Monthly Archives: August 2026

E = mc² – The Tiny Equation That Changed the Universe

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Just a quick note before you start reading. This was supposed to be a quick hit for a midweek article, however, it actually took me bloody ages!! Surprisingly, it wasn’t just the maths that caused me issues, it was getting the damned equations into WordPress that made me swear profusely! In hindsight I should have rounded the speed of light to 300,000,000 m/s as most folk would do in an article of this nature. Me? No. I wanted to keep accuracy paramount and use the actual number which is 299,792,458 m/s. Anyway, it’s done now, so on with the article (I really hope you can make sense of it all despite the WordPress formatting).

For this week’s midweek article, just for you, my faithful few, I’m going to answer a question I’ve been sent. And that question is:

“Can you please explain E = mc²?”

And the answer to that question is:

Yes, I can. So, there we go. Question answered. Don’t forget to come back on Friday for this week’s full article on Einstein’s Theory of General Relativity.

Obviously, I’m joking. I wouldn’t leave you hanging like that. Here’s the actual answer, No, I can’t.

Sorry, still joking, I’m just trying to avoid doing the maths, you all know how much I hate doing maths, after all, I’ve only mentioned it a few times.

Ok, on to the real answer, what actually is E = mc² and what does it tell us?

Well, it may be the most famous equation ever written, yet its message is surprisingly simple, and that is, matter and energy are different forms of the same thing.

So, without further ado, let’s meet the world’s most famous equation. Ready? Of course you are!

In 1905, Albert Einstein published his Special Theory of Relativity, a paper that changed physics forever, and a few months later he published a follow up paper building on it, and in that paper was:

E=mc2E = mc^2

We’ve all heard it, the majority know it’s calculation, but not everybody actually knows what it really means, and that is, this beautiful little equation tells us that every object around us contains an enormous amount of stored energy.

Your phone. Your coffee mug. A sugar cube. The classic physics paperclip. Even you. Everything with mass contains energy.

Right then, let’s break it down.

E=mc2E = mc^2

E = Energy (joules). m = Mass (kilograms). c = Speed of Light

The speed of light is defined exactly as: c = 299,792,458 m/s

But, the equation doesn’t just use c, it uses c².

And that means:

c2=c×cc^2 = c \times c

So, if we substitute the exact value (it’s maths time) we get:

c2=299,792,458×299,792,458c^2 = 299{,}792{,}458 \times 299{,}792{,}458

=89,875,517,873,681,764= 89{,}875{,}517{,}873{,}681{,}764

Or

8.9875517873681764×10168.9875517873681764 \times 10^{16}

That massive number is nearly 90 quadrillion, folks, that is 90,000,000,000,000,000. A seriously big number. And it is this gigantic number that is the reason even a tiny amount of matter contains an astonishingly humungous amount of energy.

To work it out, just imagine a single sugar cube sitting beside a cup of tea, and let’s assume a typical sugar cube has a mass of 4 grams

First let’s convert grams to kilograms by dividing it by 1,000: 4 grams = 0.004kg

Next were going to use Einstein’s equation: E = mc²

Now we are going to substitute the variables with the actual values:

E=(0.004)(89,875,517,873,681,764)E = (0.004)(89{,}875{,}517{,}873{,}681{,}764)

And in joules that calculation gives us:

E=359,502,071,494,727.056E = 359{,}502{,}071{,}494{,}727.056

Or as a simple equation we have

E≈3.60×1014 JE \approx 3.60 \times 10^{14}\ \text{J}

So, the answer is, one ordinary sugar cube contains 359,502,071,494,727 joules of energy in its mass.

It may not look that remarkable, but Einstein’s equation reveals that our tiny 4-gram sugar cube contains an almost unimaginable amount of hidden energy.

Which leads us straight into your next question.  If a sugar cube really has that much energy stored in it, why doesn’t it explode?

And that is because, fortunately, ordinary matter doesn’t automatically convert itself into energy, and most of the energy remains locked inside the matter. Only special processes can release some of it and they are nuclear fusion inside stars, nuclear fission in reactors, and matter meeting antimatter. It is only in these situations that a tiny amount of mass is converted into energy, and you get a big bang.

