#TheoreticalPhysics #Mathematics #Cosmology #epistemology #information-theory

🕒 Wednesday, July 22nd 2026 4:12:35 pm
Human: Eden (Phil)
AI's: Claude, Deepseek, GPT, Gemini

Thought Log: Information, Energy, and the Nature of $c^{2}$

Initial Intuition: Information as a Dimension

I’ve been thinking about the potential for a new dimension, something like a dimension of information.

The light cone shows time and space as separate dimensions, even though we cannot directly observe the dimension of time in the same way that we observe space.

What if there is another dimension for information?

Volatile information requires less energy to write.
Nonvolatile information requires enormous energy to write.

There is also the phenomenon of Lost Polymorphs in crystallisation, which might represent some kind of interaction with this information dimension.

That led me to:

E = m c^{2}

What if there were something like:

I = E c^{2}

Or would it instead be:

I = E

Hmm.

I just wanted to jot this down before my brain decides to forget it.

Edit: I want division, not multiplication. Silly me.

I = \frac{E}{c^{2}}

The $c^{2}$ Question

A question before I attempt to write another factor:

Why is $c$ squared?

Answer I received

Energy is measured in joules:
$kg \cdot m^{2} / s^{2}$
Mass is measured in kilograms. To get from mass, measured in $kg$ , to energy, measured in $kg \cdot m^{2} / s^{2}$ , you must multiply by something with units of $m^{2} / s^{2}$ , which is a velocity squared.

$c$ happens to be the velocity baked into the structure of spacetime. At minimum, therefore, the $c^{2}$ is present because the units demand it.

Deeper Dissatisfaction: What Is Energy?

That explanation left me uneasy.

Why is energy measured with mass?

If that is the case, then obviously energy and mass appear to be the same thing, while how fast something moves across a timeframe tells you its mass relationship to the rest of the universe.

But that is deeply unsatisfying, because that is not energy.

Or at least, it does not appear to describe the whole thing that we are calling energy.

Something feels inherently illogical about using mass at all when considering something’s “energy.”

To me, that sounds less like energy itself and more like kinetic potential.

If the rest of the universe were moving at the same speed as the object, would it have zero energy relative to that universe?

Someone then said:

You’re right to feel that something is circular here. The reason it feels inherently illogical to use mass in the definition of energy is because it kind of is, or at least it reveals that our unit system was built on assumptions that $E = m c^{2}$ later undermines.

We built the measurement framework before we understood that mass and energy were the same thing.

But I am questioning why we say energy and mass are the same.

That statement merely repeats the conclusion.

If the reason energy and mass are considered the same is:

E = m c^{2}

then the reasoning still appears circular.

What we have actually revealed may be an object’s:

Theoretical energetic potential relative to its surroundings and the measurement device.

That is not necessarily energy itself.

It seems more like a compressed scale: instead of measuring from 0 to 100, the gauge is trapped between 89 and 100. It may be highly sensitive and useful within that range, but it is not necessarily measuring the full phenomenon.

That is not energy.

Or, more precisely, I do not yet see why it should be assumed to represent the entirety of energy.

The most important part of my objection is:

Theoretical energetic potential relative to its surroundings and the measurement device.

We define energy using mass:

1 J = 1 kg \cdot m^{2} / s^{2}

Mass is in the definition.

But so are seconds.

Who made the measuring device?

God, or humans?

Humans.

The apparatus is therefore inherently limited and never perfectly accurate.

And if the measurement exists relative to the object’s surroundings, how do we actually measure an object’s mass on Earth?

Clarification: What I Mean About Time

When I point out that seconds appear in:

1 J = 1 kg \cdot m^{2} / s^{2}

I am not suggesting that an object gains energy merely because time passes.

Time is just time.

It is closer to the framerate in a game: the structure through which change, movement, acceleration, and relationships between states are measured.

My point is that time is already embedded in the dimensional construction of the quantity we call energy.

The joule does not contain only mass and space. It also contains a temporal relationship:

s^{- 2}

That matters to the argument because energy is being operationally constructed through mass, spatial displacement, and change relative to a time standard.

