A relativistic correction of natural gravitation.
In 1989 an article has been published: ‘Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field’. His result, briefly resumed: A uniform gravitational field is not flat, is quite interesting.But the way he proves the result is usable for any gravitational field; I will start with a description of his ‘method’, though in a slightly modificated way; the result will be the same.
First, a definition of a rigid reference system:
Observers (think of space-ships, provided with an ideal clock, are positioned in an arbitrary gravitational field, in the following way:
- the power of the observers (spaceships) is constant.
- the light-distance, measured by any observer to any other one is constant.
- one of the observers is designed as head observer H; his spatial coördinate will be 0; the spatial coördinate of observerA will be the light-distance, measured by H of A: xA.
-the standard clocks of the observers cannot be synchronized; theclock of H will be appointed as coördinate clock. All other observers are supplied with a coördinate clock in the following way: suppose the light-distance of observer A, measured by H, is a; the light-distance of H, measured by A, is b.
Then, b/a determines the ratio between standard clock and coördinate clock; the ratio will be α(x), depending of the spatial distance x of the observer. All coördinate clocks can be synchronized.
We consider the one-direction case.
The frame is rigid; the corresponding coördinate system will be represented by ds2=α(x)2(dt2–dx2),(the so called Weyl -form) where α(0)=1. So local velocity=dx/dt,coördinate velocity. The relationship between local acceleration (this is what every observer really ‘feel’, and are able to measure in his spaceship lab, by releasing a free falling particle alongside a rod), and coördinate acceleration, is more complex.
We have to consider the equations for geodesics. We start with the more general form:
d2xa/du2+Γabc.(dxa/du).(dxb/du) =λ(u).(dxa/du). In our special case, we have Γabc=0, except Γ001=Γ010=Γ100=Γ111=α’/α, α’ being dα(x)/dx. Considering x0=t, x1=x; we find for the first equation:
d2t/du2+2α’/α.(dt/du).(dx/du)=λ(u).(dt/du). This holds for any parameter, with corresponding λ(u). Therefore, this is also true also for t(u). As a result, the first equation delivers: 2α’/α.dx/dt=λ(t). The second equation will become: d2x/dt2+α’/α+α’/α.(dx/dt)2=2α’/α.(dx/dt)2 => d2x/dt2 + α’/α – α’/α.(dx/dt)2=0. Suppose, we release a free falling particle at moment t, so dx/dt =0 at that moment => d2x/dt2= – α’/α. In this case, the relation between local and coördinate acceleration is: local acc.=1/α. coördinate acc.= α’/α2. This holds for every gravitation field. Desloge uses the result for a uniform field; we investigate what comes out for natural gravity. Suppose there is a rigid reference frame, radially positioned with respect to a heavy body. The one- direction form of the Schwarzschild metric is:
ds2=(1–2m/x)dt2 – (1 –2m/x)—1dx2; x>2m. First, we have to transform the metric in the Weyl-form.
The transformation t̄=t, x̄=x+2m.ln(x – 2m) will do the job. The result: ds2={1 – 2m/x(x̄)}(dt̄2– dx̄2).
(There is no mathematical expression for the inverse of x̄(x) ,so we leave it: x(x̄).
There is one final task: 1–2m/x → 1 for x̄ infinite. We want the head observer at some finite x̄. We take A, with coördinate x̄A, and t̂=(1/k).t̄, x̂=(1/k).(x(x̄)–x(x̄A)); k= (1–(2m/x(x̄A)))2 will have as result: α̂=(1/k)(1–2m/x)1/2; x=x(x̄(x̂))and for x̂ =0, αA=1. Now we able to use the previous result:
d2x̂/dt̂2=α ‘/α=((1–2m/x)–1/2.2m/http://ift.tt/1iKfjJS;)/(1–2m/x)1/2=…..k2m/x2! This is the result we expect (k only dependent on the choice of A as head observer), if all observers are measuring with their coördinate clocks. However, an observer, eventually not being member of the reference system, and therefore measuring with his standard clock, will find: (1/α).2km/x2=(1–2m/x)–1/k2m/x2. Still keep in mind, x=x(x̄(x̂)).
The result is classical gravity multiplied with a ‘relativistic correction’, negligible in ‘normal’ circumstances, but considerable in the neighbourhood of the Schwarzschild radius.
A relativistic correction of natural gravitation.
