[Lewis Thomson (OP)]

Ranjit has been lost in this lunar conundrum for a while so he submitted it to The Naked Scientists...

"So, gravity and time relate. Greater the gravity slower time. Time runs faster on moon than on earth. So the question-  When the moonrocks were brought back- they must have looked older than the comparable rocks left behind on earth early on when the moon 'departed' earth? Maybe not by much given the margin of error in measurement. However if that is right (and I hesitate to ask this....) will there come a period when the rocks originated from the earth....BEFORE the earth was formed?"

Do you have any answers that can make things relatively simple? Leave them in the comments below...

[Origin]

"So, gravity and time relate. Greater the gravity slower time. Time runs faster on moon than on earth. So the question-  When the moonrocks were brought back- they must have looked older than the comparable rocks left behind on earth early on when the moon 'departed' earth? Maybe not by much given the margin of error in measurement.

The age difference like you said will be VERY small.

However if that is right (and I hesitate to ask this....) will there come a period when the rocks originated from the earth....BEFORE the earth was formed?"

No that will never happen.  Here is an analogy, assume there is a race where one racer is just a little faster than the other.  The longer the race lasts the farther the faster runner will pull ahead of the slower one.  So you are effectively asking the question, "if the race went on long enough would the slower racer at some point start going backwards and end up at the starting line"?  Obviously not!

[Janus]

Ranjit has been lost in this lunar conundrum for a while so he submitted it to The Naked Scientists...

"So, gravity and time relate. Greater the gravity slower time. Time runs faster on moon than on earth. So the question-  When the moonrocks were brought back- they must have looked older than the comparable rocks left behind on earth early on when the moon 'departed' earth? Maybe not by much given the margin of error in measurement. However if that is right (and I hesitate to ask this....) will there come a period when the rocks originated from the earth....BEFORE the earth was formed?"

Do you have any answers that can make things relatively simple? Leave them in the comments below...

First off, the time dilation is not due to a difference in the strength of gravity, but to a difference in gravitational potential.
To illustrate the difference, imagine you had a uniform gravity field (one that did not change in strength with altitude). If you place two clock at different heights in this field, the higher clock runs faster even though it feels exactly the same gravitational force as the lower clock.

Now, to the other point.  Say you have a rock on Earth, before the Moon was formed. it is broken in half and one half is blasted off and becomes part of the newly formed Moon. Billions of years later, that rock is collected and returned to Earth and put next to the other half.
Both rocks share the same "end points" They both agree that they were once part of the same rock and that they were separated at the same moment.  They agree that they were reunited at the same moment. They just disagree on how much time elapsed for them between those moments.
Using the racer analogy from the other post.  The racers start at the same starting line and end at the same finish line, but one took a different route and ended up running a further distance.

[Eternal Student]

Hi.

   I like all the previous replies.

   The original question is quite difficult to answer.  I think it's important to set out the difference between "co-ordinate time" and "proper time".   There are many texts and articles that will describe the notion of "proper time".   A clock can ONLY measure the passage of proper time and not the passage of co-ordinate time.   

    An object shows age according to the amount of proper time that has elapsed for it.   It does not age according to the co-ordinate time that has elapsed.

    Let's say this another way:   
1.  Yes it is possible then when the space explorers (of the future and from some other galaxy) travel to the moon and examine the rocks, they find they have been aging and show properties indicating they have experienced 10 billion years of proper time.   If those moon rocks had spent most of their proper time on the moon then we could say the rocks are 10 billion years old in the local time of the moon.
2.  Yes it is possible that when they travel to earth they can run some tests and find the whole planet has only experienced 9.9 billion years of proper time.  Or to phrase it another way, they might find that planet earth is only 9.9 billion years old IN THE LOCAL TIME when you are located at the earth.
3.   However, they DO NOT conclude the moon rocks couldn't have come from the earth unless they are careless and thought that local times pass at the same rates regardless of where a clock is located.   The age of some object just tells you how much proper time it has experienced, it doesn't tell you anything about how much co-ordinate time has elapsed since the object was made.