A sugar cube does release energy when you eat it, and that energy is chemical energy, and it comes from rearranging atoms. Einstein’s equation is talking about something much deeper, and that is the energy contained in the mass itself. The energy your body gets from eating sugar is only a tiny fraction of the sugar cube’s total mass-energy.

Here’s a fun fact for you, if all the mass of a sugar cube could somehow be converted directly into energy, the result would be vastly greater than the energy released by burning or digesting it, and that is because chemical reactions use only a teeny tiny part of the energy hidden inside matter.

The Sun also runs on Einstein’s equation, as deep inside it, hydrogen nuclei fuse together to form helium, although the helium produced has slightly less mass than the hydrogen that went into making it, which leads to another question. Where did the missing mass go? And you’ve guessed it, Einstein’s equation gives us the answer. The missing mass became energy and this energy eventually leaves the Sun as heat and light.

That’s not all folks, there is an even bigger equation as the famous equation is actually a simplified version of a more complete relationship from Special Relativity, and that equation is:

E2=(mc2)2+(pc)2E^2 = (mc^2)^2 + (pc)^2

E = total energy, m = rest mass, p = momentum, and c = speed of light

This equation works for everything in the universe, whether moving or stationary. Imagine our sugar cube sitting motionless on a saucer, beside a cup of tea. OK, because it isn’t moving:

p=0p = 0

Next, we’ll substitute into the full equation:

E2=(mc2)2+(0)2E^2 = (mc^2)^2 + (0)^2

E2=(mc2)2E^2 = (mc^2)^2

And then we’ll take the square root of both sides which gives us…

E=mc2E=mc^2

And there it is, folks, the world’s most famous equation is simply a special case for an object that is not moving.

Ah, but what happen if you pick up the sugar cube and throw it across the room? Quite why you would do that, I don’t know, maybe it upset you or you have saccharophobia, an irrational fear of sugar cubes.

The sugar cube now has momentum, so:

p>0p > 0

Then the full equation becomes:

E2=(mc2)2+(pc)2E^2 = (mc^2)^2 + (pc)^2

Its total energy now includes energy from its mass and energy from its motion, and the faster it moves, the larger the momentum term becomes.

What about Light? I hear you say. Well, light is special because it has zero rest mass.

m=0m = 0

And substituting into the full equation gives us:

E2=(pc)2E^2 = (pc)^2

And taking the square root gives us:

E=pcE=pc

And remarkably, the result tells us that light carries energy even though it has no rest mass. And that means that every Wi-Fi signal, radio broadcast, mobile phone signal, microwave, X-ray, and ray of sunlight carries energy because of this relationship.

Before Einstein, mass and energy were thought to be completely different things, but, after Einstein, physicists realised they are simply different forms of the same physical reality. Matter can become energy. Energy can become matter. The universe constantly converts one into the other. And all of that profound insight can be demonstrated with something as ordinary as a sugar cube. How about that!

So to summarise:

We have the famous equation:

E=mc2E=mc^2

The full equation is:

E2=(mc2)2+(pc)2E^2 = (mc^2)^2 + (pc)^2

The exact speed of light is

c=299,792,458 m/sc = 299{,}792{,}458\ \text{m/s}

And the speed of light squared is:

c2=89,875,517,873,681,764c^2 = 89{,}875{,}517{,}873{,}681{,}764

Which means the mass and energy of our single sugar cube is:

m=0.004kgm=0.004 kg

E=359,502,071,494,727 JE = 359{,}502{,}071{,}494{,}727\ \text{J}

Which tells us that our ordinary cube of sugary sweetness contains an extraordinary amount of energy, and that’s because every kilogram of matter is multiplied by one of the largest important numbers in nature:

89,875,517,873,681,76489,875,517,873,681,764

And that astonishing connection between matter, energy, stars, light, and the universe is captured in just five symbols, which is perhaps the most famous and most powerful equation that has ever been written.

E=mc2E=mc^2

Entropy, Thermodynamics & Black Holes – The Article I Promised You!

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Hello again my faithful few, it is time, to discover the hidden rules that govern everything from your morning cup of tea to the edge of the universe. Are you ready? Of course you are!

Did you know, the universe has a favourite direction?

