# Time dialation formula

## Main Question or Discussion Point

could someone tell me the formula to figure out how much relative time speed difference there is between two or more objects of unequal mass?

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HallsofIvy
Homework Helper
If you are referring to the Lorenz contraction of time with speed, it has nothing to do with mass.

LURCH
Originally posted by HallsofIvy
If you are referring to the Lorenz contraction of time with speed, it has nothing to do with mass.
Perhaps gravitational time dilation?

yes, gravitational time dilation. since an object of mass has gravity relative to its mass. more mass, more gravity.

years ago i made my own formula for this. A simple one dealing with just multiplication, division, and exponents. But it was based on an inacurate measurement of time dilation, and was only applicable to the Earths gravity, which was represented as a constant variable. I just did it out of interest. It was kinda cool cause it could tell you how much relative time difference there would be per inch per second from sea level.

so whats the real formula that I could apply to any object(s) of any mass(es)?

Janus
Staff Emeritus
Gold Member
The gravitational time dilation formula is

$$T= \frac {T_{0}}{\sqrt{1- \frac{2GM}{Rc^{2}}}}$$

R is the distance from the center of the object, or the radius of the object if you are considering a point on the surface of a spherical body.

T is time as measured from a point sufficiently removed from the gravity field of the object. (I.E. At an infinite distance form the object. )

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Originally posted by Janus
The gravitational time dilation formula is

$$T= \frac {T_{0}}{\sqrt{1- \frac{2GM}{Rc^{2}}}}$$

R is the distance from the center of the object, or the radius of the object if you are considering a point on the surface of a spherical body.

T is time as measured from a point sufficiently removed from the gravity field of the object. (I.E. At an infinite distance form the object. )
so,
T = the relative time elapsed from the center of a massive object to an observer at a distance from the massive object?
To = the time on the observers watch?
G = Gravity? how do you calculate this?
M = mass? how do you calculate this?
C = light speed? I would assume......

does this formula only work if the observer is stationary relative to the massive object? what if the observer is maintaining a constant distance, but is orbiting at a speed? what if the observer is falling towards the object?
someone please explain this further, i never had an interest in math untill recently

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Janus
Staff Emeritus
Gold Member
Originally posted by mikesvenson
so,
T = the relative time elapsed from the center of a massive object to an observer at a distance from the massive object?
To = the time on the observers watch?
G = Gravity? how do you calculate this?
M = mass? how do you calculate this?
C = light speed? I would assume......

does this formula only work if the observer is stationary relative to the massive object? what if the observer is maintaining a constant distance, but is orbiting at a speed? what if the observer is falling towards the object?
someone please explain this further, i never had an interest in math untill recently
First off, please note that I have edited my response, to correct a typo. It is T that represents the time on the observers watch.

G is the Gravitational constant. Its value is aprox 0.00000000006673

M is the mass of the body, which you have to already know, or be able to measure.

R is the distance from the center of the mass the clock you are comparing the observer's clock is.

c is the speed of light in a vacuum.

So for a for a clock sitting on the surface of the Earth:
M = the mass of the Earth = 6 x 1024 kg

and

R = the radius of the Earth = 6378000 m

Plugging these numbers in will give you how much slower a clock on the surface of the Earth runs than our distantly removed observers.

If you raise the clock to a higher altitude, you increase R by the correct amount and recalculate.

For an circularly orbiting object you would use the gravitaional time dilation formula for its height, but then also factor in the SR time dilation due to its velocity.

If the object was falling straight down, then its time dilation would be changing from moment to moment, since both its velocity and distance from the center of the mass would be changing constantly.

David
Originally posted by mikesvenson

This is not true overall “time dilation”. This is merely the rate an atomic clock will slow down in different gravitational potentials. This is based on a slow-down in the internal oscillation rates of the atoms. An atom existing in stronger gravity or accelerating will have a slower internal oscillation rate, but that rate doesn’t necessarily represent all of “time”, since time is also determined by other local factors, such as thermodynamics and mechanical motions in the same area where the atom is located. For example, a large oscillating mass, such as the bob on a pendulum clock, will oscillate faster in a strong gravity field, while an atom will oscillate internally more slowly. This is the difference between large-scale Newtonian mechanics and small-scale quantum mechanics. In addition, there is thermodynamic time, as seen in the external vibration rates of molecules and atoms. This kind of vibration rate and time often speeds up at the same places where internal atomic oscillation rates of atoms slows down, such as at the surface of a star.

