- #1
Frank Castle
- 580
- 23
As I understand it, the proper length, ##L## of an object is equal to the length of the space-like interval between the two space-time points labelling its endpoints, i.e. (in terms of the corresponding differentials) $$dL=\sqrt{ds^{2}}$$ (using the "mostly plus" signature).
Furthermore, this is only equal to the objects rest length, ##L_{0}## when the two space-time points are simultaneous, i.e in the inertial frame for which ##dt=0## and as such (in terms of the corresponding differentials) $$dL_{0}=\sqrt{dx^{2}+dy^{2}+dz^{2}}$$
I'm not sure I quite understand this last point, however. What is the intuition for why the two events have to be simultaneous?
For example, suppose one wishes to measure the length of a rod and one is in an inertial frame that is at rest with respect to the rod. Utilising the definition of proper length, ##dL_{0}=\sqrt{ds^{2}}##, why is it that the two events (labelling the two endpoints of the rod) have to be simultaneous, since if one is at rest with respect to the rod, then surely if one measures the endpoints of the rod at two different times to determine the distance this won't affect the result as the endpoints are at rest with respect to the observer?!
Is the point that the rest length of the rod is simply determined by Pythagoras's theorem: $$dL_{0}=\sqrt{dx^{2}+dy^{2}+dz^{2}}$$ and so the proper length, ##L## is equal to the rest length, ##L_{0}## only in the case where the two events labelling the endpoints of the rod are simultaneous, i.e. when ##dt=0##, such that $$dL=\sqrt{ds^{2}}=\sqrt{dx^{2}+dy^{2}+dz^{2}}=dL_{0}\;?$$ (Similar to how the proper time of an object is defined as the length of the time-like interval between two space-time points along the objects worldline, ##d\tau=\frac{1}{c}\sqrt{-ds^{2}}##, and this coincides with coordinate time, ##t## in the case where one is in the rest frame of the object, i.e. (in terms of the corresponding differentials) when ##dx=dy=dz=0##, such that ##d\tau=\frac{1}{c}\sqrt{-ds^{2}}=dt##?!)
Furthermore, this is only equal to the objects rest length, ##L_{0}## when the two space-time points are simultaneous, i.e in the inertial frame for which ##dt=0## and as such (in terms of the corresponding differentials) $$dL_{0}=\sqrt{dx^{2}+dy^{2}+dz^{2}}$$
I'm not sure I quite understand this last point, however. What is the intuition for why the two events have to be simultaneous?
For example, suppose one wishes to measure the length of a rod and one is in an inertial frame that is at rest with respect to the rod. Utilising the definition of proper length, ##dL_{0}=\sqrt{ds^{2}}##, why is it that the two events (labelling the two endpoints of the rod) have to be simultaneous, since if one is at rest with respect to the rod, then surely if one measures the endpoints of the rod at two different times to determine the distance this won't affect the result as the endpoints are at rest with respect to the observer?!
Is the point that the rest length of the rod is simply determined by Pythagoras's theorem: $$dL_{0}=\sqrt{dx^{2}+dy^{2}+dz^{2}}$$ and so the proper length, ##L## is equal to the rest length, ##L_{0}## only in the case where the two events labelling the endpoints of the rod are simultaneous, i.e. when ##dt=0##, such that $$dL=\sqrt{ds^{2}}=\sqrt{dx^{2}+dy^{2}+dz^{2}}=dL_{0}\;?$$ (Similar to how the proper time of an object is defined as the length of the time-like interval between two space-time points along the objects worldline, ##d\tau=\frac{1}{c}\sqrt{-ds^{2}}##, and this coincides with coordinate time, ##t## in the case where one is in the rest frame of the object, i.e. (in terms of the corresponding differentials) when ##dx=dy=dz=0##, such that ##d\tau=\frac{1}{c}\sqrt{-ds^{2}}=dt##?!)