c = 1

. If

B

is in the future lightcone of

A

and

C

in that of

B

, then

t_{B} - t_{A} \geq ∣ x_{B} - x_{A} ∣, t_{C} - t_{B} \geq ∣ x_{C} - x_{B} ∣.

(1)

Adding and using the triangle inequality gives

t_{C} - t_{A} \geq ∣ x_{B} - x_{A} ∣ + ∣ x_{C} - x_{B} ∣ \geq ∣ x_{C} - x_{A} ∣.

(2)

Thus the statement is true.

ii

 0  0

Solution

 0  0

False. Take

A = (0, 0), B = (1, 1), C = (- 1, 1),

(1)

with coordinates written as

(x, t)

. Both

B

and

C

lie on the future lightcone of

A

, but

B

and

C

are simultaneous and spatially separated, so neither is in the future lightcone of the other.

iii

 0  0

Solution

 0  0

True. The assumptions give

t_{B} - t_{A} \geq ∣ x_{B} - x_{A} ∣, t_{A} - t_{C} \geq ∣ x_{A} - x_{C} ∣.

(1)

Adding and applying the triangle inequality proves

t_{B} - t_{C} \geq ∣ x_{B} - x_{A} ∣ + ∣ x_{A} - x_{C} ∣ \geq ∣ x_{B} - x_{C} ∣,

(2)

B

is in the future lightcone of

C

iv

 0  0

Solution

 0  0

False. With

(x, t)

coordinates, take

A = (0, 0), B = (1, 1), C = (- 1, 1) .

(1)

Both separations from

A

are lightlike, but

B - C = (2, 0)

is spacelike.

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