Gravity inside the Earth

[Gerinski]

It seems common sense that as you go deeper into the Earth (eg with your feet towards the center) you will feel increasing pressure from the matter on top of you, that's what happens in deep sea.

It may also seem common sense that the very centre of the Earth must be an ingravity place, because you have just the same amount of matter in any direction surrounding you. Is this correct?
But if so, even being in zero gravity does anyway still mean that you have huge pressure from every direction? Shouldn't gravity from the surrounding matter "pull you outward in every direction" instead of pulling you down as it does in the surface?

Is there a relationship between gravity and pressure in this context?

Thanks!

[yourdadonapogostick]

the force is zero, so i am assuming the pressure would be zero as well.

[Doc Al]

While your "weight" (the gravitational force exerted on you by the mass of the Earth) will be zero at the center, the pressure you would experience (the weight of the earth pressing down on you) would be quite high.

[krab]

It seems common sense that as you go deeper into the Earth (eg with your feet towards the center) you will feel increasing pressure from the matter on top of you, that's what happens in deep sea.
But if you are in a rigid vessel (as in a submarine), this pressure won't be felt. The deeper you go in this vessel, the less gravity you feel. At the centre of the earth, you would be weightless as in free fall in space. But it is doubtful that you could design a vessel that could withstand the pressure at the centre of the earth; it's thought to be liquid there and very hot.

[Gerinski]

Thanks
But rather than just feeling weightless, shouldn't u actually feel a pull outwards in every direction, kind of getting swollen?

[Gerinski]

I remember reading in Brian Greene's "The fabric of the cosmos" something about pressure contributing also to gravity, but don't know if this has anything to do in this context

[krab]

Thanks
But rather than just feeling weightless, shouldn't u actually feel a pull outwards in every direction, kind of getting swollen?
No. Even the tiniest mass increment is pulled in all directions at once, so the forces simply cancel. You are thinking as analogous a situation where one end of something is pulled one way and the other end is pulled the opposite way. That's not the case here.

[BobG]

It seems common sense that as you go deeper into the Earth (eg with your feet towards the center) you will feel increasing pressure from the matter on top of you, that's what happens in deep sea.

It may also seem common sense that the very centre of the Earth must be an ingravity place, because you have just the same amount of matter in any direction surrounding you. Is this correct?
But if so, even being in zero gravity does anyway still mean that you have huge pressure from every direction? Shouldn't gravity from the surrounding matter "pull you outward in every direction" instead of pulling you down as it does in the surface?

Is there a relationship between gravity and pressure in this context?

Thanks!
Gravity and pressure are two separate things, but they are somewhat indirectly related.

Gravity is due to the mass below you. Any mass further away from the center of Earth than you will be cancelled out. The only way you could experience any outward gravity is if left side of your body was only gravitationally affected by the mass on the left side of your body and the right side of your body was only affected by the mass on the right side. It doesn't work that way. Each and every molecule in your body would be affected by a balanced gravitational field, so your body wouldn't feel any outward gravity.

Pressure is due to the weight of anything above you. In the sea, you feel the weight of all the water above you trying to replace your personal space with water. If you had a tunnel to the center of the Earth, it wouldn't be the weight of rocks above you that would increase the pressure - it would be the weight of the air above you. Air's not very dense, so the pressure increase wouldn't be as extreme as diving underwater. Of course, eventually you'd hit molten rock hot enough to melt your tunnel and then you would suddenly feel the pressure of all the matter above you trying to replace your personal space with molten rock. In other words, pressure depends on what's above you and wouldn't necessarily match the decrease in gravity.

[RandallB]

Although the idea of getting to the center of earth to safely test in a one atmosphere environment can only be done in a thought experiment. We can confirm the direction of these changes for real in a deep submersible like “The Challenger” that reached depth of 11 Km in Mariana Trench. I’m sure it measure greatly increasing pressure outside, and would measure the weight of a mass inside becoming lighter at depth.

