GR time dilation from SR by EP

[Austin0]

As I understand it from what I have read the gravitational time dilation concept was arrived at through the equivalence principle. That the differential dilation at different locations in an accelerating system were derived from SR and included in GR by EP.

But I have never encountered the basis in SR principles that produced that calculation of differential.
The only thing I can see would be relative velocity between front and back due to contraction but that does not seem to track because the proper differential is constant while contraction is not.
SO does anyone either know or have a link to a description of the process that Einstein employed?

Thanks

 

[tom.stoer]

Let's summarize; time dilation can be derived from the rather general proper time formula für a curve C and a metric g.

\tau[C] = \int_C d\tau = \int_C \sqrt{g_{\mu\nu}\,dx^\mu\,dx^\nu}

SR follows when restricting to a flat metric, e.g diag(+1,-1,-1,-1). Then the only effect for time dilation is du to the curve C, i.e. due to velocity and acceleration.

Time dilation can be derived via comparing two curves for two different observers with intersecting world lines.

Does this help?

Or are you looking for a construction or derivation 'a la Einstein'?

 

[Austin0]

Let's summarize; time dilation can be derived from the rather general proper time formula für a curve C and a metric g.

\tau[C] = \int_C d\tau = \int_C \sqrt{g_{\mu\nu}\,dx^\mu\,dx^\nu}

SR follows when restricting to a flat metric, e.g diag(+1,-1,-1,-1). Then the only effect for time dilation is du to the curve C, i.e. due to velocity and acceleration.

Time dilation can be derived via comparing two curves for two different observers with intersecting world lines.

Does this help?

Or are you looking for a construction or derivation 'a la Einstein'?

Your math with the sub and super script notation is outside my understanding but I do have a grasp of integrating the curve of a worldline.
So are you saying that in the launch frame the method was to chart separate world lines for the front and back of the accelerating system and then integrate them separately?
That the internal differential was calculated as the ration of these values?

In this context it would seem that acceleration was not a factor at all , only instantaneous velocity,yes?.
Thanks for your response.

 

[tom.stoer]

In SR you can rewrite the integral as

\tau[C] = \int_C d\tau = \int_{t_a}^{t^b} dt \sqrt{1-\vec{v}^2(t)}

where one specific reference frame with coordinates (t,x) is used; t is the time coordinate and v(t) is the velocity in this reference frame w.r.t. to the coordinates x(t). The reference frame is an inertial frame whereas the two objects moving along two curves C₁ and C₂ need not define an inertial frame (b/c they may be accelerated).

Using two curves C_i with i=1,2 you will find two proper times τ_i for two observers.

\tau[C_i] = \int_{C_i} d\tau = \int_{t_a}^{t^b} dt \sqrt{1-\vec{v}_i^2(t)}

The two curves are defines such that the they intersect at t_a and t_b. At t_b they can compare their proper times τ_i.

 

[PAllen]

Here is one very elementary derivation. I don't know if this has anything to do with Einstein's reasoning:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

The key point they use is the velocity change of the rocket base between emission from the front and arrival at the back. They don't reference it to a rocket, but the reasoning and math would apply.

There are much more rigorous derivations assuming a Born rigid rocket, that derive interesting things like for any length rocket there is a maximum acceleration possible consistent with Born rigidity; the longer the rocket , the smaller the maximum acceleration. The limitation arises because the for a sufficient rocket length + Born rigidity condition, the front of the rocket becomes required to have a relative velocity >=c compared to the back, in the instantly co-moving inertial frame of the back.

 

[harrylin]

http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1911_35_898-908.pdf
Regretfully in German.

The main point (§3): In first approximation* you can ignore length contraction and just consider the Doppler effect from the change in velocity.

g=gravitational acceleration.
Take K₀ as inertial rest system at the instant that a signal is sent from one clock to the other (let's say at t=0).
At arrival the other clock has speed v=g(h/c) because t=h/c.