Originally posted by David
For example, a large oscillating mass, such as the bob on a pendulum clock, will oscillate faster in a strong gravity field, while an atom will oscillate internally more slowly.
does this mean that in a strong gravity field, a pendulum clock will tick faster and an atomic clock will tick slower?

David
Originally posted by mikesvenson
does this mean that in a strong gravity field, a pendulum clock will tick faster and an atomic clock will tick slower?
Yes. This was known about pendulum clocks 400 years ago. It was not known about atomic clocks slowing down in a gravity field until Einstein deduced it in 1911. However, Lorentz, in 1895, deduced that atomic clocks would slow down when they moved rapidly through fields. So, now, with the combined theories of Einstein and Lorentz, we know that atomic clocks slow down in a gravity field and they slow down due to acceleration and when they move rapidly through fields.

This is basically “quantum mechanics”, the inner workings of atoms on the very small scale.

The speed-up of pendulum clocks in a strong gravity field works on a different principle. It is a large-scale phenomenon, so different laws govern the workings and tick rates of pendulum clocks.

It is generally only physicists and astronomers who think of “atomic time” as being “true time”. But if you go to Google and type in [biology “thermodynamic time”] you will see that biologists generally deal with thermodynamic time, i.e. heat energy time, rather than atomic time.

russ_watters
Mentor
Originally posted by David
This is not true overall “time dilation”. This is merely the rate an atomic clock will slow down in different gravitational potentials.
I maybe haven't been paying enough attention to your other posts (maybe you've discussed this before), but are you saying that an atomic clock sees its rate change for the same reason as a pendulum, ie. simply a matter of mechanical force and not real time dilation? If so, how do you reconcile that with the fact that the GPS system works and is based on Einsteinian time dilation, not mechanical clock rate issues?

Also, not all oscillations are affected by gravity. A spring-mass system perpendicular to a gravitational force is not affected by the strength of the force.

DW
Originally posted by mikesvenson
could someone tell me the formula to figure out how much relative time speed difference there is between two or more objects of unequal mass?
Time dilation in general relativity is a bit more complicated than an equation that can be fully explained in a brief posting to a message board. The cliff note version of the answer is that the invariant interval or line element ds in $$ds^2 = g_{\mu}_{\nu}dx^{\mu}dx^{\nu}$$ is the length of time $$(ds = dc\tau)$$ that goes by for something following that path. Your coordinate time is in the right hand side of the equation as $$dct = dx^0$$ and that will give you a differential equaion relating the times that is valid even if a real force is applied so that the motion is not geodesic. The metric $$g_{\mu}_{\nu}$$ for arbitrary numbers of gravitational sources is related by second order nonlinear differential equations to those sorces through Einstein's field equations. Finding exact solutions isn't always feasible so usually a linearized weak field approximation is made in which case one can simply input the Newtonian gravitational potential into places in the metric and get an approximate answer. If the motion is geodesic, then one may also refer to the equation of geodesic motion which will in some few cases yield the result strait away or eliminate coordinate variables from the expression for the line element.

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HOLY SH*T!!!!!!!!!!!

DID YOU BREATH AT ALL WHILE YOU TYPED THAT???????????
THIS IS SOOOOOOOOOO OVER MY HEAD I DONT EVEN KNOW !!!!!!!!!!
I THOUGHT I HAD THIS DOG ON A LEASH BUT IT GOT AWAY!!!!!!!!!
WHEW!!!!!
TIME FOR ME TO GO BACK TO COLLAGE AND MAJOR IN PHYSICS SO I CAN COMPREHEND WHAT YOU ARE TALKING ABOUT!!!!!!!!PEACE!

David
Originally posted by russ_watters
I maybe haven't been paying enough attention to your other posts (maybe you've discussed this before), but are you saying that an atomic clock sees its rate change for the same reason as a pendulum, ie. simply a matter of mechanical force and not real time dilation? If so, how do you reconcile that with the fact that the GPS system works and is based on Einsteinian time dilation, not mechanical clock rate issues?
Well, Russ, it’s kind of complicated and difficult to explain briefly. First, it was H.A. Lorentz, not Einstein, who first invented the concept of “atomic” time dilation. The concept of “atomic time” first came from Maxwell in 1873. You can find it mentioned on page 3 of the first volume of his famous “Treatise on Electricity and Magnetism”. Atomic time is based on the frequency of light given off by an oscillating atom, and the frequency of that light is determined by the oscillation frequency of the atom. Lorentz hypothesized in his 1895 book, “Versuch Einer Theorie Der Elektrischen Und Optischen Erscheninungen In Bewegten Körpen,” that the oscillation frequency of an atom could slow down if the atom moved through certain fields and if it was subjected to acceleration. In 1911 Einstein deduced that motion-related acceleration was quite similar to gravitational acceleration, so he extended Lorentz theory to include a slow-down in the oscillation rates of atoms that existed in a gravity field. We can’t say “atoms resting in a gravity field” since atoms don’t “rest”, they move around rather quickly because of heat energy, i.e. molecular vibrations.