But a good question would be is it lighter because of LESS gravity?
Or is it lighter even though gravity is the SAME or even GREATER just that as it is now being applied from multiple directions the weight decreases but maybe “gravity” does not ‘decrease’.

That is to ask : What is the affect on General Relativity here?
For example in a GPS system adjustments must be made for clocks in satellites running faster than the Master Clock on the surface due to GR and weaker gravity at altitude causing the orbiting clock to run faster. (Even larger SR adjustments for the speed of the orbiting clock slowing its time down must also be made).
But will the lower altitude at 11 Km below the surface continue the GR affect of slowing time down as we near the center?
Somehow that does not seem right to me. I’d think that the “gravity” for GR would be getting smaller inside the Earth and clocks would start to run faster again as they go deeper into the mass of Earth. (Meaning the GR time effect at the center of earth would be the equivalent of a deep space vacuum).
How does the details of GR address this?
Has it been confirmed by any direct tests of clocks taken to depth to confirm expectations by anyone??
I’d think the GPS system in place now would be sensitive enough to detect which type of correction is required.

Randall B

[Gerinski]

But will the lower altitude at 11 Km below the surface continue the GR affect of slowing time down as we near the center?
Somehow that does not seem right to me. I’d think that the “gravity” for GR would be getting smaller inside the Earth and clocks would start to run faster again as they go deeper into the mass of Earth. (Meaning the GR time effect at the center of earth would be the equivalent of a deep space vacuum).

That's interesting. If so, at what depht would the time-slowing reach its peak and start running faster again while moving towards the center?

[alex caps]

Keep in mind, gravity ONLY attracts, it does not repel. Therefore, at the center of the earth, you cannot be pushed away in every direction because gravity only attracts.

[Gerinski]

Keep in mind, gravity ONLY attracts, it does not repel. Therefore, at the center of the earth, you cannot be pushed away in every direction because gravity only attracts.

I got that, I was not thinking about gravity repeling. I was thinking about gravity attracting from allover around you, instead of the familiar attraction just from one direction.
At the center, gravity certainly attracts you "outwards", each lump of the earth's matter is exerting gravity pulling you away from the center.
What I missed to realise is that all this attraction anyway cancels eachother out so the net effect you feel is zero.

[Gerinski]

While your "weight" (the gravitational force exerted on you by the mass of the Earth) will be zero at the center, the pressure you would experience (the weight of the earth pressing down on you) would be quite high.

In any case, if our own weight reduces to zero as we move towards the center, that should also apply to all the other matter forming the earth.

The "weight of the earth pressing down on you" will then actually be much less than what common sense suggests when we think about all that mass in terms of the weight it would have in our familiar environment at the surface, right?

Just by the way, applied to black holes this reasoning must not work because all their matter is concentrated at the very center, as you approach the center you are never "in between" part of the matter and the center. We can not say if the very center is zero gravity or infinite gravity, or whathever else ....

[ZapperZ]

In any case, if our own weight reduces to zero as we move towards the center, that should also apply to all the other matter forming the earth.....

Yes, but there's still the weight closer to the surface that's still pressing down on the mass below it. So even if the weight of the earth that's closer to the center is getting smaller, it is still being pushed inwards by the mass on top of it.

This means that if you only consider gravitational attraction as the source of your force, then we can say, via the Gauss's Law equivalent, that gravitational force acting on you at the center of the earth is zero. However, if you want to include the pressure of the earth surrounding you due to its weight, then this part is no longer zero due to all the weight of the earth from the surface down.

Zz.

[RandallB]

This means that if you only consider gravitational attraction as the source of your force, then we can say, via the Gauss's Law equivalent, that gravitational force acting on you at the center of the earth is zero. Doesn’t this mean using General Relativity that time would be running more in sync with deep space at the center. Since the effect of GR and gravity slowing time down as you approach the earth down to the surface level, would reverse as you go below the surface, deeper into the earth until the direct force of gravity is cancelled out completely for space at the center.
Randall B

[Doc Al]

Since the effect of GR and gravity slowing time down as you approach the earth down to the surface level, would reverse as you go below the surface, deeper into the earth until the direct force of gravity is cancelled out completely for space at the center.
Why would gravitational time dilation reverse as you go below the surface? As you go beneath the surface, gravitational potential continues to decrease. Therefore I would expect a clock beneath the surface to run slowly as measured by a clock on the surface.

