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# 物理代写|广义相对论代写General Relativity代考|PHYS764 The Metric hquat10111

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## 物理代写|广义相对论代写General Relativity代考|The Metric hquat10111

The Metric Equation. The spacetime interval $\Delta s$ between two events is defined by the metric equation
$$\Delta s^{2}=-\Delta t^{2}+\Delta x^{2}+\Delta y^{2}+\Delta z^{2}$$
The squared spacetime interval $\Delta s^{2}$ corresponds to the negative square of the time interval between two events measured in an IRF where they occur at the same place ( $\Delta x=\Delta y=\Delta z=0$ ) or the squared spatial distance between those events in an IRF where they occur at the same time $(\Delta t=0)$. The spacetime interval is important because it is a frame-independent measure of the events’ separation: observers in all IRFs agree on the value of $\Delta s$ between a given pair of events (see box $2.5$ )! The spacetime interval and the metric equation are to spacetime what distance between two points and the Pythagorean theorem are to ordinary plane geometry. This crucial equation provides the main link between special and general relativity.

The choice of overall sign in equation $2.3$ follows a well-established convention in general relativity that highlights the spatial aspect of the interval (and its analogy to distance) over its time aspect.

## 物理代写|广义相对论代写General Relativity代考|Categories of Spacetime Intervals

Categories of Spacetime Intervals. Spacetime intervals are called
spacelike if $\Delta s^{2}>0 \Rightarrow \Delta x^{2}+\Delta y^{2}+\Delta z^{2}>\Delta t^{2}$
lightlike if $\Delta s^{2}=0 \Rightarrow \Delta x^{2}+\Delta y^{2}+\Delta z^{2}=\Delta t^{2}$
timelike if $\Delta s^{2}<0 \Rightarrow \Delta t^{2}>\Delta x^{2}+\Delta y^{2}+\Delta z^{2}$
These distinctions arise only because of the minus sign in the metric equation: since there is no minus sign in the corresponding Pythagorean theorem for ordinary space, there is only one kind of distance. Since observers in all IRFs agree on the sign of $\Delta s^{2}$, all will agree on how to classify the interval between a given pair of events.

These classifications are important because events with a spacelike interval cannot be causally connected, because the order in time of such events is frame-dependent (see box 2.6). This means that no particle, message, or other causal effect can travel faster than the speed of light in an IRF.

## 物理代写|广义相对论代写General Relativity代考|The Metric hquat10111

$$\Delta s^{2}=-\Delta t^{2}+\Delta x^{2}+\Delta y^{2}+\Delta z^{2}$$

## MATLAB代写

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