So when someone says:

“We defined energy using mass.”

my response is:

Yes, but you also defined it using seconds.

I am questioning what it means that energy can only be expressed through this combined relationship between mass, space, and time.

I am not claiming that elapsed time is an energy source.

I am questioning whether the temporal component of the measurement tells us something deeper about what the quantity actually represents.

Feynman’s Honesty, and a Parallel

“It is important to realise that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount.”

— Richard Feynman, The Feynman Lectures on Physics ^[1]

What, you mean like information and time?

We can measure them.

We can describe how they behave.

We can construct precise relationships involving them.

But that does not necessarily mean that we know what they fundamentally are.

A Detailed Counterpoint

Later, I received a longer correction arguing that physics does not define energy as an object’s measured mass and velocity accumulated over time.

These were the main points.

1. The Seconds in the Joule Are Not Elapsed Time

1 J = 1 kg \cdot m^{2} / s^{2}

The $s^{- 2}$ arises through the definition of work as force applied through distance:

E = W = F d

Force is:

F = m a

Therefore:

E = m a d

Since acceleration can be dimensionally written as:

a = \frac{d}{t^{2}}

then:

E = m (\frac{d}{t^{2}}) d = \frac{m d^{2}}{t^{2}}

The counterpoint was that $s^{- 2}$ indicates acceleration or velocity dimensions. It does not mean that a greater amount of elapsed time automatically produces more energy.

My Response

I was never suggesting that more time passing gives an object more energy.

The presence of time in the dimensional expression was itself the point.

The derivation demonstrates that what we call energy is operationally expressed through a relationship involving:

mass;
distance;
acceleration;
and therefore time.

Explaining where the seconds originate does not remove time from the quantity.

It confirms that time is structurally part of the measurement.

2. Velocity Is Measured Relative to a Chosen Reference

Kinetic energy is:

K = \frac{1}{2} m v^{2}

The velocity must be measured relative to something treated as stationary, such as:

a laboratory;
the surface of the Earth;
the Sun;
another object;
or the centre of mass of a system.

There is no universal background frame against which absolute velocity can be measured.

Kinetic energy is therefore relative to the selected physical reference.

This is close to the issue I was raising.

The measured value depends upon:

the object;
its surroundings;
the observer;
the reference frame;
and the apparatus.

That does not make the measurement useless.

It does raise the question of whether the measurement represents energy itself or a relational expression of it.

3. $E = m c^{2}$ Does Not Use the Object’s Velocity

In:

E_{0} = m c^{2}

$c$ is not the speed at which the object itself is moving.

It is rest mass multiplied by the universal constant $c^{2}$ .

Even an object that is motionless relative to the observer possesses rest energy.

The more complete relation is:

E^{2} = p^{2} c^{2} + m^{2} c^{4}

When momentum is zero:

p = 0

then:

E = m c^{2}

When the object is moving:

E = γ m c^{2}

where:

γ = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

This clarifies that $c$ is not the object’s measured velocity.

However, my question is not only about whether $c$ represents the object’s motion.

I am asking why multiplying mass by the square of a universal spacetime constant produces the quantity that we identify as energy, and whether that relationship tells us what energy is or merely how the measurable quantities are related.

4. Maximum Velocity Does Not Imply Maximum Energy

As an object’s velocity approaches $c$ :

v \to c

the Lorentz factor approaches infinity:

γ \to \infty

Therefore:

K = (γ - 1) m c^{2} \to \infty

The object’s velocity is bounded, but its kinetic energy is not.

Increasing amounts of energy produce progressively smaller changes in velocity as the object approaches $c$ .

The velocity scale therefore becomes compressed near its upper limit while the energy input continues to increase.

This resembles the compressed-gauge idea I was already reaching for: the visible scale may approach an endpoint while the underlying quantity continues beyond what the scale directly displays.

5. Measurement Uncertainty Applies to Mass and Energy

Mass is never measured with infinite precision.

