Gravity at Earth's Core: Human Impact Hypothesis

[tanzanos]

Hypothetically: If a hollow sphere with a human inside were placed exactly at the centre of the Earth's core; then how would gravity affect the human (or any mass for that matter).

 

[Stonebridge]

The human would be pulled in all directions equally by the mass distributed evenly all around him/her.
As far as the person was concerned he/she would feel nothing and would float freely at the centre of the hollow sphere.
[It might get a bit hot though! :) ]

 

[Maxwell's Demon]

That's true. The point at the center of mass exhibits no vector force from the Earth's gravity, and within the volume of a spherical shell at the center of a massive body, there is no gravitational gradient at all - spacetime is 'flat' there. Here's a fascinating paper about how the laws of General Relativity describe conditions like this inside of a body so massive that it essentially stops time (with respect to a distant observer) known as a "stasis chamber" http://arxiv.org/PS_cache/gr-qc/pdf/0701/0701084v2.pdf


Note also that the time dilation is at a maximum value at the center of gravity, because the gravitational potential continues to rise beneath the surface of the planet. The magnitude of the time dilation at the center would be approximately the square root of 1.5 times greater than the time dilation at the surface of the Earth (however, because the Earth is more dense toward the center, and not of uniform density, the time dilation at the center would be very slightly greater than that).

 

[Drakkith]

How does gravity pulling in different directions affect time dilation i wonder? If at all that is...

 

[inflector]

Note also that the time dilation is at a maximum value at the center of gravity, because the gravitational potential continues to rise beneath the surface of the planet.

That doesn't jive with the idea that there would be no acceleration combined with the equivalence principle. If there is no acceleration then the gravitational potential is also zero by the equivalence principle, therefore there should be no time dilation. The gravitational potential does not continue to rise beneath the surface of the planet. As you get closer to the center it drops to zero.

The pressure? Another story. It does indeed rise as you get closer to the center.

 

[pallidin]

Time dilation will always ocurr within a changing gravitational field experience.
However, whether this dilation is noticeable, or even important, is another matter.

 

[Vanadium 50]

If there is no acceleration then the gravitational potential is also zero by the equivalence principle

No, force is the derivative of potential. No force means a constant potential, not zero potential.

 

[sub_zero]

theoretically, he will stuck for ever in that place, i recommend you to read physics for entertainment, author yakov Perelman, it's very popular and you can find it easily.
in that book yakov discussed that very situation by details.

 

[D H]

That doesn't jive with the idea that there would be no acceleration combined with the equivalence principle. If there is no acceleration then the gravitational potential is also zero by the equivalence principle, therefore there should be no time dilation.

As Vanadium already noted, zero acceleration does not mean zero potential. It merely means that the potential is constant. What Vanadium left out is that gravitational time dilation is a consequence of gravitational potential not acceleration. Time is dilated inside the sphere.

The gravitational potential does not continue to rise beneath the surface of the planet. As you get closer to the center it drops to zero.

Assuming the Earth is of uniform density and arbitrarily setting the potential at infinity to zero, the gravitational potential at some distance r<R from the center of the Earth is given by

$$\phi = -\frac{GM}{2R^3}(3R^2-r^2)$$

The potential at the center of the Earth is not zero.

 

[inflector]

Hmmm, I seem to have completely misunderstood the Equivalence Principle then.

No, force is the derivative of potential. No force means a constant potential, not zero potential.

and

As Vanadium already noted, zero acceleration does not mean zero potential. It merely means that the potential is constant. What Vanadium left out is that gravitational time dilation is a consequence of gravitational potential not acceleration. Time is dilated inside the sphere.

This doesn't seem to me to jive with:

According to General Relativity, gravitational time dilation is copresent with the existence of an accelerated reference frame.

from http://en.wikipedia.org/wiki/Gravitational_time_dilation#Important_things_to_stress".

Is this one of this instances where Wikipedia is wrong or misleading? Or have I misinterpreted the statements there? Or are you saying that at the exact center of mass for the Earth there is not an inertial frame but instead there is an accelerated frame?