Question from the title of this thread:   Does relativity make rocks on the moon older than Earth rocks?
Answer:    Yes.

Question from the main body of the OP:

However if that is right (and I hesitate to ask this....) will there come a period when the rocks originated from the earth....BEFORE the earth was formed?"

   No.   The rocks on the moon can appear to be older than the planet earth.   That just means they have experienced more proper time.   
   Only a careless space explorer who didn't know about relativity would just assume they can wind back the local clock of the moon at the same rate as they wind back the local clock of planet earth.

Best Wishes.

[gem]

HI all
Janus

First off, the time dilation is not due to a difference in the strength of gravity, but to a difference in gravitational potential.
To illustrate the difference, imagine you had a uniform gravity field (one that did not change in strength with altitude). If you place two clock at different heights in this field, the higher clock runs faster even though it feels exactly the same gravitational force as the lower clock.

Has this been shown experimentally, if so could you provide information please.

[Origin]

Has this been shown experimentally, if so could you provide information please.

Janus is correct.  If you were at the center of the earth (and were some how to manage not to be crushed or burned up)  you would feel no gravitational force and so you would be weightless. The center of the earth is at the lowest gravitational potential however so you would have a larger time dilation that at the surface of the earth.

[Janus]

HI all
Janus

First off, the time dilation is not due to a difference in the strength of gravity, but to a difference in gravitational potential.
To illustrate the difference, imagine you had a uniform gravity field (one that did not change in strength with altitude). If you place two clock at different heights in this field, the higher clock runs faster even though it feels exactly the same gravitational force as the lower clock.

Has this been shown experimentally, if so could you provide information please.

It is demonstrated every time we measure gravitational time dilation and the results match those predicted by Relativity.
So, for instance on a planet like the Earth, we can calculate both the gravitational force and the predicted gravitational time dilation between two points.  Gravitational force changes by the inverse of the square of the distance from the center of the Earth, and gravitational potential changes by the simple inverse of the distance.
So, by measuring time dilation differences between various points, we can determine which of these two matters. And we always get the answer that it is the gravitational potential difference, not the gravitational force difference.

As far as the uniform field goes:  Once we have determined that the accepted gravitational time dilation formula give the correct results, we can do the following thought experiment.
Start with the Earth and two clocks, one at sea level, and the other at 1000 km above it.  Work out both the gravitational force difference between the two and the gravitational time dilation.
Now double the Earth's radius and quadruple its mass.  The sea level clock feels exactly the same gravitational force as it did before.  But, since 1000 km is a smaller percentage of the total new Earth radius, the difference in gravitational force between the two clocks is less than what it was before.  However, the time dilation difference(as per the previously confirmed formula), increases.
keep doubling the radius and quadrupling the mass.  At each step the gravitational force difference between the clocks decreases, and the gravitational time dilation difference increases. And as the gravitational force difference approaches 0, the gravitational time dilation approaches a finite maximum.

[Eternal Student]

Hi.

   I quite like the explanation given by @Janus  but there may be a slight problem.   It's not too important because the first paragraph seems sufficient on it's own to justify the statement that time dilation depends on gravitational potential not on gravitational field strength.

Anyway, you (@Janus) have written something that was interesting and I've spent the time re-reading it and trying to work it through.  If I point out an issue it's only because the article was interesting.  I've never seen it explained that way and I thought it might actually be a useful way to explain it to others.

....(In each step)......keep doubling the radius and quadrupling the mass.

   One problem is the growth of the Schwarzschild radius.    r_s =  2GM/c²  so it quadruples in each step.   However, the radius of the planet, r, was only doubling on each step.
If you kept doing this the Schwarzschild radius would extend beyond the surface of the planet.  At that point strange things happen and I'm not sure we have a time-dilation formula we can use.
- - - - - - - -
It doesn't matter too much.  We are in agreement that time dilation seems to be driven by gravitational potential and not gravitational field strength.