In 1989 an article has been published: ‘Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field’. His result, briefly resumed: A uniform gravitational field is not flat, is quite interesting.But the way he proves the result is usable for any gravitational field; I will start with a description of his ‘method’, though in a slightly modificated way; the result will be the same.
First, a definition of a rigid reference system:
Observers (think of space-ships, provided with an ideal clock, are positioned in an arbitrary gravitational field, in the following way:
- the power of the observers (spaceships) is constant.
- the light-distance, measured by any observer to any other one is constant.
- one of the observers is designed as head observer H; his spatial coördinate will be 0; the spatial coördinate of observerA will be the light-distance, measured by H of A: xA.
-the standard clocks of the observers cannot be synchronized; theclock of H will be appointed as coördinate clock. All other observers are supplied with a coördinate clock in the following way: suppose the light-distance of observer A, measured by H, is a; the light-distance of H, measured by A, is b.
Then, b/a determines the ratio between standard clock and coördinate clock; the ratio will be α(x), depending of the spatial distance x of the observer. All coördinate clocks can be synchronized.
We consider the one-direction case.
The frame is rigid; the corresponding coördinate system will be represented by ds2=α(x)2(dt2–dx2),(the so called Weyl -form) where α(0)=1. So local velocity=dx/dt,coördinate velocity. The relationship between local acceleration (this is what every observer really ‘feel’, and are able to measure in his spaceship lab, by releasing a free falling particle alongside a rod), and coördinate acceleration, is more complex.
We have to consider the equations for geodesics. We start with the more general form:
d2xa/du2+Γabc.(dxa/du).(dxb/du) =λ(u).(dxa/du). In our special case, we have Γabc=0, except Γ001=Γ010=Γ100=Γ111=α’/α, α’ being dα(x)/dx. Considering x0=t, x1=x; we find for the first equation:
d2t/du2+2α’/α.(dt/du).(dx/du)=λ(u).(dt/du). This holds for any parameter, with corresponding λ(u). Therefore, this is also true also for t(u). As a result, the first equation delivers: 2α’/α.dx/dt=λ(t). The second equation will become: d2x/dt2+α’/α+α’/α.(dx/dt)2=2α’/α.(dx/dt)2 => d2x/dt2 + α’/α – α’/α.(dx/dt)2=0. Suppose, we release a free falling particle at moment t, so dx/dt =0 at that moment => d2x/dt2= – α’/α. In this case, the relation between local and coördinate acceleration is: local acc.=1/α. coördinate acc.= α’/α2. This holds for every gravitation field. Desloge uses the result for a uniform field; we investigate what comes out for natural gravity. Suppose there is a rigid reference frame, radially positioned with respect to a heavy body. The one- direction form of the Schwarzschild metric is:
ds2=(1–2m/x)dt2 – (1 –2m/x)—1dx2; x>2m. First, we have to transform the metric in the Weyl-form.
The transformation t̄=t, x̄=x+2m.ln(x – 2m) will do the job. The result: ds2={1 – 2m/x(x̄)}(dt̄2– dx̄2).
(There is no mathematical expression for the inverse of x̄(x) ,so we leave it: x(x̄).
There is one final task: 1–2m/x → 1 for x̄ infinite. We want the head observer at some finite x̄. We take A, with coördinate x̄A, and t̂=(1/k).t̄, x̂=(1/k).(x(x̄)–x(x̄A)); k= (1–(2m/x(x̄A)))2 will have as result: α̂=(1/k)(1–2m/x)1/2; x=x(x̄(x̂))and for x̂ =0, αA=1. Now we able to use the previous result:
d2x̂/dt̂2=α ‘/α=((1–2m/x)–1/2.2m/http://ift.tt/1iKfjJS;)/(1–2m/x)1/2=…..k2m/x2! This is the result we expect (k only dependent on the choice of A as head observer), if all observers are measuring with their coördinate clocks. However, an observer, eventually not being member of the reference system, and therefore measuring with his standard clock, will find: (1/α).2km/x2=(1–2m/x)–1/k2m/x2. Still keep in mind, x=x(x̄(x̂)).
The result is classical gravity multiplied with a ‘relativistic correction’, negligible in ‘normal’ circumstances, but considerable in the neighbourhood of the Schwarzschild radius.
A relativistic correction of natural gravitation.