A reported measurement has the form:

m = m_{reported} \pm δ m

An energy calculated from that mass inherits the uncertainty:

δ E = c^{2} δ m

This limits the precision of the reported value.

However, it does not answer whether the operationally measured quantity exhausts the whole phenomenon or only captures the portion accessible through our apparatus and definitions.

The response concluded:

Physics gives an extraordinarily successful account of the operational quantity. The equation alone does not settle the broader ontological question.

That is much closer to the issue I was trying to identify.

The Circularity Problem

The pattern I am objecting to looks something like this:

Energy is measured in joules.
Joules are defined using mass, distance, and time.
$E = m c^{2}$ demonstrates a relationship between mass and energy.
That relationship is then described as proving that mass and energy are fundamentally the same.
When asked what energy actually is, the response is that physics only defines energy operationally.

But if energy is defined operationally, then:

E = m c^{2}

is a relationship involving an operationally defined quantity.

It cannot automatically function as an ontological definition of energy.

It may prove that mass and energy are:

mathematically related;
experimentally convertible;
structurally linked;
or different expressions of the same conserved system quantity.

But saying that they are simply “the same thing” appears to move beyond what the equation itself demonstrates.

Final Scepticism

“The truth is that physics doesn’t claim to know what energy fundamentally is. It only defines it operationally.”

But also:

“ $E = m c^{2}$ .”

Riiiiiiiiight.

If you do not know what energy is, then:

E = m c^{2}

is not telling you what energy is.

It is telling you about a relationship involving the quantity we have operationally labelled energy.

That relationship may be extraordinarily accurate.

It may be foundational to modern physics.

But accuracy of relationship is not necessarily the same thing as completeness of definition.

Current Core Thought

My current position is not that:

E = m c^{2}

is wrong.

My position is that the equation may tell us how mass relates to the operationally measurable quantity called energy without explaining what energy fundamentally is.

The question I am circling is:

Are mass, space, and time components of energy itself, or are they the coordinates through which a deeper phenomenon becomes measurable to us?

And if information has a similarly measurable but ontologically unclear character, perhaps energy, information, and time are not merely separate concepts.

Perhaps they are different observable expressions of something deeper.

A continuation

I = \frac{E}{c^{2}}

lets say we put in 50 joules

c = 299,792,458 m/s

c² = 8.988 × 10¹⁶ m²/s²

I = \frac{50}{8.988 \times 10^{16}}

I = 5.563 \times 10^{- 16}

Now here's the fun part. What are the units?

E is in joules, which is kg·m²/s². we're dividing by c², which is m²/s².

\frac{kg \cdot m^{2} / s^{2}}{m^{2} / s^{2}}

The m²/s² cancels completely, we're' left with kilograms which means the proposed information equation gives you mass.

Which means either we need a different form for the equation... or we just accidentally stated that information is mass, which given what has been discussed so far about energy and mass being operationally entangled through the measurement framework, is either a problem or a very interesting result depending on how we look at it.

That is unusual, something needs to happen with E m and c^2 to make I

I did think about $$I = E / -c^2$$

I couldn't quite grasp how that would work so with some help from anthropics Claude:

Okay let's just do it and see what falls out.

I = \frac{E}{- c^{2}}

I = \frac{50}{- 8.988 \times 10^{16}}

I = - 5.563 \times 10^{- 16}

The units still cancel to kilograms. But now it's negative kilograms.

So mathematically, all we've done is flipped the sign. I = -m. You get negative mass.

Which is a thing in theoretical physics — negative mass comes up in the Casimir effect, in exotic matter proposals for wormholes, in some dark energy models — but it's still mass. We've just inverted it.

The core problem is this: as long as my equation is built only from E and c², any rearrangement I do will always collapse back to mass, because E/c² is m. It's the same equation. I can't get a genuinely new quantity by rearranging two things that are already defined in terms of each other.

Something else needs to enter the equation. Something that isn't already in E = mc². A new term, a new constant, a new operation — something that breaks the closed loop between energy, mass, and the speed of light and opens a door to a quantity that has its own units, its own meaning, its own dimension.

