Relativity of rotating system(s)

[Cleonis]

It's already known that GR predicts frame-dragging effects inside a rotating shell of matter. The http://en.wikipedia.org/wiki/Frame-dragging" discusses it in the section on the Lense-Thirring effect inside a rotating shell. These frame-dragging effects will appear as "centrifugal potential" to observers that are not rotating with the shell.

About the thought experiment of frame-dragging effects inside a rotating spherical shell:

It's only purpose is to explore a specific implication of GR. I don't know any details, I just assume the inside space of the spherical shell must be at least the size of a planet, and the thickness of the shell must be at least the diameter of the inside space.

The rotating spherical shell scenerio is physically impossible; such a shell would collapse under its own self-gravitation. Presumably this physical impossibility is tolerated because it doesn't impact the thought experiment's actual purpose.

In 1917, in the course of corresponding with Einstein, Thirring commenced calculations to obtain an approximation for the effects inside a rotating spherical shell. Frame dragging effects were found, but they fell far short of complete frame dragging inside. The result was an unwelcome surprise to Einstein, for at the time his expectation was that GR was an implementation of a strong version of Mach's principle. To pass as an expression of Mach's principle the inside frame dragging effects would have had to be far stronger than what was found. (See also from http://www.tc.umn.edu/~janss011/" )

In the same period, around 1918, other evidence surfaced that GR isn't an implementation of Mach's principle (and in particular not an implementation of strongly demanding versions of Mach's principle.)

(Of course this doesn't diminish the importance of the concept of frame dragging in its own right. I avidly followed the evaluations of the Gravity Probe B experiment as they trickled out.)

If GR isn't an implementation of a strong version of Mach's principle, then is it perhaps an implementation of a comparatively weak version of Mach's principle?
For instance, what if the distribution of inertial mass in the universe can be thought of as bringing forth the spacetime itself? Then the existence of inertial mass and GR-spacetime can be thought of as mutually dependent; GR-spacetime allowing existence of inertial mass, and inertial mass bringing forth GR-spacetime.

It has been pointed out that in order to obtain the Schwarzschild solution from the Einstein Field Equations the following condition is imposed: that towards spatial infinity the solution must approach asymptotically to Minkowski spacetime.
It has been argued (and I find it compelling) that if GR would be an implementation of a Mass/spacetime mutual dependence, then the GR equations would not allow a solution like the Schwarzschild solution. The Schwarzschild solution describes a infinite universe with a single lump of inertial mass, and inertia everywhere.

I find this reasoning compelling. It appears to me that GR is not an implementation of a weak version of Mach's principle either. In itself this does not exclude versions of Mach's principle, it just means that GR doesn't seem to be an instrument that can help in assessing whether our Universe is in some form a Machian universe.

Cleonis

<Addendum>
Reading the article by Herbert Pfister http://philsci-archive.pitt.edu/archive/00002681/" made me realize that while it's tempting to write about what I find compelling, I'm just out of my depth.

Pfister writes that in 1913 Einstein had performed similar evaluations as Thirring's, on the basis of what today is referred to as the 'Entwurf theory', a version of GR that Einstein in 1913 regarded as the finished GR. So Einstein could not have been surprised by Thirring's results.
</Addendum>

 

[yuiop]

It has been argued (and I find it compelling) that if GR would be an implementation of a Mass/spacetime mutual dependence, then the GR equations would not allow a solution like the Schwarzschild solution. The Schwarzschild solution describes a infinite universe with a single lump of inertial mass, and inertia everywhere.

It might be worth mentioning that the Kerr solution for a black hole is distinctly non Machian too, because it defines rotation of the black hole without any reference to any other mass in the universe. The black hole is simply spinning with respect to spacetime.

 

[PeterDonis]

The rotating spherical shell scenerio is physically impossible; such a shell would collapse under its own self-gravitation.

Wouldn't that depend on the specific characteristics of the shell?

