How Does Einstein's Theory Explain Time Dilation in Gravitational Fields?

[Bjarne]

What brought Einstein to the conclusion that time goes slower the deeper into a gravitational field one get.

And has that ever been proven that time goes slower.

Sorry if I am not writing perfect English.

KR
Bjarne

 

[tonyb1969]

What brought Einstein to the conclusion that time goes slower the deeper into a gravitational field one get.

And has that ever been proven that time goes slower.

Sorry if I am not writing perfect English.

KR
Bjarne

Not sure if I can relate this directly to a gravitational field; however, the theory is that of the four known dimensions (3 of space, east-west/north-south/up-down (or x/y/z), for example and 1 of time) that all objects in the universe are moving along one or more of these dimensions at the speed of light. If an object, like a photon, is moving in one direction (say east-west or the x-axis), and by definition is moving at the speed of light, there is no "time" left for it to traverse the time dimension/axis, and wouldn't age; therefore, a photon emitted billions of years ago is exactly the same age as it was when it was emitted. Moving a little slower--perhaps 99% speed of light with the other 1% ticking along the time dimension, then time would be drastically reduced relative to what we experience.

So I would assert that being pulled deeper into a gravitational field means the object is moving more quickly, and, hence, less time would elapse (because less "speed of light" is left over for the time dimension/axis). This phenomenon could occur since we get a little closer to the sun and speed up slightly, but the change is very minute, and since we are all on earth, the relative time for all of our calculations would not show any difference.

Very sophomoric response with no calculations, but I hope it helps.

 

[Bjarne]

Tonyb1969

So I would assert that being pulled deeper into a gravitational field means the object is moving more quickly

Yes a object would move faster, but not EM valves.
I do not see the point

When an object travels with greater speed, then yes both time and distance change, this has been proven by time measurement. – (This has nothing with the question to do)
Also gravitational red shift has been proven. (This also has nothing with the question to do)
But that time goes slower deeper inside a gravitational field have that really been proven?

According to my understanding it should not have anything with speed to do, but only where a watch is situated (where time is measured) for instance by the event horizon (by a black hole), - time “should be” zero. No one has off course measured it there. So where and when have this claim been confirmed.

For instance a watch at first floor “should” go slower than a watch at 100 th. floor? – Really ?

Which kind of logic are we here talking about?
What brought Einstein to such strange concluding?

KR
Bjarne

 

[A.T.]

What brought Einstein to the conclusion that time goes slower the deeper into a gravitational field one get.

He tried to bring together two things.
- Light bending at big masses, as the sun
- Constant velocity of light for every local observer
In a bend light ray the photons on the inner side of the bend, have less distance to travel. So time has to run slower there, to keep the local speed of light constant.

And has that ever been proven that time goes slower.

http://en.wikipedia.org/wiki/Gravitational_time_dilation#Experimental_confirmation

 

[Jonathan Scott]

Firstly, be very wary about the Wikipedia entry on Gravitational Time Dilation, as it contains at least one major error and is somewhat confusing.

Basically, relative time dilation in GR is closely related to potential in Newtonian gravity, and gives the potential energy difference per amount of rest energy.

For example, if an object in Newtonian gravity would change in potential energy by mgh when lifted through height h in a field g, then in GR its time rate will change by a fraction mgh/mc², which is equal to gh/c². This means that its clock rate would change to (1+gh/c²) of the original clock rate. The change in time rate has the effect of giving the same change in energy relative to the original location as occurs with potential energy in Newtonian theory.

When the change in height is sufficient that the field cannot be considered constant, more general Newtonian expressions for the potential can be used, such as -Gm(1/r₂-1/r₁) for the difference in potential at two distances from a central object.

This approximation holds provided that the potential energy is small compared with the rest energy, which certainly applies everywhere in the Solar System.

 

[Naty1]

What brought Einstein to the conclusion that time goes slower the deeper into a gravitational field one get.

Great question...I have been wondering for a long time how Einstein made many of his conjectures for general relativity...how did he get so much crammed into his "einstein tensor" and how did he pick from among different possibilities...