If you place a hot cup of tea on a table and it cools, if you drop an ice cube into a drink and it melts,  and if you spray air freshener into a room and it spreads out, none of these events require any effort, they just happen naturally.

Yet you’ve never seen the reverse occur on its own, have you? Cold tea becoming hot, melted ice spontaneously reforming, or air freshener spray gathering back in to the can.

There’s a reason for this, which lies in one of the deepest principles of nature, and that is, you’ve guessed it, the clue is in the title, it is entropy. A subject I love so much, it gets bolded.

Entropy is the central idea of thermodynamics, the branch of physics that governs energy, heat, and work, along with the direction of physical processes.

And what’s remarkable is, entropy doesn’t just influence everyday events, it plays a crucial role in understanding stars, galaxies, time itself, and even, as I’ve touched on in a previous article or three, black holes.

And now, my faithful readers, I am going to attempt to explain the science of energy without writing a full-blown scientific paper, or a book. Are you ready? Here goes!

Now then. Thermodynamics began during the Industrial Revolution as scientists attempted to build better steam engines, and as is usual when building these types of engineering feats, what they discovered was far more important.

The same laws governing steam engines also govern chemical reactions, batteries, human metabolism, planetary atmospheres, nuclear fusion in stars, black holes, and the evolution of the universe itself.

Thermodynamics became the universal rulebook of energy.

OK. There are four laws of thermodynamics. The first is the Zeroth Law. Now, this law was established by Robert H. Fowler in the 1930s quite some time after the other three laws were established, which is why it isn’t the first law. And it’s a simple one.

If object A is in thermal equilibrium with B, and B is in equilibrium with C, then A and C are in equilibrium with each other. Simples, right? Told you it was. And as simple as this sounds this law allows temperature to be measured. Without it thermometers couldn’t exist.

Then we get the First Law.

The First Law states that energy cannot be created or destroyed, energy only changes form, chemical energy becomes motion, motion becomes electricity, electricity becomes light. However, the total amount of energy remains unchanged.

After the first law, unsurprisingly we get the Second Law

The Second Law is the famous one, you often hear about in the movies, or albums by Muse, and is where entropy enters the story.

The Second Law states that the entropy of an isolated system never decreases. While entropy may remain constant in an ideal reversible process, it naturally tends to increase, and it is this law that explains why so many processes happen in one direction only.

Next? You’ve guessed it, we have the Third Law.

And that is, as temperature approaches absolute zero, entropy approaches its minimum possible value.

So, that takes us to the first question, what is entropy?

People often hear entropy described as a measure of disorder, and as useful as that description is, it’s incomplete.

There is a deeper meaning, and that is, entropy measures the number of possible microscopic arrangements of a system. Physicists call these arrangements microstates. Which, put simply, means the more possible ways something can exist while appearing the same overall, the higher it’s entropy.

Think of it like a jigsaw puzzle of an idyllic country cottage, surrounded by blue sky with fluffy clouds, a cute picket fence and dozens of flowers growing in the garden. One arrangement produces the completed image, whereas millions of arrangements produce a random mess. Nature overwhelmingly favours the states that are statistically more likely. And that tendency is entropy.

Still with me? Good. Let’s have a look at why ice melts. Ice contains water molecules arranged in an organised crystal structure and when ice melts, those molecules gain freedom. They can occupy vastly more possible arrangements which means the liquid state therefore possesses higher entropy than the solid state. Nature generally moves toward states with more possible configurations.

Next, we have one of physics’ greatest mysteries, the arrow of time, that is, why time only moves forward.

The fundamental equations of physics often work equally well forward or backward, yet reality doesn’t. Glass shatters, eggs break, and people age etc. We never witness these events naturally reversing themselves, and it’s entropy that provides us with the explanation.

As systems evolve, they overwhelmingly move from less probable states to more probable ones, and this increase in entropy creates the direction we perceive as the flow of time.

In the nineteenth century, Ludwig Boltzmann had an incredible insight. He had connected entropy to probability with one famous equation:

𝐒=𝐤ln⁡𝛀\mathbf{S = k \ln \Omega}

Where: S = Entropy, k = Boltzmann’s constant, and  Ω = the number of possible microstates. This equation links the visible world to the invisible world of atoms and molecules, and to many physicists (and amateurs like me), it is considered one of the most beautiful equations ever discovered.