Ok, so, since atomic oscillation rates tend to be very steady, if a collection of atoms are not moving rapidly through fields, not changing RMS acceleration rates due to temperature changes, and not changing altitudes or gravitational potentials, atomic clocks eventually became our “time base standard” for measuring very accurate and short time durations.

The work of both Lorentz and Einstein had already predicted that moving atomic clocks and atomic clocks that change altitudes would change their oscillation rates, their “tick rates”, so this was no surprise when the GPS satellites were first sent up. But there are also additional factors that contribute to atomic oscillation rate changes, such as temperature changes within the collection of atoms that are being monitored inside the clock. This is because higher temperatures cause higher RMS speeds of the bouncing atoms, and this causes greater accelerations when the bouncing atoms change directions. Also, they tend to change their rates when they move through magnetic and electric fields, and thus they have to be shielded as much as possible from these fields. But they can’t be shielded from gravity fields.

An atomic oscillation rate does not change for the same reason a pendulum bob changes its swing rate. That’s because there are different laws of physics at work in both kinds of clocks. A pendulum clock is more of a “macro-sized” object and its bob tends to obey the “macro-sized” laws of Newtonian mechanics. But an atom is micro-sized and, internally, works under slightly different laws. This is because there are tiny little fields inside atoms, and the particles of the atoms have to deal with and react to these fields constantly. So, this is where the laws of “quantum mechanics” come into play. A pendulum bob doesn’t have to deal with the laws of quantum mechanics, since it is a large object that is made up of billions of atoms, and small weak electric and magnetic fields generally don’t affect its swing rate, whereas deep inside the bob, the small fields do affect the oscillation rates of the atoms that make up the bob.

For example, a hot bob will generally not change its swing rate, but the hot atoms will change their internal harmonic oscillation rates inside the bob. So, if we could put some kind of “atomic probe” into the bob, we would see that the “atomic clocks” (the atoms) inside the bob speed up their internal oscillation rates when the bob is hot, while the swing rate of the bob is not affected by the temperature.

It took a long time for me to learn that there are more kinds of “time” than just one, and that “time” is tied in different ways to different kinds of motion and vibration rates of physical matter.

If you were flying in a GPS satellite, all you, as a biological being, would notice is the lack of a gravitational “pull” on your body, and you would notice, from messages sent to you from earth, that earth-based atomic clocks are ticking a little more slowly than your GPS sat clock. But you would feel no difference in “time” since your satellite would be nice and warm inside, so your body temperature would not go below 98.6 degrees F. Your own biological time is based on thermodynamic time, not atomic time.

Your brain is designed to go unconscious if your body temperature drops by 5 to 10 degrees, so you would go unconscious before you began to notice any biological “time” rate difference as a result of the lowering of your brain temperature.

There is nothing amazing about atomic oscillation rate slow-downs and speed-ups. This is just a normal function of nature. All kinds of clocks slow down and speed up for a variety of physical reasons, but no single kind of clock rate change represents a total “time” rate change at that clock. This concept is an old misconception that is based on Newton’s old definition of “absolute time”. But an oscillating atom no more represents all of “absolute time”, any more than a pendulum clock or a mechanical balance-wheel clock does. These are different kinds of click that operate by different laws of physics, and they tick out and measure different types of time. Some can slow down while other speed up. Some physical parts of different kinds of clocks can change "time" or "aging rates" while other prarts of the same clock do not change rates.

ok, this is a heck of a lot more easier to understand than what DW said

russ_watters
Mentor
Originally posted by mikesvenson
HOLY SH*T!!!!!!!!!!!