Best Wishes.

[gem]

Hi all,

Thank you Janus for that response, it sits better with me than the previous analogy of a gravity field not following the inverse square law.

If I may borrow part of your explanation to respond, and explain why/how I understand your statement is correct :

Janus
 

First off, the time dilation is not due to a difference in the strength of gravity, but to a difference in gravitational potential.

So firstly if you calculate the strength of g and escape velocity and time dilation at the Earths surface gives

                                    g  = 9.82 m/s^2
             escape velocity  =  11.19 x10^3 m/s
                  time dilation  =  + 6.97 x10^-10 sec/sec

Then to borrow your example:
 "Now double the Earth's radius and quadruple its mass"

Lets call this plant B gives the following equivalent values.

                                g  = 9.82 m/s^2
             escape velocity  =  15.82 x10^3 m/s
                  time dilation  =  + 1.39 x10^-9 sec/sec

As can be seen the identical value of g for Earth and planet B at their surface, but a different value for escape velocity and therefore gravitational time dilation.

 

[Janus]

Hi all,

Thank you Janus for that response, it sits better with me than the previous analogy of a gravity field not following the inverse square law.

If I may borrow part of your explanation to respond, and explain why/how I understand your statement is correct :

Janus
 

First off, the time dilation is not due to a difference in the strength of gravity, but to a difference in gravitational potential.

So firstly if you calculate the strength of g and escape velocity and time dilation at the Earths surface gives

                                    g  = 9.82 m/s^2
             escape velocity  =  11.19 x10^3 m/s
                  time dilation  =  + 6.97 x10^-10 sec/sec

Then to borrow your example:
 "Now double the Earth's radius and quadruple its mass"

Lets call this plant B gives the following equivalent values.

                                g  = 9.82 m/s^2
             escape velocity  =  15.82 x10^3 m/s
                  time dilation  =  + 1.39 x10^-9 sec/sec

As can be seen the identical value of g for Earth and planet B at their surface, but a different value for escape velocity and therefore gravitational time dilation.

 

Because escape velocity is a function of gravitational potential. Specifically, the difference between the gravitational potential at the point you start and a point an infinite distance away.
If we take the planet Uranus for example, its "surface" gravity is 0.889g, less than Earth's. Yet the escape velocity from that same surface is 21.3 km/s which is nearly double that of the Earth's.  Thus, just considering the gravity of each planet alone( factoring out the effect of their different orbits around the Sun), a clock on the surface of Uranus would run slower than one on the Earth, despite the fact that the surface gravity of Uranus is weaker.

[gem]

HI all,

Ok to have a stab at putting an approx value to the original question for gravitational time dilation due only to their gravity well.

using the analogy of a rock split in half from the surface of the earth and each half remained on the surface of their respective bodies, from an impact event occurring 4.51 billion years ago

Earth time dilation = 6.965 x 10^-10 sec/sec

Moon time dilation = 3.15 x 10^-11 sec/sec

I get the Earth to be 1095 days younger than the moon which may seem like a lot but is only

                                            66.51 x 10^-9 % different

I am happy to be corrected if anyone else wants to chip in.
 

[Eternal Student]

Hi.

I am happy to be corrected if anyone else wants to chip in.

   I haven't done the calculations. Actually putting the numbers into a calculator isn't fun.   However, it looks about right as an estimate.   Well done.

Earth time dilation = 6.965 x 10^-10 sec/sec

    I'm not sure if there is just one universally agreed upon way of describing time dilation.  Since there isn't a well established way of describing time dilation it's worth taking a line of text to say what you've calculated there instead of just labelling it "Time dilation".
   You seem to have calculated the additional amount of co-ordinate time (in seconds) that would have passed for each 1 second of the local time that passes.   So your values are positive and just above 0.   Let's call this  quantity δt.
   
