# Proper distance, Area and Volume given a Metric

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1. Feb 14, 2015

### unscientific

1. The problem statement, all variables and given/known data

(a) Find the proper distance
(b) Find the proper area
(c) Find the proper volume
(d) Find the four-volume

2. Relevant equations

3. The attempt at a solution

Part (a)

Letting $d\theta = dt = d\phi = 0$:

$$\Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 - \frac{AR}{2} \right)$$

Part (b)

At $r=R$:

$$A = R^2 \int_0^{2\pi} d\phi \int_0^{\pi} sin \theta d\theta$$
$$A = 4\pi R^2$$

Part (c)

$$V = \int_0^R r^2 \left( 1 - Ar^2 \right) dr \int_0^{\pi} sin \theta d\theta \int_0 ^{2\pi} d\phi$$
$$V = \frac{4}{3} \pi R^3 \left( 1 - \frac{3}{5} AR^2 \right)$$

Part (d)

$$V_4 = c\int_0^R r^2 \left( 1 - Ar^2\right)^2 dr \int_0^T dt \int_0^{\pi} sin \theta d\theta \int_0^{2\pi} d\phi$$

$$V_4 = \frac{4}{3} \pi R^3 \cdot cT \cdot \left( 1 - \frac{6}{5}AR^2 + \frac{3}{7}A^2R^4 \right)$$

Is this the correct method? This question seems a little too straightforward..

2. Feb 16, 2015

### DEvens

I would guess that, if you look forward in your text, you will find this metric gets a lot of work. For example, you may be doing some interesting work studying the place where r^2 = 1/A.

So quite likely this is just a warm-up getting some interesting features of the metric.