In 1989 an article has been published: ‘Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field’. His result, briefly resumed: A uniform gravitational field is not flat, is quite interesting.But the way he proves the result is usable for any gravitational field; I will start with a description of his ‘method’, though in a slightly modificated way; the result will be the same.
First, a definition of a rigid reference system:
Observers (think of space-ships, provided with an ideal clock, are positioned in an arbitrary gravitational field, in the following way:
- the power of the observers (spaceships) is constant.
- the light-distance, measured by any observer to any other one is constant.
- one of the observers is designed as head observer H; his spatial coördinate will be 0; the spatial coördinate of observerA will be the light-distance, measured by H of A: xA.
-the standard clocks of the observers cannot be synchronized; theclock of H will be appointed as coördinate clock. All other observers are supplied with a coördinate clock in the following way: suppose the light-distance of observer A, measured by H, is a; the light-distance of H, measured by A, is b.
Then, b/a determines the ratio between standard clock and coördinate clock; the ratio will be α(x), depending of the spatial distance x of the observer. All coördinate clocks can be synchronized.
We consider the one-direction case.
The frame is rigid; the corresponding coördinate system will be represented by ds2=α(x)2(dt2–dx2),(the so called Weyl -form) where α(0)=1. So local velocity=dx/dt,coördinate velocity. The relationship between local acceleration (this is what every observer really ‘feel’, and are able to measure in his spaceship lab, by releasing a free falling particle alongside a rod), and coördinate acceleration, is more complex.
We have to consider the equations for geodesics. We start with the more general form:
d2xa/du2+Γabc.(dxa/du).(dxb/du) =λ(u).(dxa/du). In our special case, we have Γabc=0, except Γ001=Γ010=Γ100=Γ111=α’/α, α’ being dα(x)/dx. Considering x0=t, x1=x; we find for the first equation:
d2t/du2+2α’/α.(dt/du).(dx/du)=λ(u).(dt/du). This holds for any parameter, with corresponding λ(u). Therefore, this is also true also for t(u). As a result, the first equation delivers: 2α’/α.dx/dt=λ(t). The second equation will become: d2x/dt2+α’/α+α’/α.(dx/dt)2=2α’/α.(dx/dt)2 => d2x/dt2 + α’/α – α’/α.(dx/dt)2=0. Suppose, we release a free falling particle at moment t, so dx/dt =0 at that moment => d2x/dt2= – α’/α. In this case, the relation between local and coördinate acceleration is: local acc.=1/α. coördinate acc.= α’/α2. This holds for every gravitation field. Desloge uses the result for a uniform field; we investigate what comes out for natural gravity. Suppose there is a rigid reference frame, radially positioned with respect to a heavy body. The one- direction form of the Schwarzschild metric is:
ds2=(1–2m/x)dt2 – (1 –2m/x)—1dx2; x>2m. First, we have to transform the metric in the Weyl-form.
The transformation t̄=t, x̄=x+2m.ln(x – 2m) will do the job. The result: ds2={1 – 2m/x(x̄)}(dt̄2– dx̄2).
(There is no mathematical expression for the inverse of x̄(x) ,so we leave it: x(x̄).
There is one final task: 1–2m/x → 1 for x̄ infinite. We want the head observer at some finite x̄. We take A, with coördinate x̄A, and t̂=(1/k).t̄, x̂=(1/k).(x(x̄)–x(x̄A)); k= (1–(2m/x(x̄A)))2 will have as result: α̂=(1/k)(1–2m/x)1/2; x=x(x̄(x̂))and for x̂ =0, αA=1. Now we able to use the previous result:
d2x̂/dt̂2=α ‘/α=((1–2m/x)–1/2.2m/http://ift.tt/1iKfjJS;)/(1–2m/x)1/2=…..k2m/x2! This is the result we expect (k only dependent on the choice of A as head observer), if all observers are measuring with their coördinate clocks. However, an observer, eventually not being member of the reference system, and therefore measuring with his standard clock, will find: (1/α).2km/x2=(1–2m/x)–1/k2m/x2. Still keep in mind, x=x(x̄(x̂)).
The result is classical gravity multiplied with a ‘relativistic correction’, negligible in ‘normal’ circumstances, but considerable in the neighbourhood of the Schwarzschild radius.
P.S. To my opinion, the way Desloge uses the concept of a rigid reference frame, together with the link with acoördinate system is an important contribution to the theory. Still there are several questions, open to me, which I would like to discuss with those who are interested in the matter.
