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# 物理代写|相对论代写Theory of relativity代考|CRN9074 THE SCHWARZSCHILD METRIC

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## 物理代写|相对论代写Theory of relativity代考|THE SCHWARZSCHILD METRIC

Introduction. This chapter begins a long set of chapters that explore the physical meaning of the Schwarzschild metric. This metric tensor is a spherically symmetric, time-independent solution to the Einstein equation in a vacuum, and is thus suitable for describing the spacetime in the empty space surrounding a spherical, static object. This metric therefore plays much the same role in general relativity that the formula for the gravitational field of a point mass does in Newtonian gravity and the formula for the electric field of a point charge does in electrostatics.

Spherical Coordinates for Flat Spacetime. According to box 5.6, the metric equation for latitude-longitude coordinates $\theta, \phi$ on a $2 \mathrm{D}$ spherical surface is given by
$$d s^2=R^2 d \theta^2+R^2 \sin ^2 \theta d \phi^2$$
where $R$ is the radius of the spherical surface (in three dimensions). We can label any event in an ordinary flat spacetime with spherical coordinates $r, \theta$, and $\phi$, where $r$ (now a variable) specifies the radius of the sphere around the origin on which the event lies, and $\theta$ and $\phi$ specify the event’s latitude and longitude coordinates on that surface. Therefore, the metric for spherical coordinates in flat spacetime should be
$$d s^2=-d t^2+d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$

## 物理代写|相对论代写Theory of relativity代考|The :Sc.11war:lsc:n1Jlct Metric

The Schwarzschild Metric. In 1916, Karl Schwarzschild (who at the time was fighting in the trenches of World War I and who died shortly thereafter) discovered that the following metric satisfied the Einstein field equations in empty space:
$$d s^2=-\left(1-\frac{r_s}{r}\right) d t^2+\left(1-\frac{r_s}{r}\right)^{-1} d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$
where $r_s$ is a constant with units of length called the Schwarzschild radius. (We will verify that this metric is indeed a solution to the Einstein equation later in the course.) This metric is spherically symmetric (note that surfaces of constant $r$ have the same metric as that for latitude-longitude coordinates on the surface of a sphere and so have the geometry of a sphere, and the other components of the metric depend only on $r$ ), time-independent, and becomes the flat space metric in the limit as $r \rightarrow \infty$. This metric is therefore a suitable candidate for that describing the spacetime in the empty space surrounding a spherical, static object.

## 物理代写|相对论代写Theory of relativity代考|THE SCHWARZSCHILD METRIC

$$d s^2=R^2 d \theta^2+R^2 \sin ^2 \theta d \phi^2$$

$$d s^2=-d t^2+d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$

## 物理代写相对论代写Theory of relativity代考|The :Sc.11war:Isc:n1Jlct Metric

$$d s^2=-\left(1-\frac{r_s}{r}\right) d t^2+\left(1-\frac{r_s}{r}\right)^{-1} d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$

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