Exploring Time Inside the Earth: A Comparison of Time at Different Depths

[Bjarne]

Does time go slower or faster halfway to the center of the earth?
and in the center of the Earth ?
(compared to the time at the surface of the earth)

(If so does anyone know how much?)

 

[Theinvoker]

Hiya,

No time would be the same. Standard time as we know it is just the period it takes for the Earth to rotate and orbit the Sun. One Earth year will still be 365 Earth days and an Earth day will still be 24 hours whether you are in the center of the Earth or in another Galaxy.

However if you were to "set" time according to the center of the Earth at an infinitively small point (much like Mars' year is longer that ours) the year would remain the same as it would still orbit the Sun at the same rate but there would be an infinate number of days in a year! :P
However as its not possible to get that small, if you stood in the center, it would still take 365 days for you to rotate through 360 degrees to back your original point.

Time in relation to the days, years etc is a man made concept and is based upon the characteristics of the Earth to provide order ie. we all turn up for work at the same time. However, a period of time in the non man made sense, ie. between event A and even B is much more complex as we have to relate it to the man made version for most people to make sense of it eg. The universe was born 3.7 billion years ago, but man wasn't about then, neither was our sun, so how could their be a year at that point? ;)

 

[Bjarne]

I mean according to the theory of relativety

 

[George Jones]

I mean according to the theory of relativety

If an observer on Earth's surface uses a telescope to look down a tunnel to a clock at the Earth's centre, he will see his clock running faster than the clock at the Earth's centre.

By how much?

I'm in the process of calculating this, but since my 16-month old daughter thinks this is a complete waste of good playing time, it might take a while.

 

[Laura1013]

This thread from the archives may help: https://www.physicsforums.com/archive/index.php/t-82309.html (link fixed).

If I'm reading it correctly, I think the consensus was that time would run more slowly inside the Earth.

Edited to add: Or, to agree with how George phrased it, the observer on the surface would see his/her clock running faster.

The universe was born 3.7 billion years ago, but man wasn't about then, neither was our sun, so how could their be a year at that point? ;)

Left out a 1 there in the age of the universe, didn't you? ;)

 

[Theinvoker]

Left out a 1 there in the age of the universe, didn't you? ;)

Oops typos are great. I still like the idea of an infinate number of days in a year lol :)

Ps. that link seems to just go round to this one, whet abouts is it found as it sounds quite interesting read :)

 

[Laura1013]

Ps. that link seems to just go round to this one, whet abouts is it found as it sounds quite interesting read :)

Sorry about that! I fixed it.

 

[George Jones]

If the Earth is modeled as a constant density, non-rotating sphere, then Schwarzschild's interior solution can be used. When $G=c=1$,

$$d\tau^{2}=\left( \frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2Mr^{2}}{R^{3}}}\right) ^{2}dt^{2}-\left( 1-\frac{2Mr^{2}}{R^{3}}\right) ^{-1}dr^{2}-r^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}\right),$$

where $R$ is the $r$ coordinate at the surface of the Earth.

If an observer on the Earth's surface uses a telescope to look down a tunnel to a clock at the Earth's centre, he will see his clock running faster than the clock at the Earth's centre.

Consider two dentical clocks, one moving around the Earth once a day on the Earth's surface at the equator ($\theta = \pi/2$) and one at the Earth's centre. Both clocks have constant $r$ values, so $dr=0$ for both clocks, and, after factoring out a $dt^2$, the above equation becomes

$$\left( \frac{d\tau }{dt}\right) ^{2}=\left( \frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2Mr^{2}}{R^{3}}}\right) ^{2}-v^{2},$$

where $v=rd\phi/dt$ is, approximately, the speed of something moving along a circular path. At the centre, $v=r=0$, and, on the surface, $r = R$ and $v = 1.544 \times 10^{-6}$, which is one Earth circumference in one day.

Then, with $G$ and $c$ restored,

$$\frac{d\tau_{centre}}{d\tau_{surf}}=\left( \frac{d\tau_{centre}}{dt}\right) \left( \frac{d\tau_{surface}}{dt}\right)^{-1} =\frac{\frac{3}{2}\sqrt{1-\frac{2GM}{c^{2}R}}-\frac{1}{2}}{\sqrt{1-\frac{2GM}{c^{2}R}-v^{2}}}$$.

Running, the numbers, I get

$$\frac{d\tau_{centre}}{d\tau_{surf}} = 1 - 3.5 \times 10^{-10}.$$

Lots of places errors could have crept in, though.

 

[Theinvoker]

Interesting workings. Would be interesting to run this across a variety of distances from the surface and see if there's a relationship between them and weather the center is a special case due to the 0 distance :)

Finding a way to factor in the non constant density of the Earth would also be a good challenge, but think that's way over my head for the moment lol.

 

[Bjarne]

George Jones

Thank you for the nice calculations.
Sorry I do not understand a mathematically language, (and English is not perfect).
I understand that it is 2 causes for the time difference.
One is cause due to the different speed of rotation of the earth, and one is due to that the force of gravity is different.