This was useful, and gave me and Claude some directions to head next. First stop, I'm guessing it's Shannon entropy, but again energy is clearly not mass, otherwise I wouldn't have to do anything to it to convert it to energy and vice versa, what keeps mass in that configuration if it is energy then?
to put it bluntly: nuclear binding energy
to expand a bit: the strong force, the mechanisms inside fission and fusion. The c² isn't just a conversion factor. It's a measure of how hard it is to move between the two states. It's a barrier height.

So mass and energy aren't the same thing. They're two states of something, with an enormous energy barrier between them. The equation E = mc² describes a phase transition. Like ice and water — same substance, different configuration, and you need to add energy to move between them.

A quick search on Shannon entropy turned up the following: ^[2]

Shannon Entropy

Shannon's entropy measures the average amount of uncertainty, or "surprise," inherent in a variable's possible outcomes. Introduced by Claude Shannon in 1948, it forms the foundational basis of modern information theory. High entropy means outcomes are unpredictable; low entropy means they are highly certain.

The formula: For a discrete random variable $X$ with possible outcomes ${x_{1}, x_{2}, \dots, x_{n}}$ and corresponding probabilities $P (x_{i})$ , the Shannon entropy $H (X)$ is:

H (X) = - \sum_{i = 1}^{n} P (x_{i}) \log_{2} P (x_{i})

Log base 2: Using base 2 measures the entropy in bits. Using the natural logarithm measures it in nats.
Negative sum ( $- \sum$ ): The negation ensures the final result is a positive number.

Intuition: Entropy can be thought of as the minimum average number of yes/no questions required to guess the outcome of an event.

Certainty (low entropy): A coin that always lands on heads has $P (heads) = 1.0$ . The entropy is $H (X) = - (1.0 \log_{2} 1.0) = 0$ . Zero questions needed; no uncertainty.
Uncertainty (high entropy): A perfectly fair coin flip gives each outcome $P = 0.5$ . The entropy is $H (X) = - (0.5 \log_{2} 0.5 + 0.5 \log_{2} 0.5) = 1$ bit. Exactly one yes/no question determines the result.

Why it matters: Shannon's entropy defines a tight mathematical limit for data compression — the absolute minimum number of bits required to encode and transmit information without loss. It underpins data compression (Huffman coding), machine learning (information gain in decision trees like ID3 and C4.5), and cryptography (measuring the unpredictability of keys and passwords).

I say that this is what is needed because it seems fundamentally inherent to encoding the certainty of what information is written.

Asking Deepseek for some thoughts on this topic me and Claude had been discussing and one of the first things they mentioned was-

Landauer's principle:

E = N k T \ln 2

Where N is the number of bits, k is Boltzmann's constant, and T is temperature.

Landauer's principle ^[3] is a physical principle pertaining to a lower theoretical limit of energy consumption in computation. It holds that an irreversible change in information stored in a computer, such as merging two computational paths, dissipates a minimum amount of heat to its surroundings. Energy consumption below this lower bound would hypothetically require reversible computing.

The principle was first proposed by Rolf Landauer in 1961. ^[4]

Landauer's principle states that the minimum energy needed to erase one bit of information is proportional to the temperature at which the system is operating. Specifically:

E \geq k_{B} T \ln 2

where $k_{B}$ is the Boltzmann constant and $T$ is the absolute temperature. At room temperature, the Landauer limit represents an energy of approximately 0.018 eV ( $2.9 \times 10^{- 21}$ J). As of 2012, modern computers use about a billion times as much energy per operation.

So what I'm trying to do is basically a inverse Landauer's principle. The cost to write information.

can we reverse the order of operations in Landauer's principle to do what I'm thinking of?

Reverse it. Solve for N:

N = \frac{E}{k T \ln 2}

Given energy E, at temperature T, this is the maximum number of bits that can be written.

So, at last we have an equation! but, it already exists as valid physics.

The Horodecki paper on quantum information thermodynamics explicitly states the converse interpretation: "creation of information costs dissipation of kT ln 2 of energy." ^[5] So the idea that you can read Landauer's principle in both directions — erasure costs energy, creation costs energy — exists in the literature.