[RandallB]

Why would gravitational time dilation reverse as you go below the surface? As you go beneath the surface, gravitational potential continues to decrease. Therefore I would expect a clock beneath the surface to run slowly as measured by a clock on the surface.I don’t understand your logic, As compared to a surface clock, clocks at higher altitudes further away from the earth run faster than the surface clock due to the gravitational potential decrease. So how do you reason that when going below the surface also results in a gravitational potential decrease, but results in time running more slowly there???

It’s the surface clock that is experiencing the maximum gravitational potential thus it should have the Max time dilation.
RB

[Janus]

Doesn’t this mean using General Relativity that time would be running more in sync with deep space at the center. Since the effect of GR and gravity slowing time down as you approach the earth down to the surface level, would reverse as you go below the surface, deeper into the earth until the direct force of gravity is cancelled out completely for space at the center.
Randall B

No. Gravitational time dilation is related to difference in Gravtational potential not local Gravitational force. Sitting at the center of the Earth you are deeper in Earth gravity well than you are at the surface (IOW, it takes work to lift you from the center to surface), thus time runs slower at the center than at the surface.

[Janus]

I don’t understand your logic, As compared to a surface clock, clocks at higher altitudes further away from the earth run faster than the surface clock due to the gravitational potential decrease. So how do you reason that when going below the surface also results in a gravitational potential decrease, but results in time running more slowly there???

It’s the surface clock that is experiencing the maximum gravitational potential thus it should have the Max time dilation.
RB

I think you are having a problem with what "Gravitational Potential" means.

Quite simply, a difference in gravitational potential is proportional to how much engery it take to move from one point to another.

Example: you have a uniform gravity field of one g. (one that does not decrease in strength with height. You have two clocks of equal mass sitting at different heights in that field. The two clocks will have different gravitational potentials even though they both experience exactly 1g, and according to GR, the one that is above the other will run faster.

Thus a clock sitting at the center of the Earth is not at the same potential as one in space, even though they experience the same gravitational force locally.

[RandallB]

Quite simply, a difference in gravitational potential is proportional to how much engery it take to move from one point to another. Thanks Janus,
I think I see the point; it’s the total energy required to move back out to deep space that is always increasing as you move towards the center the per unit increase may get smaller below the surface as compared to the rate above the surface but still an increase. I’ll ponder on it a bit but I think I see the point.

Still be nice if someone has documented experiments done at deep water depths demonstrating time dilation running slower at depth vs. surface.
Do you know if anyone has tried?

Plenty have been done confirming faster time at higher altitudes.
RB

[Gerinski]

No. Gravitational time dilation is related to difference in Gravtational potential not local Gravitational force. Sitting at the center of the Earth you are deeper in Earth gravity well than you are at the surface (IOW, it takes work to lift you from the center to surface), thus time runs slower at the center than at the surface.

I guess I see more or less the point but, when we consider just a simple stationary clock right at the center of the Earth, therefore in a zero gravity location, and not trying to move at all, how the hell does the clock tell the difference compared to another clock sitting also in zero gravity in outer space?

[pervect]

I guess I see more or less the point but, when we consider just a simple stationary clock right at the center of the Earth, therefore in a zero gravity location, and not trying to move at all, how the hell does the clock tell the difference compared to another clock sitting also in zero gravity in outer space?

All clocks run at the rate of 1 second/second as far as they themselves are concerned. They don't have to "tell the difference". However, we still need to define the issue of how clocks at different locations in space are to be compared. To compare a clock running at the bottom of a gravity well, we have to somehow transport it's time reading through space.

