According to the principles of special relativity, the speed of light (denoted as 'c') is the maximum speed at which information or matter can travel through space. As a result, nothing with mass can reach or exceed the speed of light. Therefore, the scenario you described, where two spaceships are each traveling at 0.6 times the speed of light (0.6c) relative to the Earth, would not be possible under the constraints of special relativity.
If both spaceships were moving away from Earth in opposite directions at speeds close to the speed of light, their relative velocities would not simply add up to 1.2c. Instead, the relativistic velocity addition formula would need to be used to calculate their relative velocity.
The relativistic velocity addition formula is given by: v' = (v1 + v2) / (1 + (v1 * v2) / c^2)
Using this formula, if two spaceships were each traveling at 0.6c relative to the Earth, their relative velocity would be:
v' = (0.6c + (-0.6c)) / (1 + (0.6c * -0.6c) / c^2) = 0 / (1 - 0.36) = 0 / 0.64 = 0
According to this calculation, the relative velocity between the two spaceships would be 0, meaning that they would be at rest relative to each other. This outcome is a consequence of the relativistic velocity addition formula and the fact that velocities close to the speed of light do not add up linearly.
In summary, based on the principles of special relativity, two spaceships traveling at 0.6c relative to Earth would not be able to approach each other at speeds greater than the speed of light (1c). Their relative velocity would be calculated using the relativistic velocity addition formula, resulting in a value of 0, indicating that they would be at rest relative to each other.