Rindler coordinates or Rindler frame is a coordinate system or reference frame used to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics, the coordinates of a hyperbolically accelerated reference frame constitute a useful coordinate chart representing flat Minkowski spacetime. In special relativity, a uniform acceleration results in hyperbolic motion, for which a uniformly accelerating frame of reference in which it is at rest can be chosen as its proper reference frame. The phenomena in this frame can be compared to effects arising in a homogeneous gravitational field. Historically, such coordinates were introduced soon after the advent of special relativity, when they were studied (fully or partially) alongside the concept of hyperbolic motion: In relation to flat Minkowski spacetime by Albert Einstein (1907, 1912), Max Born (1909), Arnold Sommerfeld (1910), Max von Laue (1911), Hendrik Lorentz (1913), Friedrich Kottler (1914), Wolfgang Pauli (1921), Karl Bollert (1922), Stjepan Mohorovičić (1922), Georges Lemaître (1924), Einstein & Nathan Rosen (1935), Christian Møller (1943, 1952), Fritz Rohrlich (1963), Harry Lass (1963), and in relation to both flat and curved spacetime of general relativity by Wolfgang Rindler (1960, 1966). For details and sources, see § History.
Formulation In this article, the speed of light is defined by c = 1, the inertial coordinates are (X, Y, Z, T), and the hyperbolic coordinates are (x, y, z, t). These hyperbolic coordinates can be separated into two main variants depending on the accelerated observer's position: If the observer is located at time T = 0 at position X = 1/α (with α as the constant proper acceleration measured by a comoving accelerometer), then the hyperbolic coordinates are often called Rindler coordinates with the corresponding Rindler metric. If the observer is located at time T = 0 at position X = 0, then the hyperbolic coordinates are sometimes called Møller coordinates or Kottler–Møller coordinates with the corresponding Kottler–Møller metric. An alternative chart often related to observers in hyperbolic motion is obtained using Radar coordinates which are sometimes called Lass coordinates. Both the Kottler–Møller coordinates as well as Lass coordinates are denoted as Rindler coordinates as well.
Characteristics of the Rindler frame
The worldline of a body in hyperbolic motion having constant proper acceleration α {\displaystyle \alpha } in the X {\displaystyle X} -direction as a function of proper time τ {\displaystyle \tau } and rapidity α τ {\displaystyle \alpha \tau } can be given by
T = x sinh ( α τ ) , X = x cosh ( α τ ) {\displaystyle T=x\sinh(\alpha \tau ),\quad X=x\cosh(\alpha \tau )}
where x = 1 / α {\displaystyle x=1/\alpha } is constant and α τ {\displaystyle \alpha \tau } is variable, with the worldline resembling the hyperbola X 2 − T 2 = x 2 {\displaystyle X^{2}-T^{2}=x^{2}} . Sommerfeld showed that the equations can be reinterpreted by defining x {\displaystyle x} as variable and α τ {\displaystyle \alpha \tau } as constant, so that it represents the simultaneous "rest shape" of a body in hyperbolic motion measured by a comoving observer. By using the proper time of the observer as the time of the entire hyperbolically accelerated frame by setting τ = t {\displaystyle \tau =t} , the transformation formulas between the inertial coordinates and the hyperbolic coordinates are consequently:
with the inverse
t = 1 α artanh ( T X ) , x = X 2 − T 2 , y = Y , z = Z {\displaystyle t={\frac {1}{\alpha }}\operatorname {artanh} \left({\frac {T}{X}}\right),\quad x={\sqrt {X^{2}-T^{2}}},\quad y=Y,\quad z=Z}
Differentiated and inserted into the Minkowski metric
d s 2 = − d T 2 + d X 2 + d Y 2 + d Z 2 , {\displaystyle \mathrm {d} s^{2}=-\mathrm {d} T^{2}+\mathrm {d} X^{2}+\mathrm {d} Y^{2}+\mathrm {d} Z^{2},}