One way of doing this transport operation is to just watch the reading of the lower clock with a telescope. One finds in this case that the clock lower in the gravity well runs slower. (Alternatively, one can say that from the viewpoint of the lower clock, the upper clock appears to run faster).

This cannot be explained away by propagation delay varying, because the propagation delay stays constant.

[Gerinski]

Clear now :-)

[anneb]

hmm, still not satisfied. How can there be weight when there is no gravity at the center of the earth? Doesn't the pressure caused by weight increase while going down to about half of the radius of the earth and then slowly start diminishing because of diminishing gravity? Or is center-earth pressure caused by something else than gravity? Same question applies to the center of the sun where pressures are supposed to be extremely high, but are they really?

Could someone give a formula for pressure and gravity as a function of depth?

[D H]

A couple of points first.
1. Welcome to PF, anneb!
2. In general it is not a good idea to revive threads that have been inactive for a long time, and this thread has been inactive for nearly four years!

hmm, still not satisfied. How can there be weight when there is no gravity at the center of the earth?

Nobody said that. They said just the opposite: You would be weightless at the center of the Earth.

Doesn't the pressure caused by weight increase while going down to about half of the radius of the earth and then slowly start diminishing because of diminishing gravity?
No. It is the gravitational force that decreases. The rate at which the pressure increases decreases as the gravitational force decreases. The pressure itself does not decrease. It keeps on building and building as depth increases, albeit at a reduced rate once the gravitational force finally starts decreasing.

A simple (far too simple) model of the Earth is to assume constant density. In this case the gravitational force will simple be a linear ramp. In reality, the density is anything but constant. The increasing pressure compresses the material that makes up the Earth, and the composition of the material varies with depth. In reality, the gravitation force decreases a bit initially, builds back up to a bit over surface gravity, and finally starts a steady (but not linear) decline at the mantle/outer core boundary. The gravitation force of course reaches zero at the center.

A simple model of the pressure is to assume hydrostatic equilibrium:

\frac{\partial p}{\partial z} = -\,\rho g

Could someone give a formula for pressure and gravity as a function of depth?
The only way to arrive at a formula is to use overly simplistic models. The real picture is purely empirical.

[anneb]

Thanks for this prompt reply after 4 years of lying dormant!

The cumulative building up of pressure while going down, even under diminishing gravity, sounds very plausible.

The forces are not linear because most of the heavier materials have probably moved to the center causing the distribution of mass in the sphere to be non-linear.

So as a thought experiment, if we would try to measure the weight of all mass surrounding us at the center of the earth, we would not measure any weight. But because of their mass - not their weight - the molecules immediately surrounding the center, exert an attracting force on the molecules that are further to the outside. The accumulated effect of these forces causes the pressure to be the highest at the center of the sphere.

[rossphysics]

If there is no gravity when you are at the center of the earth, then the rocks just above you would have next to no mass, so I would guess that pressure would also be zero.

[Janus]

If there is no gravity when you are at the center of the earth, then the rocks just above you would have next to no mass, so I would guess that pressure would also be zero.
The rocks just above you have mass, they just have little weight. However, they are supporting the weight of the rock above them which weigh a little more and which in turn support the weight of the rocks above them, etc. So the pressure at the center is due to the weight of all the rock above you and is quite large.

[Gear300]

Try to think of it like this: Force is a vector, so the net force could be said to be 0N. However, pressure is a scalar - the forces pile up; its sort of like pressing against a balloon from multiple directions - it will eventually pop. Also note that we're not referring to point particles, so a net force of 0N doesn't necessarily mean there is no deformation (though it does mean the acceleration is 0m/s^2).

[rossphysics]

In Newtons gravity the center of the earth would be like a piece of iron surrounded by magnets. The pull would be outward, in all directions, toward the great mass. The greatest pressure would be half way between the surface and the center of the earth.
In Einstein's world it would be more like being in a car that was in a car crusher. General relativity pushes from all directions inward.

