Back to the Zoom Tool Put your scrollwheel or trackpad to good use! Check out new settings that allow you to zoom and scroll easily. Play Now. Join a Game. Marketplace Toggle Dropdown. 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.

It is worthwhile to also introduce an alternative frame, given in the Minkowski chart by the natural choice. Transforming these vector fields using the coordinate transformation given above, we find that in the Rindler chart in the Rinder wedge this frame becomes. 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.

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.

For the moment, we simply consider the Rindler horizon as the boundary of the Rindler coordinates. 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. The geodesic equations in the Rindler chart are easily obtained from the geodesic Lagrangian ; they are.

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. This gives the fourth first integral, namely.

The complete seven parameter family giving any null geodesic through any event in the Rindler wedge, is. See the figure. The fact that in the Rindler chart, the projections of null geodesics into any spatial hyperslice for the Rindler observers are simply semicircular arcs can be verified directly from the general solution just given, but there is a very simple way to see this. A static spacetime is one in which a vorticity-free timelike Killing vector field can be found.

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Sebastian R. Rinderle - Josephine A. Joseph P Rinderle - Anton Rinderle - Maria Anna Rinderle - Ida Anna Rinderle - Otto Marttmet Rinderle - Henry A Rinderle - Edmund George Rinderle - PubMed: Oie HK, et al. Documentation Basic Documentation Product Sheet. Certificate of Analysis. Cell Micrograph. Login Please enter a username. Please enter a password. Invalid username or password. Rattus norvegicus , rat. Culture Properties.

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