the metric in the hyperbolically accelerated frame follows as
These transformations define the Rindler observer as an observer that is "at rest" in Rindler coordinates, i.e., maintaining constant x, y, z, and only varying t as time passes. The coordinates are valid in the region 0 < X < ∞ , − X < T < X {\displaystyle 0<X<\infty ,\;-X<T<X} , which is often called the Rindler wedge, if α {\displaystyle \alpha } represents the proper acceleration (along the hyperbola x = 1 / α {\displaystyle x=1/\alpha } ) of the Rindler observer whose proper time is defined to be equal to Rindler coordinate time. To maintain this world line, the observer must accelerate with a constant proper acceleration, with Rindler observers closer to x = 0 {\displaystyle x=0} (the Rindler horizon) having greater proper acceleration. All the Rindler observers are instantaneously at rest at time T = 0 {\displaystyle T=0} in the inertial frame, and at this time a Rindler observer with proper acceleration α i {\displaystyle \alpha _{i}} will be at position X = 1 / α i {\displaystyle X=1/\alpha _{i}} (really X = c 2 / α i {\displaystyle X=c^{2}/\alpha _{i}} , but we assume units where c = 1 {\displaystyle c=1} ), which is also that observer's constant distance from the Rindler horizon in Rindler coordinates. If all Rindler observers set their clocks to zero at T = 0 {\displaystyle T=0} , then when defining a Rindler coordinate system we have a choice of which Rindler observer's proper time will be equal to the coordinate time t {\displaystyle t} in Rindler coordinates, and this observer's proper acceleration defines the value of α {\displaystyle \alpha } above (for other Rindler observers at different distances from the Rindler horizon, the coordinate time will equal some constant multiple of their own proper time). It is a common convention to define the Rindler coordinate system so that the Rindler observer whose proper time matches coordinate time is the one who has proper acceleration α = 1 {\displaystyle \alpha =1} , so that α {\displaystyle \alpha } can be eliminated from the equations. The above equation has been simplified for c = 1 {\displaystyle c=1} . The unsimplified equation is more convenient for finding the Rindler Horizon distance, given an acceleration α {\displaystyle \alpha } .
t = c α artanh ( c T X ) ≈ X ≫ c T c 2 T α X ⇒ X ≈ c 2 T α t ≈ T ≈ t c 2 α {\displaystyle {\begin{aligned}&t={\frac {c}{\alpha }}\operatorname {artanh} \left({\frac {cT}{X}}\right)\;{\overset {X\,\gg \,cT}{\approx }}\;{\frac {c^{2}T}{\alpha X}}\\&\Rightarrow X\approx {\frac {c^{2}T}{\alpha t}}\;{\overset {T\,\approx \,t}{\approx }}\;{\frac {c^{2}}{\alpha }}\end{aligned}}}
The remainder of the article will follow the convention of setting both α = 1 {\displaystyle \alpha =1} and c = 1 {\displaystyle c=1} , so units for X {\displaystyle X} and x {\displaystyle x} will be 1 unit = c 2 / α = 1 {\displaystyle =c^{2}/\alpha =1} . Be mindful that setting α = 1 {\displaystyle \alpha =1} light-second/second2 is very different from setting α = 1 {\displaystyle \alpha =1} light-year/year2. Even if we pick units where c = 1 {\displaystyle c=1} , the magnitude of the proper acceleration α {\displaystyle \alpha } will depend on our choice of units: for example, if we use units of light-years for distance, ( X {\displaystyle X} or x {\displaystyle x} ) and years for time, ( T {\displaystyle T} or t {\displaystyle t} ), this would mean α = 1 {\displaystyle \alpha =1} light year/year2, equal to about 9.5 meters/second2, while if we use units of light-seconds for distance, ( X {\displaystyle X} or x {\displaystyle x} ), and seconds for time, ( T {\displaystyle T} or t {\displaystyle t} ), this would mean α = 1 {\displaystyle \alpha =1} light-second/second2, or 299 792 458 meters/second2).