Are there some references or papers you can point me to that illustrate this divergence in the ties between acceleration and time dilation?

 

[inflector]

One further item that doesn't make sense to me now:

The strong equivalence principle suggests that gravity is entirely geometrical by nature (that is, the metric alone determines the effect of gravity) and does not have any extra fields associated with it. If an observer measures a patch of space to be flat, then the strong equivalence principle suggests that it is absolutely equivalent to any other patch of flat space elsewhere in the universe.

From http://en.wikipedia.org/wiki/Equivalence_principle#The_strong_equivalence_principle".

Or is this statement:

The point at the center of mass exhibits no vector force from the Earth's gravity, and within the volume of a spherical shell at the center of a massive body, there is no gravitational gradient at all - spacetime is 'flat' there.

in error?

Or perhaps I have got this completely wrong and time dilation has nothing to do with spacetime curvature.

 

[D H]

You need to learn to walk before you can learn to run, inflector. Physics students really should learn about Newtonian mechanics in general and Newton's shell theorem in particular before delving into general relativity. There are many ways to prove Newton's shell theorem.

1. Use Newton's convoluted geometric arguments. One word: Yech.
2. Integrate the Newton's expression for gravitational acceleration over a ring and then integrate that over the sphere. This is a rather messy double vector integral. Still yech.
3. Integrate the gravitational potential and find that it is constant inside a sphere. A freshman in college could do this integral.
4. Invoke Gauss law for gravitation and be done with it. Easy, but freshman don't know Gauss' law for gravitation.

With option 3 it is easy to show that the gravitational potential inside a spherical shell, while constant, is not the same as the gravitational potential at some point outside of and far removed from the shell.

General relativity must agree with Newtonian mechanics in the domain in which Newtonian mechanics has been well-tested. This is the domain of smallish masses, smallish relative velocities, and not-so-small distances between masses. Note that one of the key post-dictions of general relativity, the relativistic precession of Mercury, is a combination of a not-so-small, a not-so-small relative velocity, and somewhat small distances. Even then, the difference between Newtonian mechanics and general relativity is very, very small.

Regarding the two wikipedia statements you cited regarding the equivalence principle: First off, it's wikipedia. No justification, no explanation, no math, just raw statements. Do learn to take things in wikipedia about controversial subjects or science beyond that taught to freshmen with a big grain of salt.

The first statement applies to clocks in a spaceship. Suppose a spaceship is accelerating and has two clocks in it, one near the tail and one near the nose. That acceleration results in the equivalent of a gravitational potential difference between the clocks. The clock in the nose will be time dilated compared to the clock in the tail of the rocket.

The second statement misses the mark in my mind. In particular, it is missing the word "local". Comparing conditions inside a spherical shell of mass to those outside the shell is not a local experiment.

 

[inflector]

You need to learn to walk before you can learn to run, inflector. Physics students really should learn about Newtonian mechanics in general and Newton's shell theorem in particular before delving into general relativity.

...[snip]...

General relativity must agree with Newtonian mechanics in the domain in which Newtonian mechanics has been well-tested.

How can we address time dilation using Newtonian mechanics? Is there some sort of conversion factor like the Lorentz factor that allows one to go from a Newtonian gravitational potential to time dilation?

Regarding the two wikipedia statements you cited regarding the equivalence principle: First off, it's wikipedia. No justification, no explanation, no math, just raw statements. Do learn to take things in wikipedia about controversial subjects or science beyond that taught to freshmen with a big grain of salt.

I didn't think general relativity, the equivalence principle or time dilation was particularly controversial.

As far as the Wikipedia quotes. I'm happy to believe they are wrong or misleading but I need some references, if possible, where I can learn the correct application. In particular, anything that shows specifically how time dilation is not related to acceleration or space curvature. I just don't yet understand how we can discuss this in terms of Newtonian gravitational potential since Newtonian mechanics doesn't allow for time dilation.

