You Are Moving Through TIME At The Speed Of Light: Time Dilation Intuition

Monday, September 24, 2018

You Are Moving Through TIME At The Speed Of Light: Time Dilation Intuition

Time Dilation Intuition

In the post discussing what space-time is, I said that it is as if we move through space-time at the speed of light.
If we are stationary in space, then we move through time at the speed of light.
As we begin to move through space, some of that motion through time is diverted to motion through space and this is why time dilation occurs. With what we know about the space-time interval; we can have a deeper think about what this really means.

Lets consider 2 frames of reference. Our frame of reference is one watching Bob moving at speed v. The other is Bob's frame where we move in the opposite direction at speed v.

Lets look at the space-time interval between 2 events in those frames. The events we will look at are Bob's watch going tick and Bob's watch going tock.
From Bob's POV
The location of his watch never changes. This means for him the space-time interval is
Δx'² - c²Δt'² = Δs'²
where Δx' is 0
- c²Δt'² = 0
and remember this time is the proper time as it is the time between two events in the frame in which they happen in the same location
- c²Δτ² = 0

From Our POV
The space-time interval (distance in space-time between tick and tock) is
Δx² - c²Δt² = Δs²

The space-time interval is invariant between reference frames and so we can write the following
Δs'² = Δs²
Δx² - c²Δt² = - c²Δτ²
Δx² + c²Δτ² = c²Δt²
Δx²/Δt² + c²Δτ²/Δt² = c²
(Δx/Δt)² + c²Δτ²/Δt² = c²

The distance between the tick and tock divided by the time between them is the speed of Bob. He travels this distance in that time

v²+ c²Δτ²/Δt² = c²
This equation tells us why we can think about time dilation in the intuitive way described above. c² is a constant and so the value never changes.

When v = 0
Bob is stationary in space and moving purely through time.
The equation becomes c²Δτ²/Δt² = c²
The only way that this can be true is if Δτ²/Δt²= 1. Since Bob is stationary in space, our time (Δt) and his time (Δτ) elapse at the same rate and so Δτ²/Δt²= 1 is true.

As v increases
c²Δτ²/Δt²must decrease to keep c² the same.
For this to happen Δτ/Δt must decrease.
The proper time between a tick and tock never changes. Bob will measure the same time no matter how fast he moves because the clock is always stationary relative to him.
The time we measure can change Δt however. As the speed of Bob increases, time slows down from our perspective. This means the time between each tick and a tock increases from our POV. Δt increases and this means Δτ/Δt decreases like we said. This way we the constant c² does not change.
We conclude that increasing Bob's speed results in the slowing down of time.

To summarise this simply
Motion Through Space + Motion Through Time = A Fixed Constant
This means the only way to increase your motion through space is to decrease your motion through time.

The Sad Life Of A Photon Part 3

When v = c; Δτ/Δt must be 0. The only way this can be the case is if the time we measure between a tick and tock on his clock tends to ∞. If the time between a tick and tock is infinite; Bob does not experience time. This is what we predicted earlier when we first spoke about this concept of moving through time at the speed of light.

You now have two ways to think of time dilation. You can derive it from the thought experiment where light is moving up and down in one frame and diagonally in the another. In this example, the time between each tick and tock is greater when you see the light clock moving by you. This means the time elapses slower when you see the light clock moving by you and so time dilation occurs for moving clocks.

You can also think about it in terms of moving through space-time and that as we begin to move through space, some of that motion through time is diverted to motion through space.

You can also derive the equation for time dilation from the equation above showing
Motion through space + Motion through time = A constant

This concludes the posts involving the space-time interval. In the next set of posts we will work towards deriving E = mc².

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