“It’s all relative.” If you could have a Google search for the term relative back in 1905, you’d say without a doubt that we can thank Albert Einstein for that. From it, you may have heard that space informs time, while time informs space, a unified form of spacetime. Let’s dive into what this means and how it works.
We’ve already explained how we have different dimensions expressed in the form of different coordinates. We know what three dimensions are: length, width, & height. We would represent them as follows (x, y, z). We call this Euclidean space, because we are able to apply the Pythagorean theorem to uncover the properties of a triangle. To uncover this internal space, we would use the following formula:
Why are we using Δ for each dimension? That’s to express distance between two different objects. We would actually represent this as a vector, so movement from one point to another would be the squaring of two vectors, as we have it shown above in our refined Pythagorean theorem.
The notion of time as the fourth dimension was also something we discussed and so we may naturally be inclined to express it as follows: (x, y, z, t). This is putting time on the same footing as the the three spatial dimensions – so if we use the Pythagorean theorem to determine what this internal space would look like, we would use:
This would shift us from a spatial coordinate system to a spacetime coordinate system. However, this is incorrect. Think about it, we’re measuring x, y, and z in some spatial unit of measurement, but time wouldn’t be measured in that way. How do we overcome this?
When Albert Einstein was working in a patent office, he was thinking about how reality would be experienced at very fast speeds. In one of his thought exercises, he thought about how a train moving fast on train tracks would look like it is moving fast to someone standing on the platform, but that when a train running parallel in the same direction is observed from his train, it would look as though it isn't moving. This is the relativity effect. Measuring time would be a difficult thing to do between two objects unless you have some base measurement to relate the other object to. What happens when there are multiple bodies moving at different speeds? Without the appropriate properties to measure them, you’ll never get a distinct read on the real speed. To the person standing on the platform, they’re stationary so that would be a good frame of reference, but how do you resolve this everywhere in space?
Let there be light! In 1676, the Danish astronomer Ole Roemer discovered the speed of light by timing the eclipses of Jupiter’s moon, Io. He discovered that light’s speed is constant, around 186,000 miles/second. Einstein used this as a reliable ruler to baseline all measurements in the universe against. In a world where everything is relative to another, the speed of light can be the anchoring needed.
So, we revisit the equation from above and find that it would need light to be anchored to time
Now, our time component is anchored to the speed of light, which can be measured in miles, meters, kilometers, however we choose. If we say, meters, then we could reformulate our variables as follows:
But now that we have our 4 dimensions represented cleanly within the Pythagorean theorem, let’s see if the statement we have makes sense. We can assume that our three spatial dimensions in our formula above equate to a distance of 10 meters for argument’s sake. Rather than adding each of the dimensions, we’ll subtract so we can try and uncover the time that would lapse. We’re left with the following:
In order to find the minimum allowed time for time to propagate between the two distances, we’ll make Δs2 = 0. If there is a minimum amount of time for movement between two points, then that would also imply a maximum velocity, since
With a denominator that is as small as possible, the velocity increases, hence the maximum velocity uncovered through the minimum time interval. Let’s take a second look at our equations again.
If the speed of light is the maximum velocity for any possible shortest distance, then we have a speed limit in our universe, and it’s light! Not only is the speed of light a great constant, but we can now use it reliably throughout the universe as well. So looking at our old equation below, it wouldn’t make sense that we would add the distance from the time interval that uses the speed of light.
Instead, it would make more sense to subtract from it and get the following:
With this equation, we find that Euclidean space isn’t what we’re dealing with here, instead it is Minkowski space. Because time and space are now intricately woven, we can play with our parameters and find that space influences time, which influences space; we have our spacetime which is a single fabric in our universe!
Transformation between time and space acts as its own symmetry, which are called boosts in physics. So in our example with the two trains moving in parallel at the same speed, it seems as though they aren’t moving. This is the relativity effect, where two coordinate systems move at the same velocity. It’s a symmetry because you can’t tell if you’re moving or not, which matches our definition of applying symmetry and it seems as though there is no change.
There are other symmetries as well, known as translations. Moving from one location to another is a spatial translation (x → x + 𝝐) and moving from one point in time to another is a temporal translation (t → t + 𝝐). These translations are part of the symmetry group P(1, 3), which is known as the Poincare group and it is used in Minkowski space, along with rotations and boosts. If you liked what you read, stay tuned for my book Manifest, coming soon!