I have some hints, a few tidbits, but not an answer: He theorized a number of mathematical formulations and ended up eliminating several when he "discovered" his equivalence principle; the math he used was based on the work and insights of others: Lorentz and Fitzgerald for time dilation and length contraction in special relativity, Ricci and Weyle Tensors and Riemann geometry for GR, and I believe Minkowski for four dimensional spacetime considerations after GR was initially formulated. And his hypothesis for the bending of light (energy) was NOT accepted until the Eddington eclipse experiments (maybe 1916?) confirmed it...and eliminated a competing theory from another physicist which did NOT predict bending of light.

I have read that his original notes are available,have been studied and that he made many,many mistakes, drew incorrect inferences, made math errors...but he always ended up correcting them...but that took from about 1905 to 1915 or so...it was a masterpiece but not an obvious nor quick undertaking! And the cosmological constant is a well known example of uncertainty in his theorizing...

But the clues he might have come across have not been revealed in the references I've used...

 

[Al68]

What brought Einstein to the conclusion that time goes slower the deeper into a gravitational field one get.

And has that ever been proven that time goes slower.

Sorry if I am not writing perfect English.

KR
Bjarne

The simple answer is the equivalence principle. Since he calculated with SR that two clocks at rest relative to an accelerated reference frame will run at different rates, ie, a clock at the rear of a spaceship will run slow relative to a clock at the front, and that according to the equivalence principle, it doesn't matter if the ship is accelerating at 1 G in deep space, or 10 feet off the ground (stationary with earth) at Earth's surface, or just sitting on the launchpad. It's 1 G acceleration either way. The "lower" clock will run slow relative to the "upper" clock either way.

Another way to look at it is, according to an inertial observer (in freefall) the Earth's surface is accelerating upward at 1 G, and from the inertial frame, a clock at rest with Earth's surface will be running slow, and a lower clock at rest with Earth's surface will be running even slower, since relative to the inertial frame, the lower clock has a higher velocity.

Al

 

[rcgldr]

What brought Einstein to the conclusion that time goes slower the deeper into a gravitational field one get. Has that ever been proven that time goes slower.

It is known that the clocks in the GPS satellites run faster in space than equivalent clocks at sea level on earth, and a planned for correction was made to compensate for this effect on GPS satellites.

The simple answer is the equivalence principle. Since he calculated with SR that two clocks at rest relative to an accelerated reference frame will run at different rates, ie, a clock at the rear of a spaceship will run slow relative to a clock at the front

This differs from what I have read elsewhere. The equivalence princple makes a comparson between an imaginary constant strength gravitational field (such as one generated by a infinitely large plane), and acceleration. There is no correlation between a varying strength gravitational field and acceleration. The rate of decrease versus distance for a graviational field depends on the distance from the source of the gravitational field.

gravity and speed of light

My own question, to an observer in a space does a light beam in a very strong gravtational field appear to move at a different speed than a light beam in a very weak gravitational field? It's already known that an observer can see that 2 otherwise identical clocks run at different rates depending on the strength of the gravitational field they operate in. Would light based clocks operate any differently?

 

[Jonathan Scott]

This differs from what I have read elsewhere. The equivalence princple makes a comparson between an imaginary constant strength gravitational field (such as one generated by a infinitely large plane), and acceleration. There is no correlation between a varying strength gravitational field and acceleration. The rate of decrease versus distance for a graviational field depends on the distance from the source of the gravitational field.

The equivalence principle applies over any region where the direction and strength of the field is sufficiently uniform not to affect the results. This certainly works well enough for the clock rate to vary in the same way in a gravitational field and an accelerating frame of reference. Basically, by Special Relativity, the clock rates at front and back of an accelerating spaceship vary, because the definition of simultaneity between the front and back is changing with velocity, so for example an object dropped at the front of the spaceship appears to have potential energy which is converted to kinetic energy (relative to the spaceship) as it falls towards the back. By the principle of equivalence, the same effect is expected in a gravitational field.

My own question, to an observer in a space does a light beam in a very strong gravtational field appear to move at a different speed than a light beam in a very weak gravitational field? It's already known that an observer can see that 2 otherwise identical clocks run at different rates depending on the strength of the gravitational field they operate in. Would light based clocks operate any differently?