1.) I have understood that times run slower at the surface of the earth, compared to further out in space, because of is gravity stronger here, - correct?
2.) Gravity inside the Earth's is decreasing gradually until 0 inside the earth, - so this mean that gravity difference should become less in inwards direction and cause time to runs faster the further we reach the centre of earth?– Correct? – (But it is the opposite you have calculated – why)
3.) Also the rotation of the Earth has an influence, - how much wills this cause time to increase / decrease (for instance per day) at the surface of the earth?

Actually I need to know: - how much will a clock gain or lose per day when it was places 3.000 Meter below the surface of the earth, for instance in a goldmine in South Africa ? – (compared to a clock at the surface of the earth)

(Have such experiments been done?)
(Is sensitive atomic clocks suitable to such experiments - expensive to buy / rent )

sorry for the bad english..

Regards
Bjarne

 

[Janus]

George Jones

1.) I have understood that times run slower at the surface of the earth, compared to further out in space, because of is gravity stronger here, - correct?
2.) Gravity inside the Earth's is decreasing gradually until 0 inside the earth, - so this mean that gravity difference should become less in inwards direction and cause time to runs faster the further we reach the centre of earth?– Correct? – (But it is the opposite you have calculated – why)

Regards
Bjarne

This is a common misconception about gravitational time dilation. Time runs slower on the surface than in space because the surface is deeper in the Earth's gravity field, not because the force of gravity is stronger. Even though the force of gravity decreases as you approach the center, you are still going deeper into the Earth's gravity field. One way to look at it is that it takes work to climb out from the center of the Earth, just as it takes work to climb from the surface to further out in space.

Another example

 

[marcus]

Bjarne and others, you might possibly be interested in knowing about the TCG (Geocentric Coordinate Time)
http://en.wikipedia.org/wiki/Geocentric_Coordinate_Time
this time is used for some astronomical purposes.
It is the time that would be told by a clock that is co-moving with the CENTER OF THE EARTH, but far enough removed so that it is not slowed down by being in Earth's gravity well.

There is also something analogous for the sun, called the TCB (Barycentric Coordinate Time) which is used in solar system astronomy because it is the time that the solar system can be imagined to be running on. It is the time that would be told by a clock co-moving with respect to the center of the sun---actually not the center of the sun, but the barycenter of the solar system. This is the center of gravity around which all the planets actually orbit----they do not orbit the sun exactly but rather the barycenter of the whole system. Moving in parallel with it, but far enough away not to be influenced by the sun's graviational field.

http://en.wikipedia.org/wiki/Barycentric_Coordinate_Time

One keeps track of these time standards, for example of TCG (referenced to the center of the earth), by modifying an average of readings from various atomic clocks. All the REAL clocks are distorted by their various motions and locations in gravitational field, so their time must be CORRECTED (using formulas similar to what George Jones was showing us, I would imagine) to give the true earth-center time.

As Wikipedia says, the TCG clock "ticks faster" than clocks on the Earth surface, by some fraction which is estimated in the article. Likewise the TCB clock ticks faster still, because it does not suffer the time dilation associated with the Earth's motion, or the dilation associated with the Earth's position deep in the gravity well of the inner solar system.
One can picture the TCB clock located somewhere in the Oort cloud in the outer reaches of the solar system, but always stationary with respect to the barycenter.

 

[Mixolydian]

Time is not real, it is a series of moments. 1 second would still have the same duration in the center of the Earth as the surface of the earth. This clock you guys speak of must be theoretical, or digital. Why can't we assume that clocks appear to move faster or slower based on the unfavorable conditions present in the center of the earth.

It's not a very good control when both clocks fall under different conditions, how would you expect to get the same result? Shouldn't you assume that the result would be different ><?

What would it prove if both enviroments showed the same result.

If this observer was observing this, it mostly likely has to do how the waves of light is reaching them. I would expect the light coming from the center of the Earth to differ vs. light from the surface of the earth.Moment by moment scales like seconds or minutes can't change, moving to the center of the Earth won't make you live longer, if anything you will die faster. I know I would live longer then my friend in the center of the Earth :P.

 

[Naty1]

Mixo: I can't find a single thing in your post that reflects current theory...so I will not reply ...If you are imagning another world, all well and good; but your post does not represent this one.

 

[Nabeshin]

Time is not real, it is a series of moments. 1 second would still have the same duration in the center of the Earth as the surface of the earth.


This clock you guys speak of must be theoretical, or digital. Why can't we assume that clocks appear to move faster or slower based on the unfavorable conditions present in the center of the earth.

It's not a very good control when both clocks fall under different conditions, how would you expect to get the same result? Shouldn't you assume that the result would be different ><?

What would it prove if both enviroments showed the same result.

If this observer was observing this, it mostly likely has to do how the waves of light is reaching them. I would expect the light coming from the center of the Earth to differ vs. light from the surface of the earth.


Moment by moment scales like seconds or minutes can't change, moving to the center of the Earth won't make you live longer, if anything you will die faster. I know I would live longer then my friend in the center of the Earth :P.

Thanks for the opinions supported by... nothing...

 

[russ_watters]

Note that the thread is 10 months old and has run its course. Locking.