    One common alternative way of describing time dilation is as follows:  Time dilation can be expressed as the ratio of the number of seconds that pass in the local time to the number of seconds that pass in the co-ordinate time.  Equivalently, it is the number of seconds that pass in the local time for each 1 second of co-ordinate time that passes.   So this gives a value that is just under 1.     This quantity is    1 / (1+δt).

Best Wishes.

[evan_au]

the escape velocity from (Uranus) is 21.3 km/s which is nearly double that of the Earth's

As Einstein said, it's all relative.

If you take into account the Sun's mass, Earth is deep in the Sun's gravitational well, compared to Uranus.
- Escape velocity from the Sun (at Earth's distance): 42.1 km/second
- Escape velocity from the Sun (at Uranus' distance): 9.6 km/second
- The Sun's massive gravitational potential well dwarfs the Earth's gravity potential

So it looks to me like a clock on the surface of Uranus would tick faster than an identical clock on the surface of the Earth?
https://en.wikipedia.org/wiki/Escape_velocity#List_of_escape_velocities

[Eternal Student]

Hi.

If you take into account the Sun's mass, Earth is deep in the Sun's gravitational well, compared to Uranus.

    Yes seems reasonable.   There's also different speeds of orbit around the sun and time dilation from special relativity you can throw in to the mix.
   
   I've seen a reference text for time dilations on every planet in the solar system, taking account of gravity from all major sources in the solar system,  plus orbit speed around the sun and also the rotation of the planet around its own axis.  I can't find that now with a quick Google search - but someone has done all the maths.   If I recall correctly, proximity to the sun is sufficient to put the planets in the right order of time dilation that would apply on their surface from Mercury to Pluto - their own individual mass, radius, rotation and orbit speed makes just a small contribution.   A planet is a mole-hill in the area, everything is really on the slopes of Mount Everest anyway.

Best Wishes.

*LATE EDITING:   I don't recall correctly:  See post #16 from Halc for more complete info.

[gem]

Hi all,

I think we maybe wondering a little from the original question, however given eternal student's last comment, it maybe relevant to flag up you have to be careful with your comparisons/analogy's


ES

A planet is a mole-hill in the area, everything is really on the slopes of Mount Everest anyway.

     

I believe using the word slope in this context is misleading and will bring you back to the same issue Janus raised regarding gravitational time dilation error in the wording of the original question.

Given the slope of the well tells you how hard the pull of gravity is at that point.

The depth of the well tells you how much energy it takes to escape.

https://en.wikipedia.org/wiki/Gravitational_potential#/media/File:GravityPotential.jpg

 

[Eternal Student]

Hi.

I believe using the word slope in this context is misleading

   Fair enough.  It's happening "in the shadow of Mount Everest".  (Hmmm... I'm not strong on poetry either).

I think we maybe wondering a little from the original question

    The original question was answered in a podcast by the nakedscientists nearly a week ago.
https://www.thenakedscientists.com/podcasts/question-week/does-relativity-affect-age-moon-rocks

   The person who asked the question doesn't seem to have joined the forum discussion either.   So I wouldn't have thought there's not much danger of hijacking the Original Post or causing any problem to the OP. 
    I don't own the website but you (gem) can probably move the thread in any direction without causing a problem.

Best Wishes.

[Halc]

If I recall correctly, proximity to the sun is sufficient to put the planets in the right order of time dilation that would apply on their surface from Mercury to Pluto - their own individual mass, radius, rotation and orbit speed makes just a small contribution.

Rotation speed is completely insignificant compared to orbital speed, and even that makes terribly little difference compared to the varying potentials.

Sorted by slowest to fastest clocks on the surface of each planet, the list is:
Mercury, Jupiter, Venus, Earth, Saturn, Mars, Neptune, Uranus, Pluto

The standard reference for this data, including many of the moons, is Randall Munroe who yes, did all the mathematics for xkcd