The strength of a gravitational field does not affect the speed of light or the rate of clocks locally. The gravitational field is the gradient of the potential, and it is the relative potential which affects the rate of clocks and the apparent size of rulers, regardless of the field or acceleration.

 

[atyy]

And has that ever been proven that time goes slower.

Apart from its routine use in GPS nowadays that Jeff Reid mentioned, gravitational time dilation is related to gravitational red shift observed in the Pound-Rebka experiment.

My own question, to an observer in a space does a light beam in a very strong gravtational field appear to move at a different speed than a light beam in a very weak gravitational field? It's already known that an observer can see that 2 otherwise identical clocks run at different rates depending on the strength of the gravitational field they operate in. Would light based clocks operate any differently?

Gravitational red shift as observed by Pound and Rebka is a light based clock, so it would be the same. The way general relativity is set up, every little piece of spacetime is sort of flat, like in special relativity, so there are preferred coordinates and the speed of light is always the same. Curved spacetime is built by smoothly pasting together lots of these little pieces, which is why Carlip and Gibbs say "Finally, we come to the conclusion that the speed of light is not only observed to be constant; in the light of well tested theories of physics, it does not even make any sense to say that it varies." Over large regions of space, there are no preferred coordinates, and the speed of light depends on the coordinates you choose.

http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html

 

[Bjarne]

According to cosmologically redshift, we know that space expand.
According to gravitational redshift (Pound-Rebka experiment) space around the Earth must also either expand or contract, - right ?
1.) But how must we understand expanded or contracted space?
2.) What happens with a 1 meter stock when space expand let's say 100%, will it now be 2 meter long ? - I mean is distance in this case relative?
3.) What happens with the 1 meter stick when it moves towards the sun, - will it become relative shorter ? – (But still 1 meter for those that follow the stick and watching it).

 

[atyy]

According to cosmologically redshift, we know that space expand.
According to gravitational redshift (Pound-Rebka experiment) space around the Earth must also either expand or contract, - right ?

Usually not, but maybe, maybe not. Expanding space in cosmological redshift is a coordinate dependent description, and the usual non-expanding space around the Earth is also a coordinate dependent description. So I'm not sure there are no sensible coordinates in which space around the Earth is not expanding.

1.) But how must we understand expanded or contracted space?
2.) What happens with a 1 meter stock when space expand let's say 100%, will it now be 2 meter long ? - I mean is distance in this case relative?
3.) What happens with the 1 meter stick when it moves towards the sun, - will it become relative shorter ? – (But still 1 meter for those that follow the stick and watching it).

Try sections 2.6.2 - 2.6.4 of:
Expanding Space: the Root of all Evil?
Matthew J. Francis, Luke A. Barnes, J. Berian James, Geraint F. Lewis
http://arxiv.org/abs/0707.0380

 

[Bjarne]

Atyy wrote:

Usually not, but maybe, maybe not

If "Usually not" – then we have a big problem with the theory of relativity.
Now, imaging a space probe 10 mia. km. from the Sun. Let say time, out there, goes double so fast than at earth.
If distance is the same out there, - than on earth, - the space probe would move double so long a distance, compared to the distance it would have close to earth.
Maybe we have no problem to accept that.

Let us imaging a car our there, - speed is 100 km/h
This car competes with a car on earth. Speed is the same 100 km/h.
The car “out there” will win the race. According to our time and our distance the space-car seems to have traveled double so fast than the car on earth.
BUT both cars drove only 100 km/h. something is wrong. (We are able to explain it, - time is different)
Let’s now have a new competition, this time with the speed of light. Again the space-car seems to be double so fast.
Now we have an even bigger problem because now not only speed seems to be the double that it really was, - BUT now speed seems to be double so fast than speed of light (!)

Compared to our time, light must now travel through an area (out there) with the double speed, not 300.000 Km/s (our time) but 600.000 km/s (our time).
Well still because time out there is double so fast than our time, - BUT this is impossible, this violate our realety - right?
Simple because according to our realety light must now travel throug an area with "double speed".

