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Austin0
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Are accelerating lines of simultaneity correct??
This is an idea and question that I have been considering for a long time but put on hold while I sought a firmer grasp of the geometry of Minkowski spacetime graphs.
My current understanding is this:
Lines of simultanieity wrt inertial frames are a graphing of the interrelationship between the clocks and positions of one frame with another.
As such the points of intersection represent accurate colocations of clocks and ruler points. These are both quantitatively and geometrically (with tranformation) valid both spatially and temporally.
They are equivalent to the clocks and rulers of the frames themselves , whether actual or virtual.
This can be taken as consistent with reality as observers from both frames at these points of intersection will agree on the spatial coordinates and times of these events.
But in the case of accelerating frames this appears to no longer be valid.
The spacetime locations of a point in the accelerating frame as graphed as the worldline, is of course accurate in the coordinates of the rest frame, but the resulting lines of simultaneity no longer conform. They represent a dynamic non-uniform metric mapped onto what is essentially a Euclidean matirx (with single transform).
Taken in sum they perhaps represent varying degrees of curvature into the z plane.
SO the spatial distance between the point of the worldline and a point of intersection is no longer geometrically valid. Neither is the direction.
This is most glaringly obvious where these lines intersect. This represents the simultaneous colocation of two temporally separated points of a single frame.
As these lines can be taken to represent an extension of the frame itself this would also mean colocation of disparate clocks and observers. Clearly this can not be consistent with reality.
Looking outward past the intersection at the diverging lines it is clear that there is the representation of temporal reordering and causality reversal.
It may be suggested that this simply means that the lines are only accurate up to the points of intersection but I think this is not the case. I think they are spatially and temporally inaccurate throughout, with the degree of error a function of spatial and temporal distance from their origens at the worldline and their temporal separation on that worldline.
I have not done the math to confirm this for two reasons
1) Time
2) I unfortunately lack the calculus to derive instantaneous velocities from the slope of worldline tangents.
I am aware that due to this lack of mathematical corroboration , many will dismiss this out of hand but I am hoping that someone with the math skills will find the question interesting enough to run some numbers and put it to rest (or not).
If anybody either does not understand or disagrees with my idea of the equivalance of hyperplanes of simultaneity and the frame itself there is a recent thread addressing this and I welcome all objections and criticisms https://www.physicsforums.com/showthread.php?t=415501"
Thanks
This is an idea and question that I have been considering for a long time but put on hold while I sought a firmer grasp of the geometry of Minkowski spacetime graphs.
My current understanding is this:
Lines of simultanieity wrt inertial frames are a graphing of the interrelationship between the clocks and positions of one frame with another.
As such the points of intersection represent accurate colocations of clocks and ruler points. These are both quantitatively and geometrically (with tranformation) valid both spatially and temporally.
They are equivalent to the clocks and rulers of the frames themselves , whether actual or virtual.
This can be taken as consistent with reality as observers from both frames at these points of intersection will agree on the spatial coordinates and times of these events.
But in the case of accelerating frames this appears to no longer be valid.
The spacetime locations of a point in the accelerating frame as graphed as the worldline, is of course accurate in the coordinates of the rest frame, but the resulting lines of simultaneity no longer conform. They represent a dynamic non-uniform metric mapped onto what is essentially a Euclidean matirx (with single transform).
Taken in sum they perhaps represent varying degrees of curvature into the z plane.
SO the spatial distance between the point of the worldline and a point of intersection is no longer geometrically valid. Neither is the direction.
This is most glaringly obvious where these lines intersect. This represents the simultaneous colocation of two temporally separated points of a single frame.
As these lines can be taken to represent an extension of the frame itself this would also mean colocation of disparate clocks and observers. Clearly this can not be consistent with reality.
Looking outward past the intersection at the diverging lines it is clear that there is the representation of temporal reordering and causality reversal.
It may be suggested that this simply means that the lines are only accurate up to the points of intersection but I think this is not the case. I think they are spatially and temporally inaccurate throughout, with the degree of error a function of spatial and temporal distance from their origens at the worldline and their temporal separation on that worldline.
I have not done the math to confirm this for two reasons
1) Time
2) I unfortunately lack the calculus to derive instantaneous velocities from the slope of worldline tangents.
I am aware that due to this lack of mathematical corroboration , many will dismiss this out of hand but I am hoping that someone with the math skills will find the question interesting enough to run some numbers and put it to rest (or not).
If anybody either does not understand or disagrees with my idea of the equivalance of hyperplanes of simultaneity and the frame itself there is a recent thread addressing this and I welcome all objections and criticisms https://www.physicsforums.com/showthread.php?t=415501"
Thanks
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