- #1
Dylan H
- 14
- 0
I'm trying to understand the relationship between the spacetime metric [itex]\Delta s^2 = \Delta x^2 - c^2\Delta t^2[/itex] and the simple formulas for time dilation and length contraction in special relativity [itex]x = \frac{1}{\gamma} \bar{x}[/itex] and [itex]t = \gamma \cdot \bar{t}.[/itex]
Suppose from an inertial reference frame, I observe an object passing through two points [itex](x_1, ct_1)[/itex] and [itex](x_2, ct_2)[/itex], and I want to consider how these two points appear from a different inertial reference frame. I suppose that a change in reference frame amounts to a particular structure-preserving transformation of Minkowski space, and while I'm not sure exactly which structure ought to be preserved, at least spacetime separation ought to be preserved. (So if the original points were separated by [itex]\Delta s^2[/itex] they should remain so in any reference frame.)
Depending on the sign of [itex]\Delta s^2[/itex], there is either a reference frame in which the two events occur at the same time, or one in which they occur at the same place. Proper distance is the spatial separation between two events from a reference frame in which they are simultaneous, and proper time is the temporal separation between two events from a reference frame in which they occur in the same place.
From what I understand, length contraction implies that the spatial separation between two events is never more than the proper distance, and similarly time dilation implies that temporal separation never less than the proper time.
I have trouble reconciling this with the following: if in one reference frame I measure the separation between the two events as [itex](\Delta x_A, c\Delta t_A)[/itex] and in another as [itex](\Delta x_B, c\Delta t_B)[/itex], then
$$\Delta x_A^2 - (c\Delta t_A)^2 = \Delta s^2 = \Delta x_B^2 - (c\Delta t_B)^2.$$
But if the measurements are simulatenous in reference frame A, so that [itex]\Delta t _A = 0[/itex] and[itex]\Delta x_A [/itex] is the proper length, it seems like we get
$$\Delta x_A^2 = \Delta x_B^2 - (c\Delta t_B)^2$$
which suggests that the spatial separation [itex]\Delta x_B [/itex] measured in any other reference frame is always longer than the proper length. Something's wrong --- please advise.
Suppose from an inertial reference frame, I observe an object passing through two points [itex](x_1, ct_1)[/itex] and [itex](x_2, ct_2)[/itex], and I want to consider how these two points appear from a different inertial reference frame. I suppose that a change in reference frame amounts to a particular structure-preserving transformation of Minkowski space, and while I'm not sure exactly which structure ought to be preserved, at least spacetime separation ought to be preserved. (So if the original points were separated by [itex]\Delta s^2[/itex] they should remain so in any reference frame.)
Depending on the sign of [itex]\Delta s^2[/itex], there is either a reference frame in which the two events occur at the same time, or one in which they occur at the same place. Proper distance is the spatial separation between two events from a reference frame in which they are simultaneous, and proper time is the temporal separation between two events from a reference frame in which they occur in the same place.
From what I understand, length contraction implies that the spatial separation between two events is never more than the proper distance, and similarly time dilation implies that temporal separation never less than the proper time.
I have trouble reconciling this with the following: if in one reference frame I measure the separation between the two events as [itex](\Delta x_A, c\Delta t_A)[/itex] and in another as [itex](\Delta x_B, c\Delta t_B)[/itex], then
$$\Delta x_A^2 - (c\Delta t_A)^2 = \Delta s^2 = \Delta x_B^2 - (c\Delta t_B)^2.$$
But if the measurements are simulatenous in reference frame A, so that [itex]\Delta t _A = 0[/itex] and[itex]\Delta x_A [/itex] is the proper length, it seems like we get
$$\Delta x_A^2 = \Delta x_B^2 - (c\Delta t_B)^2$$
which suggests that the spatial separation [itex]\Delta x_B [/itex] measured in any other reference frame is always longer than the proper length. Something's wrong --- please advise.