THIS IS SOOOOOOOOOO OVER MY HEAD I DONT EVEN KNOW !!!!!!!!!!
I THOUGHT I HAD THIS DOG ON A LEASH BUT IT GOT AWAY!!!!!!!
Easy boy [woof], its not that bad, just take it slow. If you're really interested, pick up a laymans' book on the subject. You'll get it.
If you were flying in a GPS satellite, all you, as a biological being, would notice is the lack of a gravitational “pull” on your body, and you would notice, from messages sent to you from earth, that earth-based atomic clocks are ticking a little more slowly than your GPS sat clock. But you would feel no difference in “time” since your satellite would be nice and warm inside, so your body temperature would not go below 98.6 degrees F. Your own biological time is based on thermodynamic time, not atomic time.
This is the part I was looking for: it is not correct. Yes, there are different ways of defining/measuring time, but there is one that affects all the others: relativistic time. The effects on atomic clocks are not simply mechanical clock rate effects they are manifestations of the rate of time passage itself changing.

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David
Originally posted by russ_watters
Yes, there are different ways of defining/measuring time, but there is one that affects all the others: relativistic time. The effects on atomic clocks are not simply mechanical clock rate effects they are manifestations of the rate of time passage itself changing.
No, that is not correct. That is a common myth that was probably started by Lorentz in his 1895 book, and it was continued by Einstein in his 1905 SR version of the 1895 Lorentz theory. It was Lorentz who invented time dilation, slow atomic clock “tick” rates, mass increase, two relatively moving “systems”, length contraction, the Lorentz Transformation, the S1 and S2 systems, and the speed limit of “c”, but Lorentz’s theory involved fields, acceleration, atomic clocks, and real forces of nature.

The idea that an “atomic clock” might be a “true time” clock can be traced back to Maxwell’s 1873 book and his definition of “time”, using an oscillating atom as an example of a perfect “clock”. When Lorentz, in the late 19th Century, theorized that atomic vibration rates would slow down under certain conditions, and when Einstein began to talk about time dilation, then it became common in the field of physics and astronomy to think of atomic time as “true” time. But, gradually, Einstein drifted away from that point of view, somewhat. However, “atomic time” became the time-base standard for physics, but thermodynamic time (“heat time”) gradually became the time-base standard for biology and other fields. Both are equally valid, but only for their particular kinds of time. On earth, and in fact in most of the universe, thermodynamic time usually overrides atomic time and is far more important and more representative of “true time” than atomic time is.

russ_watters
Mentor
Where in the writings of Einstein and Lorenz do they specify that their theories apply only to atomic clocks?

David
Originally posted by russ_watters
Where in the writings of Einstein and Lorenz do they specify that their theories apply only to atomic clocks?

See Lorentz’s 1895 “Versuch Einer Theorie Der Elektrischen Und Optischen Erscheinungen In Bewegten Körpen.” See page 38: “Nimmt man nun diese Vertheilung für das System S2 und leitet daraus durch die oben besprochen Transformation ein System S1 ab, so besteht auch in diesem ein Ueberschuss positiver Ionen nur an einer gewissen Oberfläche E, während in allen inneren Punkten die electrische Kraft x verschwindet.”

See his chapter titled, “Abschnitt III, UNTERSUCHUNG DER SCHWINGUNGEN, WELCHE VON OSCILLIRENDEN IONEN ERREGT WERDEN,” starting on page 48. This is where he introduced atomic time dilation in 1895.

You can find Einstein’s introduction of atomic clocks in his 1911 paper. That’s when he switched over from mechanical to specifically atomic clocks.

See also Charles Steinmetz’s “Four Lectures on Relativity and Space,” 1923, Dover 1967 Edition, page 67, where he explains Einstein’s use of atomic clocks. See also Maxwell’s 1873 definition of an atomic clock in Volume 1 of his “Treatise on Electricity and Magnetism”. Also See Einstein’s 1918 paper, “Dialog über Einwände gegen die Relativitätstheorie,” from Die Naturwissenschaften 6 (1918). In this paper, he adds atomic clocks, gravity fields, and acceleration to the 1905 SR theory, as the result of his 1911 discovery.

Originally posted by russ_watters
... The effects on atomic clocks are not simply mechanical clock rate effects they are manifestations of the rate of time passage itself changing. [/B]
Not so Russ; in dealing with relativistic time dilation effects we talk about clock RATES. There is no reference to the absolute nature of time being changed, only the clock 'rates'.
Nowhere is this more evident than in atomic clocks.

For ex., in cesium clocks, we 'define' one second as 9,192,931,770 oscillations of the cesium133 atom (hyperfine transition). When the clock is carried into a relativistic situation it is the the number of oscillations compared to a clock at rest that changes.
It says nothing about the absolute value of the passage of time.

