Special Relativity & Space-time tradeoff

[WIP. Will appreciate feedback.]

This article tries to highlight the analogy between the concept of space-time tradeoff in computer science and the concepts of time dilation and length contraction in special relativity.

Space-time tradeoff in Computer Science

In general, for a computer program, there is a trade-off between data storage used by it and its execution time. Depending on the application, a program trades for increased execution time for decreased storage or vice versa.

Special Relativity

Einsteins theory of relativity postulates:

  1. the laws of physics are identical for all inertial frames of reference
  2. the speed of light in a vacuum is the same for all observers.

The rest of the article assumes the speed of light in a vacuum.

The concepts of length contraction and time dilation come from Lorentz transformation and these two postulates. In the simplest words, for an observer at rest, the length of a moving object appears to be smaller than its original length at rest (length contraction). In addition, the elapsed time at the moving object appears to be slower for the observer at rest (time dilation). These two concepts are fundamentally related.

Interpretation of length contraction and time dilation from computer science perspective

Let’s assume information is represented as a graph/network on the most fundamental level. We observe/store this underlying information in different forms according to our method of observation or storage. For example, a cassette tape stores the information as a chain of magnetic polarities on a magnetic material, a hard drive stores information as a 2D pattern of magnetic polarities, a brain stores information as a network of synaptic potentials, etc.

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Figure A.

Let’s assume a universe U with 1+1 dimension i.e. one length dimension x and time dimension t. Let an object A move along x-dimension in a vacuum with respect to an observer O at rest. Now we are interested in the act of observation of A by O. For the observer O, due to the movement of A, there is an increase in the information along x-dim. In different words, O is receiving more information from A along x-dim.

Here,

Computer = Universe

A Computer Program = An act of observation

There is a space-time trade-off to incorporate this increase in the information or more demand for data storage. For a computer program, one can either:

  1. simply increase the usage of data storage or equivalently store the data in a more efficient/compact way (length contraction)
  2. increase the execution time (time dilation)
  3. or some trade-off point in-between

Our universe seems to follow Option 3. However, our universe seems to be smarter and does one more thing—it optimizes for the storage of information via length contraction to compensate for the increase in information, in addition to an increase in execution time via time dilation. Unlike a computer program, it is probable that the universe has a finite storage/processing capacity and cannot simply use more storage. Hence, it has to resort to optimizing the representation of information. Here, the observer O is reducing the size of the symbol representing the basic unit of information along x-dim. If the object A moves a distance of delta_x, observer O registers an interval smaller than delta_x which optimizes its information representation for storage/processing. For example, you can represent an English text on 26-dimensional space where each letter is an orthogonal dimension. In this case, it is efficient to represent the most frequent letters like ‘e’ (i.e. more information flow from e-dimension) by smallest/shortest symbols such as ‘0’ or ‘1’ and represent infrequent words like ‘q’ by longer symbols such as ‘0000’ or ‘0001’. Our universe does the exact same thing via length contraction— it decreases the length of the symbol for dimension with more information flow. However, this efficient representation of information is not enough to handle the increased information flow and resulting in time dilation or increase in execution time.

Length contraction vs. Time dilation

To summarize,

Computer = Universe

A Computer Program = An act of observation

Decrease the symbol size of the frequent symbol= Length contraction

Increase in execution time = Time dilation/increase in observation time

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Figure B. Source: http://www.zamandayolculuk.com/html-1/timedilation-2.htm
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Figure C. Length contraction vs. Time dilation

In Figure B and C, the relative amount of length contraction (red in B) and time dilation (blue) compared to the initial length L0 and time T0 at rest, are similar for small v/c. However, the amount of time dilation increases faster and is dominant in high velocities. Which means, for smaller velocities compared to the speed of light c, the universe favors length contraction. In other words, it tries its best to tackle the increased information flow by efficient information representation and minimizes the time dilation. Which makes sense! Especially if you don’t want to deteriorate the observation experience by increasing observation time! However, as the speed increases, it’s the opposite — the amount of time dilation dominates length contraction. One way to explain this is that the additional improvement in information representation via length contraction gets smaller and smaller compared to the significant increase in information flow due to extremely high velocity, or alternatively, it becomes more and more difficult to do so. And, finally, there is an upper limit for it determined by the speed of light and the only way to meet the demand.

Lorentz Transformation is the cost function of the Universe

The equations of length contraction and time dilation (see Figure B) between two frames of references are derived from Lorentz transformation. Which means, Lorentz transformation is the operation/cost function of the universe (at least for the act of observation) which determines the trade-off between length contraction and time dilation. An equivalent in computer science is a cost function with variables such as cost and diminishing returns.

Few questions:

  1. If the universe is a computer, processing speed = speed of light in vacuum. Can we imagine different universes for different speeds of light? If yes, what are the allowed values? Can it be any arbitrary values such as 0.1c, 1.3c, etc. or has to be integer multiples of c?
  2. Why the speed of light has to be a constant? Can we simulate similar universe like ours by removing this constraint while making necessary changes in other variables? For example, maybe we can derive a different set of transformations than Lorentz transformation which allows for variable speed of light and the end results are still consistent with all the experiments so far.
  3. One proposal. Let’s say there is a huge information space. This information space when sampled at different frequencies gives different universes with different maximum speeds of light or causality. However, all information in this space (even outside the sampling bandwidth) affects these universes. For example, information density affects as a gravitational force. Dark matter of a universe will be then the information that is outside the bandwidth of the universe and hence cannot be directly observed. However, the effect is still felt via gravity or some equivalent emergent force that emerges from the gradient in entropy.

[Note the idea presented in this article can be explained from another perspective: the information transfer/change rate (in spacetime) of the universe is constant (determined by the speed of light). To incorporate the increase in information from x-dim due to velocity along x-dim, the resolution along x-dim increases (length contraction). However, the resolution along the time dimension decreases (time dilation) to maintain the constant rate of information transfer. This should be true for at least the act of observation (similar to a computer program)]

Speed of light = processing speed of a computer = causality

References

  1. https://en.wikipedia.org/wiki/Space%E2%80%93time_tradeoff

Scientist

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