In 1989 an article has been published: ‘Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field’. His result, briefly resumed: A uniform gravitational field is not flat, is quite interesting.But the way he proves the result is usable for any gravitational field; I will start with a description of his ‘method’, though in a slightly modificated way; the result will be the same.
First, a definition of a rigid reference system:
Observers (think of space-ships, provided with an ideal clock, are positioned in an arbitrary gravitational field, in the following way:
- the power of the observers (spaceships) is constant.
- the light-distance, measured by any observer to any other one is constant.
- one of the observers is designed as head observer H; his spatial coördinate will be 0; the spatial coördinate of observerA will be the light-distance, measured by H of A: xA.
-the standard clocks of the observers cannot be synchronized; theclock of H will be appointed as coördinate clock. All other observers are supplied with a coördinate clock in the following way: suppose the light-distance of observer A, measured by H, is a; the light-distance of H, measured by A, is b.
Then, b/a determines the ratio between standard clock and coördinate clock; the ratio will be α(x), depending of the spatial distance x of the observer. All coördinate clocks can be synchronized.
We consider the one-direction case.
The frame is rigid; the corresponding coördinate system will be represented by ds2=α(x)2(dt2–dx2),(the so called Weyl -form) where α(0)=1. So local velocity=dx/dt,coördinate velocity. The relationship between local acceleration (this is what every observer really ‘feel’, and are able to measure in his spaceship lab, by releasing a free falling particle alongside a rod), and coördinate acceleration, is more complex.
We have to consider the equations for geodesics. We start with the more general form:
d2xa/du2+Γabc.(dxa/du).(dxb/du) =λ(u).(dxa/du). In our special case, we have Γabc=0, except Γ001=Γ010=Γ100=Γ111=α’/α, α’ being dα(x)/dx. Considering x0=t, x1=x; we find for the first equation:
d2t/du2+2α’/α.(dt/du).(dx/du)=λ(u).(dt/du). This holds for any parameter, with corresponding λ(u). Therefore, this is also true also for t(u). As a result, the first equation delivers: 2α’/α.dx/dt=λ(t). The second equation will become: d2x/dt2+α’/α+α’/α.(dx/dt)2=2α’/α.(dx/dt)2 => d2x/dt2 + α’/α – α’/α.(dx/dt)2=0. Suppose, we release a free falling particle at moment t, so dx/dt =0 at that moment => d2x/dt2= – α’/α. In this case, the relation between local and coördinate acceleration is: local acc.=1/α. coördinate acc.= α’/α2. This holds for every gravitation field. Desloge uses the result for a uniform field; we investigate what comes out for natural gravity. Suppose there is a rigid reference frame, radially positioned with respect to a heavy body. The one- direction form of the Schwarzschild metric is:
ds2=(1–2m/x)dt2 – (1 –2m/x)—1dx2; x>2m. First, we have to transform the metric in the Weyl-form.
The transformation t̄=t, x̄=x+2m.ln(x – 2m) will do the job. The result: ds2={1 – 2m/x(x̄)}(dt̄2– dx̄2).
(There is no mathematical expression for the inverse of x̄(x) ,so we leave it: x(x̄).
There is one final task: 1–2m/x → 1 for x̄ infinite. We want the head observer at some finite x̄. We take A, with coördinate x̄A, and t̂=(1/k).t̄, x̂=(1/k).(x(x̄)–x(x̄A)); k= (1–(2m/x(x̄A)))2 will have as result: α̂=(1/k)(1–2m/x)1/2; x=x(x̄(x̂))and for x̂ =0, αA=1. Now we able to use the previous result:
d2x̂/dt̂2=α ‘/α=((1–2m/x)–1/2.2m/http://ift.tt/1iKfjJS;)/(1–2m/x)1/2=…..k2m/x2! This is the result we expect (k only dependent on the choice of A as head observer), if all observers are measuring with their coördinate clocks. However, an observer, eventually not being member of the reference system, and therefore measuring with his standard clock, will find: (1/α).2km/x2=(1–2m/x)–1/k2m/x2. Still keep in mind, x=x(x̄(x̂)).
The result is classical gravity multiplied with a ‘relativistic correction’, negligible in ‘normal’ circumstances, but considerable in the neighbourhood of the Schwarzschild radius.
A relativistic correction of natural gravitation.
