What is the maximum ratio in the rate of change in time in reference to object $A$ which is standing still and object $B$ which is moving at the speed of light?

# [Physics] the maximum time dilation between two objects, if one is standing still and the other is moving at $c$

observersreference framesspecial-relativityspeed-of-lighttime

#### Related Solutions

Everything you have said describing the situation in your question is correct; Person A and Person B disagree about how much time elapses on person A's clock between the two events. (The first event is Person B leaving Star 1 and the second event is Person B arriving at Star 2) This is not a logical contradiction. It stems from the relativity of simultaneity and the fact that the time between two events is different in different reference frames.

The time between two events is minimized when the spatial separation between them is zero because the interval

$$\Delta s^2 = \Delta t^2 - \Delta x^2$$

is invariant (the same for everyone). Person B therefore perceives the minimum possible time between the two events, which is 5 years. Person A perceives some spatial separation between the events, and so perceives a longer time between them (10 years).

We can use this information to work out the speed $v$. For Person A, $\Delta t^2 - \Delta x^2 = 5^2$ because that's the answer for Person B, and it must be the same for A. We know $\Delta t^2 = 100$, so

$$100 - \Delta x^2 = 25$$

or

$$ \Delta x = \sqrt{75} = 5\sqrt{3}$$

$v$ is then

$$v = \frac{\Delta x}{\Delta t} = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}$$

The situation is not symmetric with respect to A and B because A is not moving relative to the stars, but B is. The existence of the stars breaks the symmetry. A symmetric situation would be if A and B start at their own stars, then meet in the middle.

Another symmetric scenario would be to let B start moving away from A. When A's clock reads 10 years, ask her what B's clock reads. When B's clock reads 10 years, ask him what A's clock reads. In that case, both would say that the other's clock reads 5 years.

So, if the setup of the problem is symmetric with respect to A and B, their answers should be, also. Because this problem does not have that symmetry, the answers A and B give do not have the symmetry.

Finally, you might be concerned that Person A thinks the time between the two events is 10 years, but according to Person B, Person A's clock reads only 2.5 years elapsed. This is due to the relativity of simultaneity. According to Person B, he is arriving at Star 2 and checking Person A's clock simultaneously. Those events have a big spatial separation, though. According to Person A, they are not simultaneous. Person A thinks Person B has checked her clock too soon.

## Best Answer

Objects, defined as things with mass, don't move at the speed of light. The time dilation factor is

$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

and it has no limit - it diverges at $v\to c$. For speeds very close to the speed of light, we could define $\epsilon = \frac{c - v}{c}$, then we'd have $\gamma \sim \frac{1}{\sqrt{2\epsilon}}$ This shows how much time slows down for speeds very near the speed of light. Here's a picture

As gamma shoots up to infinity, the time dilation factor becomes arbitrarily large. If you want the clock to go 1/100 as fast, or one millionth as fast, or one quadrillionth, that can be done by going very close to the speed of light (just solve for $\epsilon$ in the above).