It has been pointed out that in order to obtain the Schwarzschild solution from the Einstein Field Equations the following condition is imposed: that towards spatial infinity the solution must approach asymptotically to Minkowski spacetime.
It has been argued (and I find it compelling) that if GR would be an implementation of a Mass/spacetime mutual dependence, then the GR equations would not allow a solution like the Schwarzschild solution. The Schwarzschild solution describes a infinite universe with a single lump of inertial mass, and inertia everywhere.

If you take the solution in isolation, yes. But there's nothing stopping you from patching a portion of a Schwarzschild solution--say one with a central mass equal to that of the solar system, and taking the r coordinate out to a couple of light-years, which would be some 10^16 times the Schwarzschild radius--into a tiny portion of a global solution something like an FRW model. The boundary condition of asymptotic flatness for the Schwarzschild solution would just be a result of the fact that, on a distance scale of a light-year or two, the spacetime geometry of the universe as a whole *is* flat, because that scale is much too small to see any curvature effects. (Similar remarks would apply to the Kerr solution.)

Of course, if the universe as a whole is not closed, then we've just pushed the problem of what determines the boundary conditions out to the universe as a whole. That's why some physicists like models in which the universe is closed, so that there is no boundary condition required. (Hartle and Hawking's "no boundary" proposal in cosmology goes a step further by saying that the universe is closed in the time dimension, roughly speaking, as well as in the space dimensions, so there's no boundary condition required in time either.)

 

[hoiland]

Do the atoms on the surface of Earth decay slower relative to the atoms in the core of the Earth due to the velocity difference?

I'm not sure this particular question from the original post was ever adequately addressed.

Granted we are probably more concerned about the gravitational rather than the velocity difference, I'm still wondering whether this effect is significant enough to effect radioactive decay in the Earth's interior detectable over billions of years.

Despite any simplified calculations I've attempted, I'm not sure exactly what sort of gravitational potential a rock half-way-to-the-center-of-the-earth might be experiencing.

I'm particularly interested in long lived isotopic systems (like Lu-Hf, U-Pb, Rb-Sr, etc.) within the Mantle (so from around 7-70 km deep down to about 2855 km below the surface). Since in Geologic Time we're dealing with billions of years, I was wondering if these systems would need to be calibrated against the effect of relativity on their "apparent" rates of decay at depth.

For example, "mantle plumes" (which rise like lava lamp bubbles from the core-mantle boundary) are thought to be the source of hotspot volcanism (like that in Hawaii). These magmas have a distinctive isotopic signature, which is very different from magmas sourced from the upper mantle (like those of the mid-Atlantic Ridge). This is mostly due to the continued extraction of partial melts from the upper mantle, etc. due to plate tectonics over the past few billion years.

But assumptions about the lower mantle's composition are based off of measurements from undifferentiated meteorites (which were decaying far from a pronounced gravitational field) and on models from known decay rates combined with the age of the Earth (as calculated at the surface). Thus, we expect an undisturbed mantle to have a certain Lu-Hf ratio today by taking the starting ratio (from chondritic meteorites) and calculating what it ought to be today (using decay rates and time).

Specifically, would the dilation of time at ~3,000 km deep within the Earth be significant enough to affect model estimates of what today's isotopic ratios ought to be (in a closed system, and assuming meteorites do give a reliable starting ratio for 4.5 billion years ago)?

Thanks in advance for any help!

 

[D H]

Specifically, would the dilation of time at ~3,000 km deep within the Earth be significant enough to affect model estimates of what today's isotopic ratios ought to be (in a closed system, and assuming meteorites do give a reliable starting ratio for 4.5 billion years ago)?

No.

Clocks at the center of the Earth do tick slower than do clocks at sea level. A tiny, tiny bit slower.

The scale factor is 1 - 6.969290134×10^-10. Suppose a pair of ideal clocks were synchronized 4.5 billion years, with one kept at sea level for the next 4.5 billion years and the other magically placed and kept at the center of the Earth for the next 4.5 billion years. Today those two clocks would differ a grand total of a bit over 3 years. That of course is orders of magnitude smaller than the uncertainty in that 4.5 billion year figure.