[Doc Al]

The pull would be outward, in all directions, toward the great mass.
Actually, at the center there would be no outward pull at all.

[rossphysics]

Doesn't Newton say that a mass is pulled to another mass?

[D H]

Newton also derived an equation for the gravitational force on an object inside a spherical shell of mass. It is a very simple equation.

Google "Newton's shell theorem".

[rossphysics]

If we divided the earth into cubic meters, would the man in the middle be drawn to each of the cubes according to the distance to those meters?
And, I don't think that two planets is a good way of thinking because they are not one inside the other.

[DaveC426913]

Doesn't Newton say that a mass is pulled to another mass?
...
And, I don't think that two planets is a good way of thinking because they are not one inside the other.
If you have a planet on your left and a planet on your right, which way are you pulled?
Nowhere. You are weightless.

If you have half a planet on your left and half a planet on your right, which way are you pulled?
Nowhere. You are weightless.



If we divided the earth into cubic meters, would the man in the middle be drawn to each of the cubes according to the distance to those meters?

Yes. They would all balance out, resulting in zero weight.

[Dadface]

Doesn't Newton say that a mass is pulled to another mass?

Newton does say that but according to the theory every single point is pulled equally in all directions(assuming a spherical earth of uniform density) the result being that the forces cancel exactly at each point and the resultant force is zero.

[Dadface]

If you have a planet on your left and a planet on your right, which way are you pulled?
Nowhere. You are weightless.

If you have half a planet on your left and half a planet on your right, which way are you pulled?
Nowhere. You are weightless.




Yes. They would all balance out, resulting in zero weight.

I like the planet analogy but in this example I think that the person experiences a stretching force.Something that seems odd is that inside the earth there is no stretching force due to the forces acting on every point of infinitessimally small size cancelling.I have come across this somewhere else on this forum and I must admit I feel very uncomfortable with this concept.

[A.T.]

stretching force
What is a "stretching force"? There are just forces, and if you apply them nonuniformly to some body it can cause stretching, compressing, bending, etc. If you are big enough, you would actually fell compression in the center of the earth.

[Dadface]

What is a "stretching force"? There are just forces, and if you apply them nonuniformly to some body it can cause stretching, compressing, bending, etc. If you are big enough, you would actually fell compression in the center of the earth.

Of course there are just forces but stretching force is a commonly used and accepted term as are the terms resultant force, compressive force, breaking force,frictional force,surface tension force ......the list goes on.My use of the term stretching force is appropriate to the situation being described in that the gravitational force can have a stretching effect.
In your last sentence do you mean that "you would actually feel compression in the centre of the earth"?If so what causes this compression?

[A.T.]

In your last sentence do you mean that "you would actually feel compression in the centre of the earth"?If so what causes this compression?
Gravity applied non uniformly to your body. I'm thinking about a very very tall person in a tunnel resting at the center.

[quietrain]

erm, what if we are inside the earth but not exactly at the centre, then the forces experienced would be of different magnitudes? then the summation of forces will not be 0?

[Doc Al]

erm, what if we are inside the earth but not exactly at the centre, then the forces experienced would be of different magnitudes? then the summation of forces will not be 0?
Assuming the usual spherical symmetry, the gravitational force you'd feel at some distance D from the center of the earth will be due to the mass below you (r < D) and will be directed towards the center. The mass above you (r > D) will exert no force on you.

[MikeyW]

There is only one force, the net gravitational force (weight), and that always points towards the centre of the Earth, increasing linearly inside the radius, and decreasing like 1/r^2 outside.

If you are for example 1km away from the centre, you experience a force identical to when you are standing on a 1km radius planet of the same density as the core. All mass "above" you is cancelled and this is true for any spherically symmetric distribution.

edit- darn.. Doc Al has said exactly what I said in fewer words and 3 minutes earlier. At least we agree which is a good sign of my understanding!