Next, we’ll take a look at why perfect engines can’t exist

Suppose engineers build the ultimate engine, every bit of heat becomes useful work. There’s no waste, and no losses.

Thermodynamics says this is impossible. The Second Law requires that some energy inevitably becomes unavailable for useful work and is released as waste heat, so therefore, no heat engine can ever achieve 100% efficiency. Nature always demands an entropy bill.

Let’s have a look at one of my favourite examples, Black Holes, the very first article I wrote for this blog. Black holes changed everything as for much of the twentieth century, physicists believed black holes were simple objects. They appeared to have, no temperature, no entropy, and no internal complexity, and this, my friends, created an enormous problem.

Imagine throwing a book into a black hole. The book contains information and the information has entropy. If the black hole were to simply swallow everything and hide it forever, entropy would seem to disappear from the universe, and this would  violate the Second Law. That means, somewhere, something was wrong.

In the early 1970s, physicist Jacob Bekenstein had a revolutionary idea and  proposed, what at the time was, a shocking solution. He suggested that black holes themselves possess entropy.

At first, the idea seemed absurd.  How could an object that appears completely black have entropy?

Bekenstein, however, had a realisation, and that was entropy had to be related to the surface area of the black hole’s event horizon. In effect, the horizon was storing information about everything that had fallen inside. And this was one of the most important insights in modern theoretical physics.

Soon afterward, our favourite physicist, Stephen Hawking, made an even more astonishing discovery. Using quantum mechanics, Hawking showed us that black holes are not truly black. They emit tiny amounts of thermal radiation, now known as: Hawking Radiation. Regular readers will  know all about Hawking radiation, I wrote about it a couple of weeks ago, remember?

And this implied something absolutely astonishing. If black holes emit thermal radiation, they possess temperature, entropy, and thermodynamic behaviour. Black holes were no longer just gravitational objects. They were thermodynamic objects. And in some sense, they were giant cosmic heat engines.

Let’s move on to the entropy of a black hole. One of the strangest discoveries in physics is that a black hole’s entropy is proportional not to its volume, but to the area of its event horizon, and this was completely unexpected, as this meant that for ordinary objects, more volume means more storage capacity. Whereas for black holes, more surface area means more entropy. And this relationship became known as the Bekenstein-Hawking Entropy Formula, and it suggests that information about a three-dimensional region may somehow be encoded on a two-dimensional surface. This idea later inspired the famous Holographic Principle, one of the most profound concepts in modern theoretical physics, but sorry folks, once again that isn’t for now, that’ll have to be another article (which luckily for you I am working on, so it won’t be long).

What was discovered was black holes are entropy monsters. The entropy contained within a black hole is enormous. In fact, a black hole possesses vastly more entropy than an ordinary star made from the same amount of matter.

When a star collapses into a black hole, entropy increases dramatically, which means black hole formation actually satisfies the Second Law beautifully.

Far from violating entropy, black holes may represent some of the highest-entropy objects in the universe, which is why physicists sometimes describe them as the ultimate entropy engines.

And with that, it was time for another mystery to emerge from the shadows, The Information Paradox.

If Hawking Radiation causes black holes to slowly evaporate, what happens to the information that fell into them? There appears to be three possibilities to answer that question. Either information is destroyed, it escapes somehow, or, more likely, our understanding of physics is incomplete. This became known as the Black Hole Information Paradox, and it remains one of the greatest unsolved problems in physics. Entire fields of theoretical research have grown from attempts to solve it.

Today, many cosmologists believe the universe began in an extraordinarily low-entropy state and that this may govern the fate of the universe. Since then, entropy has steadily increased. Stars burn hydrogen, galaxies evolve, black holes grow, and entropy rises.

Far in the future, black holes may dominate the universe, and even they will eventually evaporate through Hawking Radiation. After unimaginable lengths of time, the cosmos may approach a state of maximum entropy known as Heat Death. This is standard cosmology, and in this state, all the stars are gone, temperature differences completely vanish, any useful work is impossible, and although energy remains, it is hopelessly spread out to be of any use to anything. The universe doesn’t run out of energy, but it does run out of opportunities to use that energy.