Variants of transformation formulas A more general derivation of the transformation formulas is given, when the corresponding Fermi–Walker tetrad is formulated from which the Fermi coordinates or Proper coordinates can be derived. Depending on the choice of origin of these coordinates, one can derive the metric, the time dilation between the time at the origin d t 0 {\displaystyle dt_{0}} and d t {\displaystyle dt} at point x {\displaystyle x} , and the coordinate light speed | d x | / | d t | {\displaystyle |dx|/|dt|} (this variable speed of light does not contradict special relativity, because it is only an artifact of the accelerated coordinates employed, while in inertial coordinates it remains constant). Instead of Fermi coordinates, also Radar coordinates can be used, which are obtained by determining the distance using light signals (see section Notions of distance), by which metric, time dilation and speed of light do not depend on the coordinates anymore – in particular, the coordinate speed of light remains identical with the speed of light ( c = 1 ) {\displaystyle (c=1)} in inertial frames:
The Rindler observers In the new chart (1a) with c = 1 {\displaystyle c=1} and α = 1 {\displaystyle \alpha =1} , it is natural to take the coframe field
d σ 0 = x d t , d σ 1 = d x , d σ 2 = d y , d σ 3 = d z {\displaystyle d\sigma ^{0}=x\,dt,\;\;d\sigma ^{1}=dx,\;\;d\sigma ^{2}=dy,\;\;d\sigma ^{3}=dz}
which has the dual frame field
e → 0 = 1 x ∂ t , e → 1 = ∂ x , e → 2 = ∂ y , e → 3 = ∂ z {\displaystyle {\vec {e}}_{0}={\frac {1}{x}}\partial _{t},\;\;{\vec {e}}_{1}=\partial _{x},\;\;{\vec {e}}_{2}=\partial _{y},\;\;{\vec {e}}_{3}=\partial _{z}}
This defines a local Lorentz frame in the tangent space at each event (in the region covered by our Rindler chart, namely the Rindler wedge). The integral curves of the timelike unit vector field e → 0 {\displaystyle {\vec {e}}_{0}} give a timelike congruence, consisting of the world lines of a family of observers called the Rindler observers. In the Rindler chart, these world lines appear as the vertical coordinate lines x = x 0 , y = y 0 , z = z 0 {\displaystyle x=x_{0},\;y=y_{0},\;z=z_{0}} . Using the coordinate transformation above, we find that these correspond to hyperbolic arcs in the original Cartesian chart.
As with any timelike congruence in any Lorentzian manifold, this congruence has a kinematic decomposition (see Raychaudhuri equation). In this case, the expansion and vorticity of the congruence of Rindler observers vanish. The vanishing of the expansion tensor implies that each of our observers maintains constant distance to his neighbors. The vanishing of the vorticity tensor implies that the world lines of our observers are not twisting about each other; this is a kind of local absence of "swirling". The acceleration vector of each observer is given by the covariant derivative
∇ e → 0 e → 0 = 1 x e → 1 {\displaystyle \nabla _{{\vec {e}}_{0}}{\vec {e}}_{0}={\frac {1}{x}}{\vec {e}}_{1}}
That is, each Rindler observer is accelerating in the ∂ x {\displaystyle \partial _{x}} direction. Individually speaking, each observer is in fact accelerating with constant magnitude in this direction, so their world lines are the Lorentzian analogs of circles, which are the curves of constant path curvature in the Euclidean geometry. Because the Rindler observers are vorticity-free, they are also hypersurface orthogonal. The orthogonal spatial hyperslices are t = t 0 {\displaystyle t=t_{0}} ; these appear as horizontal half-planes in the Rindler chart and as half-planes through T = X = 0 {\displaystyle T=X=0} in the Cartesian chart (see the figure above). Setting d t = 0 {\displaystyle dt=0} in the line element, we see that these have the ordinary Euclidean geometry, d σ 2 = d x 2 + d y 2 + d z 2 , ∀ x > 0 , ∀ y , z {\displaystyle d\sigma ^{2}=dx^{2}+dy^{2}+dz^{2},\;\forall x>0,\forall y,z} . Thus, the spatial coordinates in the Rindler chart have a very simple interpretation consistent with the claim that the Rindler observers are mutually stationary. We will return to this rigidity property of the Rindler observers a bit later in this article.
