# Gravity inside the Earth

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!

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

Mentor
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.

Gerinski said:
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.

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

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

Gerinski said:
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.

Homework Helper
Gerinski said:
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 canceled 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
General Relativity inside Earth

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

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?

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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.

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 realize is that all this attraction anyway cancels each other out so the net effect you feel is zero.

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Doc Al said:
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 ...

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Staff Emeritus
Gerinski said:
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
ZapperZ said:
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 canceled out completely for space at the center.
Randall B

Mentor
RandallB said:
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 canceled 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
Doc Al said:
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

Staff Emeritus
Gold Member
RandallB said:
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 canceled 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.

Staff Emeritus
Gold Member
RandallB said:
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.

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RandallB
Janus said:
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

Janus said:
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?

Staff Emeritus
Gerinski said:
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.

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?

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Staff Emeritus
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.

Staff Emeritus
Gold Member
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.

Mentor
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?

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

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.

Gold Member
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.