In 1989 an article has been published: ‘Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field’. His result, briefly resumed: A uniform gravitational field is not flat, is quite interesting.But the way he proves the result is usable for any gravitational field; I will start with a description of his ‘method’, though in a slightly modificated way; the result will be the same.
First, a definition of a rigid reference system:
Observers (think of space-ships, provided with an ideal clock, are positioned in an arbitrary gravitational field, in the following way:
- the power of the observers (spaceships) is constant.
- the light-distance, measured by any observer to any other one is constant.
- one of the observers is designed as head observer H; his spatial coördinate will be 0; the spatial coördinate of observerA will be the light-distance, measured by H of A: xA.
-the standard clocks of the observers cannot be synchronized; theclock of H will be appointed as coördinate clock. All other observers are supplied with a coördinate clock in the following way: suppose the light-distance of observer A, measured by H, is a; the light-distance of H, measured by A, is b.
Then, b/a determines the ratio between standard clock and coördinate clock; the ratio will be α(x), depending of the spatial distance x of the observer. All coördinate clocks can be synchronized.
We consider the one-direction case.
The frame is rigid; the corresponding coördinate system will be represented by ds2=α(x)2(dt2–dx2),(the so called Weyl -form) where α(0)=1. So local velocity=dx/dt,coördinate velocity. The relationship between local acceleration (this is what every observer really ‘feel’, and are able to measure in his spaceship lab, by releasing a free falling particle alongside a rod), and coördinate acceleration, is more complex.
We have to consider the equations for geodesics. We start with the more general form:
d2xa/du2+Γabc.(dxa/du).(dxb/du) =λ(u).(dxa/du). In our special case, we have Γabc=0, except Γ001=Γ010=Γ100=Γ111=α’/α, α’ being dα(x)/dx. Considering x0=t, x1=x; we find for the first equation:
d2t/du2+2α’/α.(dt/du).(dx/du)=λ(u).(dt/du). This holds for any parameter, with corresponding λ(u). Therefore, this is also true also for t(u). As a result, the first equation delivers: 2α’/α.dx/dt=λ(t). The second equation will become: d2x/dt2+α’/α+α’/α.(dx/dt)2=2α’/α.(dx/dt)2 => d2x/dt2 + α’/α – α’/α.(dx/dt)2=0. Suppose, we release a free falling particle at moment t, so dx/dt =0 at that moment => d2x/dt2= – α’/α. In this case, the relation between local and coördinate acceleration is: local acc.=1/α. coördinate acc.= α’/α2. This holds for every gravitation field. Desloge uses the result for a uniform field; we investigate what comes out for natural gravity. Suppose there is a rigid reference frame, radially positioned with respect to a heavy body. The one- direction form of the Schwarzschild metric is:
ds2=(1–2m/x)dt2 – (1 –2m/x)—1dx2; x>2m. First, we have to transform the metric in the Weyl-form.
The transformation t̄=t, x̄=x+2m.ln(x – 2m) will do the job. The result: ds2={1 – 2m/x(x̄)}(dt̄2– dx̄2).
(There is no mathematical expression for the inverse of x̄(x) ,so we leave it: x(x̄).
There is one final task: 1–2m/x → 1 for x̄ infinite. We want the head observer at some finite x̄. We take A, with coördinate x̄A, and t̂=(1/k).t̄, x̂=(1/k).(x(x̄)–x(x̄A)); k= (1–(2m/x(x̄A)))2 will have as result: α̂=(1/k)(1–2m/x)1/2; x=x(x̄(x̂))and for x̂ =0, αA=1. Now we able to use the previous result:
d2x̂/dt̂2=α ‘/α=((1–2m/x)–1/2.2m/http://ift.tt/1iKfjJS;)/(1–2m/x)1/2=…..k2m/x2! This is the result we expect (k only dependent on the choice of A as head observer), if all observers are measuring with their coördinate clocks. However, an observer, eventually not being member of the reference system, and therefore measuring with his standard clock, will find: (1/α).2km/x2=(1–2m/x)–1/k2m/x2. Still keep in mind, x=x(x̄(x̂)).
The result is classical gravity multiplied with a ‘relativistic correction’, negligible in ‘normal’ circumstances, but considerable in the neighbourhood of the Schwarzschild radius.
A relativistic correction of natural gravitation.