A "paradoxical" property Note that Rindler observers with smaller constant x coordinate are accelerating harder to keep up. This may seem surprising because in Newtonian physics, observers who maintain constant relative distance must share the same acceleration. But in relativistic physics, we see that the trailing endpoint of a rod which is accelerated by some external force (parallel to its symmetry axis) must accelerate a bit harder than the leading endpoint, or else it must ultimately break. This is a manifestation of Lorentz contraction. As the rod accelerates, its velocity increases and its length decreases. Since it is getting shorter, the back end must accelerate harder than the front. Another way to look at it is: the back end must achieve the same change in velocity in a shorter period of time. This leads to a differential equation showing that, at some distance, the acceleration of the trailing end diverges, resulting in the Rindler horizon. This phenomenon is the basis of a well known "paradox", Bell's spaceship paradox. However, it is a simple consequence of relativistic kinematics. One way to see this is to observe that the magnitude of the acceleration vector is just the path curvature of the corresponding world line. But the world lines of our Rindler observers are the analogs of a family of concentric circles in the Euclidean plane, so we are simply dealing with the Lorentzian analog of a fact familiar to speed skaters: in a family of concentric circles, inner circles must bend faster (per unit arc length) than the outer ones.
Minkowski observers
It is worthwhile to also introduce an alternative frame, given in the Minkowski chart by the natural choice
f → 0 = ∂ T , f → 1 = ∂ X , f → 2 = ∂ Y , f → 3 = ∂ Z {\displaystyle {\vec {f}}_{0}=\partial _{T},\;{\vec {f}}_{1}=\partial _{X},\;{\vec {f}}_{2}=\partial _{Y},\;{\vec {f}}_{3}=\partial _{Z}}
Transforming these vector fields using the coordinate transformation given above, we find that in the Rindler chart (in the Rindler wedge) this frame becomes
f → 0 = 1 x cosh ( t ) ∂ t − sinh ( t ) ∂ x f → 1 = − 1 x sinh ( t ) ∂ t + cosh ( t ) ∂ x f → 2 = ∂ y , f → 3 = ∂ z {\displaystyle {\begin{aligned}{\vec {f}}_{0}&={\frac {1}{x}}\cosh(t)\,\partial _{t}-\sinh(t)\,\partial _{x}\\{\vec {f}}_{1}&=-{\frac {1}{x}}\sinh(t)\,\partial _{t}+\cosh(t)\,\partial _{x}\\{\vec {f}}_{2}&=\partial _{y},\;{\vec {f}}_{3}=\partial _{z}\end{aligned}}}
Computing the kinematic decomposition of the timelike congruence defined by the timelike unit vector field f → 0 {\displaystyle {\vec {f}}_{0}} , we find that the expansion and vorticity again vanishes, and in addition the acceleration vector vanishes, ∇ f → 0 f → 0 = 0 {\displaystyle \nabla _{{\vec {f}}_{0}}{\vec {f}}_{0}=0} . In other words, this is a geodesic congruence; the corresponding observers are in a state of inertial motion. In the original Cartesian chart, these observers, whom we will call Minkowski observers, are at rest. In the Rindler chart, the world lines of the Minkowski observers appear as hyperbolic secant curves asymptotic to the coordinate plane x = 0 {\displaystyle x=0} . Specifically, in Rindler coordinates, the world line of the Minkowski observer passing through the event t = t 0 , x = x 0 , y = y 0 , z = z 0 {\displaystyle t=t_{0},\;x=x_{0},\;y=y_{0},\;z=z_{0}} is
t = artanh ( s x 0 ) , − x 0 < s < x 0 x = x 0 2 − s 2 , − x 0 < s < x 0 y = y 0 z = z 0 {\displaystyle {\begin{aligned}t&=\operatorname {artanh} \left({\frac {s}{x_{0}}}\right),\;-x_{0}<s<x_{0}\\x&={\sqrt {x_{0}^{2}-s^{2}}},\;-x_{0}<s<x_{0}\\y&=y_{0}\\z&=z_{0}\end{aligned}}}
where s {\displaystyle s} is the proper time of this Minkowski observer. Note that only a small portion of his history is covered by the Rindler chart. This shows explicitly why the Rindler chart is not geodesically complete; timelike geodesics run outside the region covered by the chart in finite proper time. Of course, we already knew that the Rindler chart cannot be geodesically complete, because it covers only a portion of the original Cartesian chart, which is a geodesically complete chart. In the case depicted in the figure, x 0 = 1 {\displaystyle x_{0}=1} and we have drawn (correctly scaled and boosted) the light cones at s ∈ { − 1 2 , 0 , 1 2 } {\displaystyle s\in \left\{-{\frac {1}{2}},\;0,\;{\frac {1}{2}}\right\}} .