To make a long history short, are we forced to rethink that all distances is not what we think they are, they are all relative as well as time is.?

The conclusion seems therefore (to my opinion) to be pretty simple:

1.)When time (out there) goes faster, - let say double so fast than on Earth, then also relative distance must change proportional to: double distance.
A 1 meter stick (out there) is therefore compared to a1 meter stick on Earth = 2 meter.

2.)If we think we can use relative time, and our (not relative) distance, - Yes! - We will believe that Pioneer 10 and 11 “should” reach a longer distance than they really did.
Could the cause of the pioneer Anomaly be so unbelievable simple? - Have Einstein forgot to tell us: - that if time change due to gravity, then distance must also change?
Is it only our expectation to the distance the Pioneer 10 + 11 probes “should” have reached, - that is the real problem?

Sorry, - This is properly not perfect English, - hope you will get the point.
“Double speed” was only an example because of simplicity reasons.

-

 

[George Jones]

If "Usually not" – then we have a big problem with the theory of relativity. ... Is it only our expectation to the distance the Pioneer 10 + 11 probes “should” have reached, - that is the real problem?

Professional physicists know how to work with relativity. The parametrized-post-Newtonian (PPN) framework can be used to determine how well general relativity works within the confines of our solar system. The PPN framework shows that relativity works quite well; see Chaper 10 from Gravity: An Introduction to Einstein's General Relativity by James Hartle.

 

[Bjarne]

Can someone tell me the formula to use to:

1.) Calculate the time change in a field of gravity (due to gravity)
2.) Calculate a photons attraction (bend) towards a field of gravity

 

[atyy]

If "Usually not" – then we have a big problem with the theory of relativity.

Well, I was only thinking of a coordinate change which might allow some strange language for fun. But coordinate changes do not change a theory's physical predictions, so there cannot be any problem on this basis.

 

[atyy]

Can someone tell me the formula to use to:

1.) Calculate the time change in a field of gravity (due to gravity)
2.) Calculate a photons attraction (bend) towards a field of gravity

For a specific case, try section 14.3 about photon orbits or null geodesics of Woodhouse's notes: http://people.maths.ox.ac.uk/~nwoodh/gr/gr03.pdf

 

[Jonathan Scott]

Can someone tell me the formula to use to:

1.) Calculate the time change in a field of gravity (due to gravity)
2.) Calculate a photons attraction (bend) towards a field of gravity

Provided that this is only relating to weak gravitational fields (which certainly applies for anything in the solar system:

1.) The fractional relative change in the time of a clock does not depend on the field but on the potential. (In everyday terms, it doesn't relate to the acceleration but rather to the height). The change is equal to the Newtonian potential converted to units of potential energy per rest energy. For a central mass m, the Newtonian potential at distance r is -Gm/r in units of potential energy per mass or -Gm/rc² in units of potential energy per rest energy. This means that the fractional change in the time rate when moving from distance r₁ to distance r₂ is as follows:

Gm/c²(-1/r₂ + 1/r₁)

Another way of putting this is that the time rate changes by a factor of approximately

1 + Gm/c²(-1/r₂ + 1/r₁)

2.) For something which is moving with speed v approximately tangentially to a central gravitational field with Newtonian acceleration g, the overall acceleration is as follows:

g (1 + v²/c²)

This means that the acceleration for a light beam (or anything moving at approximately c) is exactly twice the acceleration experienced locally for a static object.

Although the acceleration for a light beam is 2g, horizontal lines in a local frame are slightly curved downwards by the gravitational field, and rulers are correspondingly slightly shorter closer to the mass, so that relative to the frame of a local observer, the downwards acceleration of anything including a light beam appears to be g.

For non-tangential motion, things are complicated by the fact that relative to an external coordinate the speed of light appears to change, and the exact result depends on the choice of coordinate system. However, even in that case the rate of change of momentum of anything including a light beam is (1+v²/c²) times the usual Newtonian value, and the local value is simply g again. For light, since c appears to be constant, the reason the momentum appears to increase as seen from the local frame is that potential energy is converted to or from kinetic energy, causing the momentum to change since in the case of light the magnitude of the momentum is equal to E/c.

