# Space-Time

## Practice

### practice problem 1

#### solution

Answer it.

### practice problem 2

- Using classical euclidean geometry, determine the circumference of this orbit.
- Using classical Newtonian mechanics, determine the speed of a satellite in such an orbit.
- Using special relativity, determine the amount by which the circumference of the orbit has been shortened due to the satellite's motion. State your answer in…
- centimeters
- English inches

#### solution

Compute the circumference with a stupid amount of precision. This is the rest length.

*C*= 2π*r**C*= 2π(6,378,100 m + 642,000 m)*C*= 44108589.174932 mSet the centripetal force formula equal to Newton's law of universal gravitation.

*F*_{c}= *F*_{g}*mv*^{2}= *Gm*_{1}*m*_{2}*r**r*^{2}Solve for speed, plug in numbers, and calculate an answer.

*v*=√ *Gm**r**v*=√ (6.67 × 10 ^{−11}N m^{2}/kg^{2})(5.97 × 10 ^{24}kg)(6,378,100 m + 642,000 m) *v*= 7531.447156 m/sApply the length contraction formula to the circumference of the orbit moving at the speed of the satellite. State the answer with a stupid amount of precision. This is the contracted length.

ℓ = ℓ _{0}√⎛

⎜

⎝1 − *v*^{2}⎞

⎟

⎠*c*^{2}ℓ = 44108589.174932 m√ ⎛

⎜

⎝1 − (7531.447156 m/s) ^{2}⎞

⎟

⎠(299,792,458 m/s) ^{2}ℓ = 44108589.161013 m Subtract the contracted length from the rest length. This is part of what Gravity Probe B is trying to measure.

∆ℓ = ℓ _{0}− ℓ

∆ℓ = 44108589.174932 m− 44108589.161013 m

∆ℓ = 0.013919 mConvert to centimeters with a simple shift of the decimal point.

∆ℓ = 1.4 cm

Convert to English inches using the definition of an inch. (An inch is 0.0254 m by definition.)

∆ℓ = 0.013919 m 1 in. 1 0.0254 m ∆ℓ = 0.55 inches

OK, so it's closer to half an inch than an inch, but its of the right order. The more important thing to note is that Gravity Probe B is not looking for space-time distortion caused by

*motion*, it's looking for space-time distortion caused by*gravity*. It's not trying to confirm the predictions of*special relativity*, it's trying to confirm the predictions of*general relativity*. The two effects are of about the same size and together they make an inch.

Kip Thorne explains how to measure the "missing inch" in a talk at Stanford University (2 April 2004). | John Stewart reports on the launch of Gravity Probe B on The Daily Show (28 April 2004). |

### practice problem 3

#### solution

Answer it.

### practice problem 4

#### solution

Answer it.