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Accelerated Reference Frames
----------------------------
An uniformly accelerated observer in Special Relativity will follow
a hyperbola.
x^{2} - T^{2} = r^{2}
The PROPER ACCELERATION along a particular curve, R, is:
a = c^{2}/R
Dimensionally this is [L^{2}/T^{2}][1/L].
It is clear from this equation that as R gets smaller, a increases.
Thus,
x^{2} - T^{2} = c^{4}/a^{2}
The hyperbola is Lorentz invariant therefore a transformation will
just move the observer to a different point on the curve where he/she
will experience exactly the same accelertaion as before.
Now, we can regard the accelerated observer in a stationary reference
frame as being equivalent to a stationary observer in a unifomly
accelerated reference frame. Unfortunately, Special Relativity doesn't
hande this situation too well because it is based on inertial reference
frames (frames moving at constant velocity). Acceleration does not
enter directly into the Lorentz transform, length contraction, or time
dilation, only the velocity is important.
The general theory of relativity is a theoretical framework applicable
to any frame of reference - inertial or accelerating. In developing this
theory, Einstein wanted to produce a theory of gravitation that
incorporated the theory of special relativity and the equivalence
principle.
Comments:
- The hyperbolae are labelled r = 1, 2, 3 etc.
- Note that the worldlines bunch together, reflecting the gravitational
length contraction.
- Any observer at rest in Rindler coordinates has constant proper
acceleration, with Rindler observers closer to the Rindler horizon
having greater proper acceleration.
- The distances 23 = 34 = AB = BC = EF = FG remain the same no matter
how long the acceleration proceeds.
- Proper times intervals along two Rindler hyperbolae for the same ω
are given by the following ratio:
(r + Δr)/r
Thus, an observer travelling along 4 to C would judge time along 2
to A passing at half the speed since (2R + 2R)/2R
By analogy to the polar coordinate conversions x = rcosθ and
y = rsinθ, it is possible to write the following transformations in
hyperbolic space.
x = rcoshω and T = rsinhω
Where ω is the RINDLER TIME along the path of the observer.
x and T are referred to as RINDLER COORDINATES. Unlike the Newtonian
case, a uniformly accelerated motion in a Minkowski spacetime cannot cover
the entire spacetime. It is restricted to the wedge of spacetime shown,
bounded by the light cone. This is referred to as the RINDLER WEDGE or
RINDLER SPACE.
Note: sinh^{2}ω - cosh^{2}ω = -1 where -∞ < ω < +∞
∴ r^{2}cosh^{2}ω - r^{2}sinh^{2}ω = r^{2}
Taking the differentials we get:
dx = drcoshω + rsinhωdω
and
dT = drsinhω + rcoshωdω
Therefore,
dτ^{2} = dT^{2} - dx^{2}
= -dr^{2} + r^{2}dω^{2}
Again this is analagous to the flat spacetime metric in polar
coordinates, dS^{2} = dr^{2} + r^{2}dθ^{2}.
Now take a particular curve, R, and displace it by an infinitesimally
small amount x'. Therefore, r = R + x' and we can write:
dτ^{2} = (R^{2} + 2Rx' + x'^{2})dω^{2} - d(R + x')^{2}
= (1 + 2x'/R + x'^{2}/R^{2})R^{2}dω^{2} - dx'^{2} ... R is fixed
= (1 + 2x'/R)R^{2}dω^{2} - dx'^{2} since R >> x'
Let Rω = t ∴ R^{2}dω^{2} = dt^{2}. Thus,
dτ^{2} = (1 + 2x'/R)dt^{2} - dx'^{2}
If we pick a value of R that avoids relativistic effects, for example
a value that gives us an acceleration of g (~10 m/s^{2}) = 1/R then we
can write:
dτ^{2} = (1 + 2x'g)dt^{2} - dx'^{2}
= (1 + 2φ)dt^{2} - dx'^{2}
Where φ is defined as the GRAVITATIONAL POTENTIAL = x'g. The
gravitational potential is the gravitational potential energy per
unit mass.
This the basic version of the RINDLER METRIC in flat space time. It
shows the correspondence between a UNIFORMLY accelerated reference
frame and a UNIFORM gravitational field. This is the EQUIVALENCE
PRINCIPLE. Note: A uniform gravitational field does not produce tidal
forces. For these we need to consider the curvature of spacetime.
To see the effects of curved spacetime it is necessary to use a metric
that contains a mass term. Consider φ = -GM/r. The metric becomes:
dτ^{2} = (1 - 2MG/rc^{2})dt^{2} - (1/c^{2}){dx^{2} + .... }
In spherical coordinates this becomes:
dτ^{2} = (1 - 2MG/rc^{2})dt^{2} - (1/c^{2}){dr^{2} + r^{2}Ω^{2}}
The problem with this metric is that at some point the (1 - 2MG/rc^{2})
term changes its sign and the signature of the metric will have 4
negative terms which is a violation of spacetime. It turns out from
Einsteins equations that the correct metric is:
dτ^{2} = (1 - 2MG/rc^{2})dt^{2} - (1/(1 - 2MG/rc^{2}))dr^{2} - (1/c^{2})r^{2}Ω^{2}
This is equivalent to the form:
dτ^{2} = (1 + φ/c^{2})dt^{2} - (1/(1 + φ/c^{2}))dr^{2} - (1/c^{2})r^{2}Ω^{2}
Which can be written (after expanding the middle term in terms
of a Taylor series) as:
dτ^{2} = (1 + φ/c^{2})dt^{2} - (1 - φ/c^{2})dr^{2} - (1/c^{2})r^{2}Ω^{2}
Now when a sign change occurs, the time and spatial terms flip to
maintain the signature of the metric.
This is the Schwarzchild metric that describes the spacetime around
a spherically symmetric gravitating object such as a black hole or
earth for that matter.
Consider a light ray travelling radially. Since light follows a null
geodesic (dτ = 0). The above equation becomes:
(1 - 2MG/rc^{2})dt^{2} = (1/(1 - 2MG/rc^{2}))dr^{2}
Or,
dr/dt = (1 - 2MG/rc^{2}) = (1 - R_{s}/r)
Thus, as the light ray approaches the event horizon it slows and
eventually stops at the Scwartzchild radius.