So, at the end of the day, thermodynamics began with steam engines, and it ended up explaining reality itself! Which again, as far as I’m concerned, absolutely boggles the mind!

To put it another way, entropy tells us why my favourite Assam tea cools, why stars shine, why time moves forwards, why engines have limits, why life can exist, and why black holes may hold the key to unifying gravity and quantum mechanics.

If the First Law says energy can never be destroyed, then the Second Law says energy becomes increasingly dispersed and increasingly difficult to use.

And nowhere is that truth more dramatic than in a black hole, where gravity, quantum physics, information, and entropy collide at the edge of the known universe.

Entropy is not merely a property of matter. It may be the accounting system for the whole of reality itself!

What Cost The Night Sky? How Three Unrelated News Stories Ruined My Sleep

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There’s a particular kind of headache that comes from reading three unrelated news stories in the same week and realising they’re not unrelated at all, and this has been one of those weeks. In truth, this article exists because I couldn’t quite let those stories go, and I’ve broken my own rule of not writing at the weekend because of it. The articles had been rattling around in my head for while, keeping me awake last night, and before I realised what I was doing, this morning, I was deep into research papers, budget tables, news archives and LinkedIn posts, trying to work out whether the numbers being quoted actually added up. The deeper I dug, the stranger the picture became. What follows is the result of several hours of increasingly obsessive fact-checking, prompted in equal measure by frustration at the money governments somehow always find for wars and by the determination of certain tech billionaires to spend vast fortunes attempting to turn themselves into the first Bond villain.

Let’s start with the sky itself, because that’s where this began, with 1.7 million satellites versus the whole of astronomy. The European Southern Observatory has just done something nobody had bothered to do properly before: modelled what happens to astronomy if everything currently proposed for orbit actually gets launched.

The number ESO astronomer Olivier Hainaut arrived at for “acceptable” damage, the point at which satellite interference stays roughly comparable to routine equipment failures is 100,000 faint, naked-eye-invisible satellites.

The number currently on the table, across every proposed constellation from every operator, is over 1.7 million.

That’s not a rounding error. That’s a different universe of consequence, and the specifics are grim: ESO’s Very Large Telescope could lose up to 28% of its field of view to Starlink alone. The Vera Rubin Observatory, arguably the most ambitious survey telescope ever built, stands to lose hours of observing time every single night. And then there’s Reflect Orbital, a company literally planning to put mirrors in orbit to beam sunlight onto Earth after dark, with up to 50,000 of them proposed by 2035. Hainaut’s modelling suggests they could make the night sky itself four times brighter, even from a light-polluted city like Munich.

We are, with a straight face, discussing whether to switch the night sky off. Not metaphorically, actually switch it off!

Here’s where it stops being an abstract problem for astronomers in Chile and starts being a very local one. Meanwhile at home, the same week this ESO analysis landed, the Royal Astronomical Society reported that the UK’s Science and Technology Facilities Council is withdrawing all funding for e-MERLIN, the network of radio telescopes that includes the Lovell Telescope at Jodrell Bank. Jodrell Bank. The place that tracked Sputnik. The place that’s been quietly doing world-class radio astronomy since before most of us were born. It’s now at genuine risk of closure.

That’s not the whole of it, either. The RAS reported the UK is withdrawing from the James Clerk Maxwell Telescope entirely, cutting 20% from the Square Kilometre Array Regional Centre, cutting 20% from the UK’s own contribution to Vera Rubin, yes, the same observatory the satellites threaten to steal hours from, and cutting 40% from the BISON solar monitoring network.

So: the instrument is being switched off from above, and the people who’d use it are being defunded from within, in the same month. If you set out deliberately to sabotage a field of science, it’s hard to imagine a more efficient one-two.

Now for the number that actually made me sit down and do the maths, and look at what we are proposing to spend it on instead, because I’d heard a vague comparison somewhere and wanted to check whether it held up. It doesn’t hold up quite the way I’d remembered. It’s a lot more startling than that.

