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# 物理代写|广义相对论代写General Relativity代考|PHYS515 Spacetime Diagrams. A spacetime diagram

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## 物理代写|广义相对论代写General Relativity代考|Spacetime Diagrams. A spacetime diagram

Spacetime Diagrams. A spacetime diagram is a convenient way to display the relationship between events in spacetime. A spacetime diagram is a two or possibly threedimensional graph with a vertical time axis and horizontal spatial axis or axes. (All axes are conventionally given the same scale.) An event in spacetime is represented by a point on the diagram. We think of a particle’s trajectory through spacetime as being a connected sequence of events called the particle’s worldline, which is represented in a spacetime diagram by a connected sequence of points-i.e., a line (maybe a curve) that shows its spatial position as a function of time $t$ (along the vertical axis). An object’s speed at any given instant is the inverse slope of the curve representing its worldline on the diagram, evaluated at that instant. Problems P2.1-P2.4 will give you practice creating and reading such diagrams.

Any light emitted by a given event $E$ will expand (as time passes) into a spherical surface centered on the event. The projection of this sphere on the $x y$ plane is a circle whose radius expands with time. If we draw this on a spacetime diagram, the expanding ring looks like a cone (see figure 2.3). We call this the event’s light cone. The light cone is important because the worldline of any particle traveling through $E$ must lie within this light cone (because no particle can travel faster than light), and all events that can be caused by $E$ must lie within this cone.

Note that because the metric of spacetime is different than the Pythagorean theorem of ordinary space, the apparent separation of points on a spacetime diagram does not give us any indication of the spacetime interval between them. For example, the spacetime interval between events $E$ and $L$ in figure $2.3$ is zero!

## 物理代写|广义相对论代写General Relativity代考|Proper Time

Proper Time. We can calculate the proper time $\Delta \tau$ measured between two events $A$ and $B$ by a clock traveling along a given worldline connecting those events using a method analogous to how we would calculate the path length along a curve in ordinary space. (1) We first imagine dividing the worldline into infinitesimal segments. (2) We use the metric equation to calculate what an imaginary inertial clock would measure while traversing each segment. If the segment is small enough to be reasonably straight, the time $d \tau$ registered by the actual clock should be essentially the same. (3) We then sum $d \tau$ over all segments. The result (see box $2.7$ ) is that
$$\Delta \tau=\int \sqrt{-d s^2}=\int_{t_A}^{t_\beta} \sqrt{1-v^2} d t$$

## 物理代写|广义相对论代写广义相对论代考|固有时

$$\Delta \tau=\int \sqrt{-d s^2}=\int_{t_A}^{t_\beta} \sqrt{1-v^2} d t$$

## MATLAB代写

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