In 1989 an article has been published: ‘Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field’. His result, briefly resumed: A uniform gravitational field is not flat, is quite interesting.But the way he proves the result is usable for any gravitational field; I will start with a description of his ‘method’, though in a slightly modificated way; the result will be the same.
First, a definition of a rigid reference system:
Observers (think of space-ships, provided with an ideal clock, are positioned in an arbitrary gravitational field, in the following way:
- the power of the observers (spaceships) is constant.
- the light-distance, measured by any observer to any other one is constant.
- one of the observers is designed as head observer H; his spatial coördinate will be 0; the spatial coördinate of observerA will be the light-distance, measured by H of A: xA.
-the standard clocks of the observers cannot be synchronized; theclock of H will be appointed as coördinate clock. All other observers are supplied with a coördinate clock in the following way: suppose the light-distance of observer A, measured by H, is a; the light-distance of H, measured by A, is b.
Then, b/a determines the ratio between standard clock and coördinate clock; the ratio will be α(x), depending of the spatial distance x of the observer. All coördinate clocks can be synchronized.
We consider the one-direction case.
The frame is rigid; the corresponding coördinate system will be represented by ds2=α(x)2(dt2–dx2),(the so called Weyl -form) where α(0)=1. So local velocity=dx/dt,coördinate velocity. The relationship between local acceleration (this is what every observer really ‘feel’, and are able to measure in his spaceship lab, by releasing a free falling particle alongside a rod), and coördinate acceleration, is more complex.
We have to consider the equations for geodesics. We start with the more general form:
d2xa/du2+Γabc.(dxa/du).(dxb/du) =λ(u).(dxa/du). In our special case, we have Γabc=0, except Γ001=Γ010=Γ100=Γ111=α’/α, α’ being dα(x)/dx. Considering x0=t, x1=x; we find for the first equation:
d2t/du2+2α’/α.(dt/du).(dx/du)=λ(u).(dt/du). This holds for any parameter, with corresponding λ(u). Therefore, this is also true also for t(u). As a result, the first equation delivers: 2α’/α.dx/dt=λ(t). The second equation will become: d2x/dt2+α’/α+α’/α.(dx/dt)2=2α’/α.(dx/dt)2 => d2x/dt2 + α’/α – α’/α.(dx/dt)2=0. Suppose, we release a free falling particle at moment t, so dx/dt =0 at that moment => d2x/dt2= – α’/α. In this case, the relation between local and coördinate acceleration is: local acc.=1/α. coördinate acc.= α’/α2. This holds for every gravitation field. Desloge uses the result for a uniform field; we investigate what comes out for natural gravity. Suppose there is a rigid reference frame, radially positioned with respect to a heavy body. The one- direction form of the Schwarzschild metric is:
ds2=(1–2m/x)dt2 – (1 –2m/x)—1dx2; x>2m. First, we have to transform the metric in the Weyl-form.
The transformation t̄=t, x̄=x+2m.ln(x – 2m) will do the job. The result: ds2={1 – 2m/x(x̄)}(dt̄2– dx̄2).
(There is no mathematical expression for the inverse of x̄(x) ,so we leave it: x(x̄).
There is one final task: 1–2m/x → 1 for x̄ infinite. We want the head observer at some finite x̄. We take A, with coördinate x̄A, and t̂=(1/k).t̄, x̂=(1/k).(x(x̄)–x(x̄A)); k= (1–(2m/x(x̄A)))2 will have as result: α̂=(1/k)(1–2m/x)1/2; x=x(x̄(x̂))and for x̂ =0, αA=1. Now we able to use the previous result:
d2x̂/dt̂2=α ‘/α=((1–2m/x)–1/2.2m/http://ift.tt/1iKfjJS;)/(1–2m/x)1/2=…..k2m/x2! This is the result we expect (k only dependent on the choice of A as head observer), if all observers are measuring with their coördinate clocks. However, an observer, eventually not being member of the reference system, and therefore measuring with his standard clock, will find: (1/α).2km/x2=(1–2m/x)–1/k2m/x2. Still keep in mind, x=x(x̄(x̂)).
The result is classical gravity multiplied with a ‘relativistic correction’, negligible in ‘normal’ circumstances, but considerable in the neighbourhood of the Schwarzschild radius.
P.S. To my opinion, the way Desloge uses the concept of a rigid reference frame, together with the link with acoördinate system is an important contribution to the theory. Still there are several questions, open to me, which I would like to discuss with those who are interested in the matter.
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