The US-Israel war on Iran that began in February 2026, Operation Epic Fury, to give it its official name, has, according to the Pentagon’s own figures given to the Senate on 21 July 2026, cost $37.5 billion so far. That’s up from $29 billion in May and $25 billion in April; the number has been climbing by roughly a billion dollars a week. Defence Secretary Pete Hegseth is now asking Congress for up to $70 billion more in emergency funding, as part of an $87.6 billion supplemental request. If Congress approves that request, the total cost attributable to the conflict could exceed $100 billion.

Here’s the part that stopped me. NASA doesn’t actually cost all that much by federal-government standards. Its annual budget has been remarkably steady at around $24-25 billion a year. The $37.5bn already spent on the Iran war is therefore equivalent to roughly eighteen months of NASA: every mission, every telescope, every ongoing operation, every payroll. Already spent, already gone, no projections involved.

Add the funding now being requested and the comparison grows to roughly four years and four months of the entire US civilian space programme. Say it slowly: four-and-a-bit years of the entire United States space programme, potentially gone in five months of a war that hasn’t produced an observatory, a scientific instrument, or a single piece of usable knowledge about the universe we live in. But hey, that’s OK, it has put the cost of living through the roof.

The arithmetic is straightforward. Add together 2026 and roughly half of 2025 and you reach the equivalent of the $37.5bn already spent on the war. Continue through 2024, 2023 and part of 2022 and the total reaches roughly $107.5bn. Same source data, same calculation, just expressed in time rather than dollars.

I’m not usually one to get on my soapbox, and I do try not to be the “billionaires bad guy,” mostly because it’s boring and it’s usually not that specific a complaint. But specificity is exactly what we have here. We’re not talking about an abstract injustice, we’re talking about a precise, sourced, comparable set of numbers. A dark sky and a functioning radio telescope at Jodrell Bank, for the price of what a war spends in a matter of weeks. Actual, unglamorous, patient scientific research, the kind that doesn’t make anyone rich, doesn’t come with a rocket launch livestream, and doesn’t get you invited on a podcast, going begging for a fraction of a fraction of what gets found overnight for other purposes.

If there’s a genuine, defensible case for where the enormous fortunes of the “let’s conquer space” tech-billionaire set should go, it’s here: into keeping Jodrell Bank’s lights on, not into another few hundred thousand satellites and orbital mirrors marketed as innovation. And if there’s a case for where a nation’s money should go, the maths above makes it for me. NASA gave us the Moon landings, Hubble, and forty years of thinking a bit harder about our place in the universe, and it did all of that on this budget. The full Iran war bill, once the current request lands, could run past four years and four months of that budget. But you don’t even need the bigger number to make the point, eighteen months of NASA, gone in five months of a war that won’t produce a single new fact about a single distant galaxy.

We can afford to look at the stars. We just keep choosing not to.



A quick note on the sources: The comparisons made in this article are drawn from publicly available sources, including ESO’s satellite-impact study, the Royal Astronomical Society’s reporting on UK astronomy funding, Pentagon cost figures reported by Reuters, the BBC and The Guardian, and NASA budget data. The references are included so readers can verify the numbers for themselves and see exactly how the conclusions in this article were reached. Regular readers will know I don’t usually include a source list like this, but because this article grew out of several hours of fact-checking and number-crunching, I thought it was only fair to show my working.

Sources:
ESO, “Beyond the limit”: one million satellites and mirrors in space pose grave threat to the night sky — the original ESO/Olivier Hainaut study
New Atlas, “1.7 million planned satellites to have ‘devastating’ impact on astronomy” — corroborates the 28% VLT field-of-view figure
Royal Astronomical Society, “Mass cuts and potential Jodrell Bank closure ‘devastating’ for astronomy”
Reuters, “War in Iran has cost the US $37.5 billion so far, Pentagon says” (21 July 2026)
Reuters, “US war in Iran has cost $29 billion so far, Pentagon says” (12 May 2026)
The Guardian, “Hegseth tells Senate war on Iran has so far cost $37.5bn” (21 July 2026)
BBC, “Iran war has cost US $37.5bn so far, Hegseth says” (21 July 2026)
Wikipedia, “Budget of NASA” (figures sourced to The Planetary Society’s historical NASA budget dataset)