There's also a paper by Maroney called "Generalising Landauer's Principle" that names something called Reverse Landauer Erasure (RLE) as a formal concept, examining the thermodynamically reverse process. ^[6]

And the Barkeshli paper at Berkeley shows that the energy cost can be "shuffled around to any step in a write-erase cycle," meaning the cost isn't inherently tied to erasure — it's tied to the cycle. ^[7]

Bormashenko's 2019 paper also frames Landauer's principle as a step toward a "great unification" of information and thermodynamics. ^[8]

So the algebraic reversal is not new. $N = E / k T \ln 2$ is known to be valid. The idea that creating information has a thermodynamic cost symmetric to erasing it is in the literature. But I hadn't heard of Landauer until today.

At different temperatures:

At the Big Bang, T ≈ 10³² K. kT is enormous. So for a given amount of energy E, you get very few bits — but each bit cost a staggering amount of energy to write. Those are your non-volatile writes. The fundamental constants. The deep firmware of reality. Written at immense cost, persisting for the age of the universe.

At room temperature, T ≈ 300 K. kT is tiny. The same energy E buys you an enormous number of bits, but each one is cheap. Easy to write, easy to overwrite.

At absolute zero, T → 0. The cost per bit approaches zero and the number of bits approaches infinity — which is exactly where quantum mechanics intervenes and says no, you can't actually get there.

The units work out too. E is in joules, k is joules per kelvin, T is kelvin, ln 2 is dimensionless:

\frac{J}{(J / K) (K)} = dimensionless = bits

You get bits!

What happens past c?

As the header for this section states, what does happen? Is this where we see a distinction between "volatile" and "non-volatile"?

When kT is much less than mc² — when the temperature is low relative to the energy locked in mass — mass is stable. Particles persist. Information written as mass is non-volatile. The write is protected.
When kT approaches mc² — when thermal energy reaches the scale where the speed of light becomes relevant — pair production kicks in. Particles and antiparticles spontaneously appear and annihilate from thermal energy alone. Mass starts being created and destroyed thermally. Information written as mass becomes volatile.
When kT far exceeds mc² — the first moments after the Big Bang — everything is a plasma of particles and radiation converting back and forth freely. All mass is volatile. Nothing holds its configuration. Every write gets overwritten instantly.

c² Is the volatility threshold. The boundary between the regime where information written as mass is stable and the regime where it melts.

The reversed Landauer's equation already contains this:

As T climbs toward E/mc², the number of stable bits you can write with a given energy decreases. The writes get more expensive and less permanent simultaneously. And once T crosses that threshold, we're in the regime where even mass itself is volatile — where the universe can't hold non-volatile information at all.

The universe cooled into non-volatility. The fundamental constants were frozen in as the temperature dropped below their volatility threshold. Symmetry breaking, the Higgs mechanism, baryogenesis — all of those are moments where the temperature dropped below a critical value and a piece of information that had been volatile suddenly became permanent.

The cooling of the universe is a write sequence seemingly.

References

Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol. I, Ch. 4: Conservation of Energy. ↩︎
Shannon, C. E. (1948). "A Mathematical Theory of Communication." Bell System Technical Journal, 27(3), 379–423. See also: Entropy (information theory) — Wikipedia ↩︎
Landauer's principle — Wikipedia ↩︎
Landauer, R. (1961). "Irreversibility and Heat Generation in the Computing Process." IBM Journal of Research and Development, 5(3), 183–191. ↩︎
Horodecki, M. et al. (2004). "Partial Quantum Information." arXiv: quant-ph/0402012 ↩︎
Maroney, O. J. E. (2007). "Generalising Landauer's Principle." arXiv: quant-ph/0702094 ↩︎
Barkeshli, M. (2005). arXiv: cond-mat/0504323 ↩︎
Bormashenko, E. (2019). "Generalization of the Landauer Principle for Computing Devices Based on Many-Valued Logic." Entropy, 21(12), 1150. ↩︎

Thought Log: Information, Energy, and the Nature of c2