 

[D H]

How can we address time dilation using Newtonian mechanics? Is there some sort of conversion factor like the Lorentz factor that allows one to go from a Newtonian gravitational potential to time dilation?

For objects at rest with respect to and outside of a non-rotating object with a spherical mass distribution, the Newtonian gravitational potential is Φ=-GM/r, where r is the distance to the center of the object. The gravitational time dilation factor derived from the Schwarzschild metric is

$$\left(\frac{\Delta \tau}{\Delta t}\right)^2 = 1 - \frac {GM}{2c^2 r}$$

Substituting Φ=-GM/r in the above yields

$$\left(\frac{\Delta \tau}{\Delta t}\right)^2 = 1 + \frac{\Phi}{2c^2}$$

That negative sign on the potential can lead to confusion. Some use U=GM/r instead, so that

$$\left(\frac{\Delta \tau}{\Delta t}\right)^2 = 1 - \frac{U}{2c^2}$$

I didn't think general relativity, the equivalence principle or time dilation was particularly controversial.

I said controversial or advanced. It certainly is advanced science. It is typically first taught at the college level as a senior year undergraduate or first year graduate course.

In particular, anything that shows specifically how time dilation is not related to acceleration or space curvature.

Indirectly, yes. Directly? Saying that will lead to misconceptions.

I just don't yet understand how we can discuss this in terms of Newtonian gravitational potential since Newtonian mechanics doesn't allow for time dilation.

Look at it in terms of energy. A photon climbing out of a gravity well will lose energy. Photons can't change in velocity; they lose energy by a decrease in their frequency. This is the gravitational redshift. This decrease in frequency also happens with electromagnetic signals such as a signal based on the ticking of a clock.

We are getting way off-topic here. So, going back to the original problem, a hollow sphere. The gravitational potential inside a hollow sphere is not the same as the gravitational potential at some point far removed from the sphere. Time will be dilated inside the sphere. So how does this jibe with the equivalence principle? The equivalence principle talks about local experiments. Comparing the rate at which a clock ticks inside our hollow sphere and at a point far removed from the sphere is not a local experiment. The equivalence principle is not applicable to this non-local experiment.

 

[Dale]

That doesn't jive with the idea that there would be no acceleration combined with the equivalence principle. If there is no acceleration then the gravitational potential is also zero by the equivalence principle, therefore there should be no time dilation.

In addition to the above responses, it is important to correctly apply the equivalence principle. The equivalence principle applies only to a small region of spacetime, sufficiently small to ignore the spacetime curvature. If we are performing experiments comparing two clocks within the hollow in the center there is no gravitational acceleration and no relative time dilation and the equivalence principle holds nicely. If we are performing experiments comparing a clock within the hollow and a clock far above the surface then there is gravitational time dilation but the clocks are too far apart for the equivalence principle to hold.

 

[DaveC426913]

The way I understand the equivalence principle, an observer in a closed environment cannot perform any local test that will tell him whether he is accelerating (say, by way of a rocket) or simply in a gravity field.

Note, the tests must be local - if he could collect data from distant locations - for example, if he could look out a window, he could simply see whether the universe was whizzing by or not.

But note that, in a closed environment, he also can't tell whether he is experiencing any time dilation wrt the rest of the universe. In his reference frame, time is ticking by normally. No local experiment can tell him otherwise.

 

[inflector]

Thanks D H, DaleSpam, and DaveC,

I really thought I understood this before. Thanks for correcting me. I always enjoy learning when I've made errors in judgement or assumption.

I think I understand my error in thinking that unaccelerated frames were 100% equivalent.

Theoretically, there'd be no way for a person inside a closed capsule to know he was in the center of the Earth floating without acceleration or in an orbit or in deep space. However, if I get this right, that doesn't mean he wouldn't be experiencing time dilation as measured by an observer outside his capsule. And I can see that there'd be no way for the person in the capsule to know that time was slower for them than it would be for an observer at a point with higher gravitational potential unless they peeked outside the capsule, in which case the equivalence principle no longer holds.