Creator

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hhmmmm interesting battle......
i wonder...
has anyone ever been able to actually DEFINE what time IS?
When we regard relativistic clock rates for a number of differences, we are only explaining the EFFECTS of time, not actually defining it. Oscillation rates are the EFFECTS of time.
We KNOW the effects of a car.
We also KNOW exactly what a car IS. Therefore we are able to manipulate the car as we choose.
If we could KNOW exactly what time IS, then maybe we could easily control it, or even move throughout it.

russ_watters
Mentor
Nice, David - post your reply in German. Sorry, mine's a little rusty, but I don't see the words "cesium" or "rubidium" in there. Surely, the clock rate errors for different types of clocks must be different and the theory must specify which it applies to....
When the clock is carried into a relativistic situation it is the the number of oscillations compared to a clock at rest that changes.
That is true, Creator, but the reason the rate of that clock changes is that time itself flows at a different rate.

David and Creator, I have no real interest in arguing this as it really isn't arguable. Modern scientists accept that Einstein's Relativity is discussing the rate of the passage of time itself. Much of our modern technology uses these theories and that is how the engineers use them. If you ask a GPS engineer, he will tell you that the reason GPS satellite clock rates are calibrated down from normal prior to launch is the combined effects of SR and GR on the rate of the passage of time for an object in orbit.
has anyone ever been able to actually DEFINE what time IS?
Certainly: http://dictionary.reference.com/search?q=time

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Einstein began to talk about time dilation, then it became common in the field of physics and astronomy to think of atomic time as “true” time. But, gradually, Einstein drifted away from that point of view, somewhat. However, “atomic time” became the time-base standard for physics, but thermodynamic time (“heat time”) gradually became the time-base standard for biology and other fields. Both are equally valid, but only for their particular kinds of time. On earth, and in fact in most of the universe, thermodynamic time usually overrides atomic time and is far more important and more representative of “true time” than atomic time is.

By believing in different sorts of time (atomic, thermodynamic), you rejected the fundaments of GR and physics:

The Equivalence Principle

You go on a long trip to a near-by star taking the Rollex (read non atomic clock with you and also a atomic clock. Your spaceship, you will notice, has no windows (they had to cut the budget somewhere!), but you go anyway. You experience the effects of lift-off but after a while you appear to be at a standstill: you are then moving at a constant speed with respect to Earth. But remember we assumed that the Rollex still ticks the same way as the clocks on Earth, and sinds Einstein that the atomic clock does not. So you will see a mismatch between the Rollex and the atomic clock: this is an experiment which is done completely inside the spaceship and which determines whether you are moving. If there were such a Rollex the Principle of Relativity would be violated.
Free quoted from:
http://phyun5.ucr.edu/~wudka/Physics7/Notes_www/node78.html#SECTION03123000000000000000 [Broken]

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Originally posted by russ_watters

A useful Google: GPS and Relativity
Wow, i had a good time reading the page about relativity and GPS! It stated that at the equator acceloration and gravity dilation cancel out each other. To my pleasure, I ironically concluded yesterday that this must be a possibility regarding a hypothetical "electrical pole" that was so tall it reached out far into space. I thought that maybe since the Gravity dilation would predict that time for the pole (relative to that of the Earth) would speed up as the pole reached farther, and Acceleration dilation would predict that time for the pole (relative to that of the Earth) would slow down, that the 2 differences in relative time for the pole might actually cancel out each other and time for the pole would relatively(to the surface of the Earth) be absolute.

Athough it started to confuse me when it stated that when a traveler left the Earth at 8/1994 at the speed of .99c headed for Alpha Centauri, the (GPS)time at Alpha Centauri upon the travelers arrival would be 9/1998, the natural time for the traveler would be an ellapsed 7 months, and the syncronized Earth watch(impossible throuth SR) would only have read an ellapsed 1 month! What the heck do they mean by "natural time" Wouldnt this be the same thing as the 1 month time? This doesnt make sense to me. Maybe I dont fully understand the way in which they present this, but I would assume that only 1 month (Earth time) would have passed for the traveler upon arrival at Alpha Centauri. Then it stated that upon return to Earth, the GPS time would be 10/2002!! This even further confused me since it only took a relative (to Earth) 1 month to get back!!

I fully understand the concept that if you travel at .99c for an extended time relative to Earth that upon your return to Earth you will actually be re-entering into the future time of Earth (relative to the elapsed time you experianced while out in space traveling at .99c). It may be hard to tell what my point is in the previouse paragraph, but maybe the confusion came from the way in which it was written on the site.

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David
Originally posted by russ_watters
but I don't see the words "cesium" or "rubidium" in there
Maxwell mentioned “sodium” in his 1873 comments.