The Rindler horizon The Rindler coordinate chart has a coordinate singularity at x = 0, where the metric tensor (expressed in the Rindler coordinates) has vanishing determinant. This happens because as x → 0 the acceleration of the Rindler observers diverges. As we can see from the figure illustrating the Rindler wedge, the locus x = 0 in the Rindler chart corresponds to the locus T2 = X2, X > 0 in the Cartesian chart, which consists of two null half-planes, each ruled by a null geodesic congruence. For the moment, we simply consider the Rindler horizon as the boundary of the Rindler coordinates. If we consider the set of accelerating observers who have a constant position in Rindler coordinates, none of them can ever receive light signals from events with T ≥ X (on the diagram, these would be events on or to the left of the line T = X which the upper red horizon lies along; these observers could however receive signals from events with T ≥ X if they stopped their acceleration and crossed this line themselves) nor could they have ever sent signals to events with T ≤ −X (events on or to the left of the line T = −X which the lower red horizon lies along; those events lie outside all future light cones of their past world line). Also, if we consider members of this set of accelerating observers closer and closer to the horizon, in the limit as the distance to the horizon approaches zero, the constant proper acceleration experienced by an observer at this distance (which would also be the G-force experienced by such an observer) would approach infinity. Both of these facts would also be true if we were considering a set of observers hovering outside the event horizon of a black hole, each observer hovering at a constant radius in Schwarzschild coordinates. In fact, in the close neighborhood of a black hole, the geometry close to the event horizon can be described in Rindler coordinates. Hawking radiation in the case of an accelerating frame is referred to as Unruh radiation. The connection is the equivalence of acceleration with gravitation.
Geodesics The geodesic equations in the Rindler chart are easily obtained from the geodesic Lagrangian; they are
t ¨ + 2 x x ˙ t ˙ = 0 , x ¨ + x t ˙ 2 = 0 , y ¨ = 0 , z ¨ = 0 {\displaystyle {\ddot {t}}+{\frac {2}{x}}\,{\dot {x}}\,{\dot {t}}=0,\;{\ddot {x}}+x\,{\dot {t}}^{2}=0,\;{\ddot {y}}=0,\;{\ddot {z}}=0}
Of course, in the original Cartesian chart, the geodesics appear as straight lines, so we could easily obtain them in the Rindler chart using our coordinate transformation. However, it is instructive to obtain and study them independently of the original chart, and we shall do so in this section.
From the first, third, and fourth we immediately obtain the first integrals
t ˙ = E x 2 , y ˙ = P , z ˙ = Q {\displaystyle {\dot {t}}={\frac {E}{x^{2}}},\;\;{\dot {y}}=P,\;\;{\dot {z}}=Q}
But from the line element we have ε = − x 2 t ˙ 2 + x ˙ 2 + y ˙ 2 +