I suppose that he just used classical Doppler f/f₀= 1+v/c
and so he obtained in approximation:

f/f₀ = 1+gh/c²

That is the apparent difference in clock frequency in an accelerating frame if the observers ignore that they are accelerating, and therefore it should also be the observed frequency shift in a gravitational field.

[Hmm I now see that if I had started my post a little later I could have just commented "yes, it has" to the preceding post! ]

*That gives in practice a very good approximation: length contraction and time dilation are usually quite negligible for acceleration from rest over common distances

 

[Austin0]

Here is one very elementary derivation. I don't know if this has anything to do with Einstein's reasoning:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

The key point they use is the velocity change of the rocket base between emission from the front and arrival at the back. They don't reference it to a rocket, but the reasoning and math would apply.

There are much more rigorous derivations assuming a Born rigid rocket, that derive interesting things like for any length rocket there is a maximum acceleration possible consistent with Born rigidity; the longer the rocket , the smaller the maximum acceleration. The limitation arises because the for a sufficient rocket length + Born rigidity condition, the front of the rocket becomes required to have a relative velocity >=c compared to the back, in the instantly co-moving inertial frame of the back.

Thanks for the link. I t was helpful and food for thought.

Regarding the limits on acceleration and length in a Born rigid system , although I don't doubt the conclusion I don't follow the reasoning.
The Born acceleration schedule requires that the maximum acceleration occur at the rear and operates on the assumption of constant proper length , meaning zero relative velocity between the front and back. In a co-moving frame the only relative velocity would be from contraction relative to that frame. So it is hard to picture how an infinitesimal interval of contraction could result in faster than c motion as the contraction is spread out over the total course of acceleration.
In addition there is no reason that the front could not have a greater than c velocity relative to the back as it is only closing velocity not actual velocity.
So perhaps I am misunderstanding your description or missing some point.

Addition. Having thought a bit , yes there are limits to acceleration for any kind of acceleration distribution if the length and magnitude is carried out to unrealistic extents.
and this is not just applicable to Born acceleration. In any real world system there is the material limitation of momentum conductivity and the speed of sound which means beyond some threshold additional thrust can't propagate fast enough and must cause deformation and torque acting against the inertia of the not yet accelerated parts of the system if it is long enough.
And all other considerations aside, the ends cannot have an acceleration which results in a Lorentz contraction of the ends exceeding 1.999,,,,,c towards each other in any frame. SO I obviously was hasty in my response .SO if you were talking about some other factor I have still missed let me know. ;-)

 

[PAllen]

Thanks for the link. I t was helpful and food for thought.

Regarding the limits on acceleration and length in a Born rigid system , although I don't doubt the conclusion I don't follow the reasoning.
The Born acceleration schedule requires that the maximum acceleration occur at the rear and operates on the assumption of constant proper length , meaning zero relative velocity between the front and back. In a co-moving frame the only relative velocity would be from contraction relative to that frame. So it is hard to picture how an infinitesimal interval of contraction could result in faster than c motion as the contraction is spread out over the total course of acceleration.
In addition there is no reason that the front could not have a greater than c velocity relative to the back as it is only closing velocity not actual velocity.
So perhaps I am misunderstanding your description or missing some point.

Addition. Having thought a bit , yes there are limits to acceleration for any kind of acceleration distribution if the length and magnitude is carried out to unrealistic extents.
and this is not just applicable to Born acceleration. In any real world system there is the material limitation of momentum conductivity and the speed of sound which means beyond some threshold additional thrust can't propagate fast enough and must cause deformation and torque acting against the inertia of the not yet accelerated parts of the system if it is long enough.
And all other considerations aside, the ends cannot have an acceleration which results in a Lorentz contraction of the ends exceeding 1.999,,,,,c towards each other in any frame. SO I obviously was hasty in my response .SO if you were talking about some other factor I have still missed let me know. ;-)

You got it. It is just an amusing math fact, not really connected to reality: if you had a 1 light second long ship, even with Born rigidity, you could only accelerate it very slowly (without violating SR).