 

[Bjarne]

Hmmm.
I doesn’t understand everything, but I have try to calculate:

Gm/c² is off course simple:

When m is the sun we will have:
6,67x10^-11 x 2x10³⁰/c²

(The result of the first part of the equation, before adding the distance difference = 1482 s.)

Now the "1482 s." must also be added with the distance difference.
Let say we want to know the relative time difference between, - Saturn and Neptune.
The distance is 4.5 billion km minus 1.5 billion >>> = is 3 x10¹²

Therefore the result of the "first part" of the equation (1482 s.) must now be added with that distance difference: 1/3 x10¹² meter - right ?

If so the result is 1482s. x 1/3 x10¹² = 4.94x10^-10 s.

But this can not be right, because if added with for instance the half distance (to Neptune) , then also the time difference will be half, - This is wrong, because the time difference must (off course) be double, - right (!) ?

What am am I doing wrong?

 

[Jonathan Scott]

Hmmm.
I doesn’t understand everything, but I have try to calculate:

Gm/c² is off course simple:

When m is the sun we will have:
6,67x10^-11 x 2x10³⁰/c²

(The result of the first part of the equation, before adding the distance difference = 1482 s.)

Now the "1482 s." must also be added with the distance difference.
Let say we want to know the relative time difference between, - Saturn and Neptune.
The distance is 4.5 billion km minus 1.5 billion >>> = is 3 x10¹²

Therefore the result of the "first part" of the equation (1482 s.) must now be added with that distance difference: 1/3 x10¹² meter - right ?

If so the result is 1482s. x 1/3 x10¹² = 4.94x10^-10 s.

But this can not be right, because if added with for instance the half distance (to Neptune) , then also the time difference will be half, - This is wrong, because the time difference must (off course) be double, - right (!) ?

What am am I doing wrong?

You have to take differences of 1/r values, not r values.

Gm/c² for the sun is about 1480m (metres, not seconds).

If r₁ is size of Saturn's orbit, about 1.5 x 10¹²m, and r₂ is the size of Neptune's orbit, about 4.5 x 10¹²m, then we get the following result:

Relative change in time rate due to the Sun's gravitational potential when moving from Saturn's orbit to Neptune's is as follows:

1480m * (-1/4.5 x 10¹²m + 1/1.5 x 10¹²m)

= 6.6 x 10^-10

That is, clocks at Neptune's orbit will run faster than those at Saturn's orbit by about 6.6 parts in 10,000,000,000.

 

[Bjarne]

Jonathan, - Thank you.

Do someone know:

1.) How many bow-second does light bend towards the Sun ?
2.) How much closer wil a such ray of light come to the sun ?

 

[spikenigma]

I've just read post 19

as far as I understand it, the rate of time dilation in a gravitational field given by:

1 - (2GM / rc^2)

G (gravitational constant) - 6,67x10-11
M - mass in kilograms of the gravitational source
r - distance in meters from the gravitational source
c (the speed of light in meters per second) - 299,792,458

is this correct?

 

[Naty1]

"What brought Einstein to the conclusion that time goes slower the deeper into a gravitational field one get..."


Via the equivalence principle, Gravitational redshift implies time dilation. Einstein discovered this after publishing special relativity during his development of GR.

http://en.wikipedia.org/wiki/Gravitational_Redshift#History

The gravitational weakening of light from high-gravity stars was predicted by John Michell in 1783 and Pierre-Simon Laplace in 1796, using Isaac Newton's concept of light corpuscles (see: emission theory) and who predicted that some stars would have a gravity so strong that light would not be able to escape...

Once it became accepted that light is an electromagnetic wave, it was clear that the frequency of light should not change from place to place, since waves from a source with a fixed frequency keep the same frequency everywhere. The only way around this conclusion would be if time itself was altered--- if clocks at different points had different rates.

This was precisely Einstein's conclusion in 1911. He considered an accelerating box, and noted that according to the special theory of relativity, the clock rate at the bottom of the box was slower than the clock rate at the top.

