relative to an observer timing them between two clocks on Earth. At (close to) the speed of light the muons take 34us to travel 10km in the earth frame but in the (stationary) muon frame only 6.8us passes so either the muons think the the Earth is rushing past them at nearly 5 times the speed of light or 'something else'. The 'something else' is (has to be?) length contraction.

From http://hyperphysics.phy-astr.gsu.edu/hba.../muon.html -

The measurement of the flux of muons at the Earth's surface produced an early dilemma because many more are detected than would be expected, based on their short half-life of 1.56 microseconds. This is a good example of the application of relativistic time dilation to explain the increased particle range for high-speed particles.

What’s the difference between the length contraction and the Lorentz transformation?

So far as I know you can use (say) the length transformation formula to calculate length contraction - I suspect there is a more complex 'Lorentz transformation' which neither I nor wikipedia know about.