[D H]

There is only one force, the net gravitational force (weight), and that always points towards the centre of the Earth, increasing linearly inside the radius, and decreasing like 1/r^2 outside.
That is a simplistic, and in this case, overly simplistic model of gravity inside the Earth. Moving from the center of the Earth outward, gravitational acceleration is zero at the center of the Earth. The gravitational acceleration increases non-linearly inside the core and reaches a value 10.7 m/s2 at the core/mantle boundary. From there outward the gravitational acceleration decreases.

[Dadface]

Gravity applied non uniformly to your body. I'm thinking about a very very tall person in a tunnel resting at the center.

But when you do the calculations you find that the gravitational effects due to the hollow shell surrounding that person cancel out.

[A.T.]

But when you do the calculations you find that the gravitational effects due to the hollow shell surrounding that person cancel out.
Only if the cavity is spherical. That is why I said: 'a very very tall person in a tunnel' would feel compression. My point was, there is no way you would feel any streching at the center on the earth.

[Dadface]

If there was a tunnel through the earth then,assuming uniform density, spherical symmetry and ignoring other forces each point on the person would experience a force which is roughly directly proportional to its displacement from the earths centre the result being that the person would keep moving from one side of the earth to the other with motion that is approximately simple harmonic.
Yes,with spherical cavities there is no stretching at the centre of the earth because all forces from the surrounding shell cancel.It does seem odd though.

[A.T.]

Yes,with spherical cavities there is no stretching at the centre of the earth because all forces from the surrounding shell cancel.It does seem odd though.
Assuming uniform density, which type of cavity would cause stretching by gravity inside the planet?

[Dadface]

Assuming uniform density, which type of cavity would cause stretching by gravity inside the planet?

Nice question but it is something I have never really thought about.My initial feeling is that there is no such cavity.Take your tunnel as an example.As you said if a man comes to rest in the middle he can experience compression.What if he is held in an off centre position his feet pointing towards the centre?If he is fixed in position at his feet he can experience compression and if he is fixed in position at his head he can experience stretching.Fix him at any other point and there can be both stretching and compressing.Of course this is cheating because of the fixing forces.Can you think of a suitable cavity?
I hope I haven't given the impression that I believe there are stretching forces,since theory suggests otherwise.What I have said is that it seems odd and that it is a concept I feel uncomfortable with.Are you familiar with feelings like this ,when there is something niggling but you can't quite pin down what it is?

[MikeyW]

How about an oblate spheroid cavity at the centre of a sphere? All mass from the maximum radius of the spheroid and above cancels, but there are still two masses above and below you, so your head experiences slight attraction upwards and your feet slightly downward.

Or have I got this all wrong?

[Dadface]

How about an oblate spheroid cavity at the centre of a sphere? All mass from the maximum radius of the spheroid and above cancels, but there are still two masses above and below you, so your head experiences slight attraction upwards and your feet slightly downward.

Or have I got this all wrong?

Have you got it all wrong?Don't know I am trying to visualise the set up you describe.As I understand it there is zero force at all points within a concentric spherical shell in an idealised spherical world .How do we know this to be the case since we can't enter such a space to do the measurements?We can use theoretical predictions by summing the contributions of all points from the surrounding shell but when we come to other shaped cavities the integration can become more complex.To make it even more complicated we are dealing with extended objects the various parts of which attract eachother gravitationally.

[rossphysics]

Doesn't anyone like Einstein any more? General Relativity says that space/time is pushing from the out, in. The center of the earth would be at maximum pressure.

[DaveC426913]

Doesn't anyone like Einstein any more? General Relativity says that space/time is pushing from the out, in. The center of the earth would be at maximum pressure.
Space-time does not push. Space-time simply curves.
Einstein has nothing to say about pressure.

[rossphysics]

I thought that the earth displaced space/time and made it more dense . . . you're right, if it were hollow, it would displace the same as if it were solid.

[DaveC426913]

I thought that the earth displaced space/time and made it more dense . . . you're right, if it were hollow, it would displace the same as if it were solid.
Made what more dense? Earth?