 

[Jonathan Scott]

I've just read post 19

as far as I understand it, the rate of time dilation in a gravitational field given by:

1 - (2GM / rc^2)

G (gravitational constant) - 6,67x10-11
M - mass in kilograms of the gravitational source
r - distance in meters from the gravitational source
c (the speed of light in meters per second) - 299,792,458

is this correct?

No, there's a spurious factor of 2 in there.

If you want the accurate time dilation for both weak and strong gravity using Schwarzschild coordinates, relative to infinity, it's the square root of that expression:

sqrt(1 - (2GM / rc^2))

If you just want the weak field approximation, it's:

(1 - GM / rc^2)

 

[spikenigma]

No, there's a spurious factor of 2 in there.

If you want the accurate time dilation for both weak and strong gravity using Schwarzschild coordinates, relative to infinity, it's the square root of that expression:

sqrt(1 - (2GM / rc^2))

If you just want the weak field approximation, it's:

(1 - GM / rc^2)

ok, so just to check that I can apply it correctly.

To use an example if I'm orbiting Earth:

G (gravitational constant) - 6.67x10^-11
M (mass of the Earth) - 5.9736×10^24 kg
r (distance in meters from gravitational source) - I'm orbiting Earth at 10 miles above the surface (16093 meters)
c (speed of light in meters per second) - 299,792,458 m/s

thus (very quickly using microsoft excel), for every 1 second on Earth I would experience 0.999999725 seconds?

 

[Jonathan Scott]

ok, so just to check that I can apply it correctly.

To use an example if I'm orbiting Earth:

G (gravitational constant) - 6.67x10^-11
M (mass of the Earth) - 5.9736×10^24 kg
r (distance in meters from gravitational source) - I'm orbiting Earth at 10 miles above the surface (16093 meters)
c (speed of light in meters per second) - 299,792,458 m/s

thus (very quickly using microsoft excel), for every 1 second on Earth I would experience 0.999999725 seconds?

r is the distance from the centre of the Earth, not from the surface, so you need to add on the radius of the Earth.

When comparing, you need to subtract the 1/r values from the two different levels. If you use just one, you're effectively comparing with infinity, not the surface of the earth.

If you're higher, your clock runs faster, so it records slightly more seconds in the same time.

If you make the simplifying assumption that g is approximately constant near the surface of the earth, the relative clock rate is simply gh/c^2 where h is the height between the comparison levels. I make that about 1.8x10^-12 for 10 miles.

 

[spikenigma]

r is the distance from the centre of the Earth, not from the surface, so you need to add on the radius of the Earth.

When comparing, you need to subtract the 1/r values from the two different levels. If you use just one, you're effectively comparing with infinity, not the surface of the earth.

If you're higher, your clock runs faster, so it records slightly more seconds in the same time.

If you make the simplifying assumption that g is approximately constant near the surface of the earth, the relative clock rate is simply gh/c^2 where h is the height between the comparison levels. I make that about 1.8x10^-12 for 10 miles.

I think I've got it:

Surface:

G (gravitational constant) - 6.67x10^-11
M (mass of the Earth) - 5.9736×10^24 kg
r (distance in meters from gravitational source) - 6300000 meters from the centre of the earth
c (speed of light in meters per second) - 299,792,458 m/sOrbit

G (gravitational constant) - 6.67x10^-11
M (mass of the Earth) - 5.9736×10^24 kg
r (distance in meters from gravitational source) - 6316063 meters from the centre of the earth
c (speed of light in meters per second) - 299,792,458 m/sorbit = 1 - ((2 x 6.67x10^-11 x 5.9736×10^24) / (6316063 x 299,792,458^2))
surface = 1 - ((2 x 6.67x10^-11 x 5.9736×10^24) / (6300000 x 299,792,458^2))

the time dilation factor is then

orbit value - surface value

 

[Bjarne]

Just to be sure if I got it correct
The time difference 10 billion km. from the Sun, is less that 1 second per year ?

Do someone know:

1.) How many bow-second does light bend towards the Sun ?
2.) How much closer wil a such ray of light come to the sun ?