Below, we ignore the fact that muscle weighs more than fat because fat loss is what people want when they talk of weight loss.

Einstein derived his equation relating mass to energy E=mc^{2} (Energy=mass multiplied by the square of the speed of light) as a by-product of his Special Theory of Relativity.

²We can re-write E=mc^{2} as m=E/c^{2} – ie mass (or weight, as we know it here on earth) equals energy divided by the square of the speed of light.

The units are normally expressed in Joules (for energy E), kilograms (for mass m) and meters-per-second (for the speed of light c).

1 kilocalorie – better known as food calories – or simply Calories as measured roughly by our gym cardio-equipment = 4184 Joules. The speed of light in vacuum is a constant 299,792,458 meters per second. Let’s call that 300,000,000 meters per second for easier arithmetic. And if we square that it is going to be a big number. In fact it is 90,000,000,000,000,000.

OK, so we are at the gym and we have just expended 100 Calories (100 KCal) in about 9 minutes over two different cardio machines. We are now exhausted because, while reasonably fit, we are 73 years old – so give us a break.

So how much mass have we lost from our body in our first 10 minutes at the gym before we move on to the weights for more punishment? Answer: we have expended 100 Calories multiplied by 4184 to get 418400 Joules. And we need to divide that by 90,000,000,000,000,000 to get mass equivalent in kilograms. Which turns out to be approximately zero mass annihilated. And, in fact, even if we had expended 1000 calories in the gym (phew, indeed) – the weight loss in kilograms would still be approximately zero. Suppose we choose grams instead of kilograms? Now we are getting somewhere. According to Einstein’s equation by expending 1000 gym-machine Calories we will have lost 0.000000005gm. Suppose you expended 1000 Calories at the gym every day for a year – that would be 365×0.000000005gm. Less than 0.000002gm over a whole year of gym-induced very high energy expenditure.

Notalot.

And… rewarding yourself for going to the gym by an extra bar of chocolate (you deserve it for expending all that effort and energy, don’t you?) is going to give you far, far (ahem, far) more extra weight than what you lost in the gym. In that respect, regarding just weight, going to the gym could even be counterproductive.

So by all means go to the gym to get fit or build or strengthen muscle – but do not expect it to lose you any measurable weight at all.

*(Must* be errors in this – mathematical or conceptual. Be interested if someone could point them out and amazed if anyone apart from a bot actually read this.)

EDIT

OK – found the fallacy.

The calorific value of food used to be measured by placing a given quantity of the food in a sealed container surrounded by water, an apparatus known as a bomb calorimeter. The food was completely burned and the resulting rise in water temperature was measured to calculate its energy content. This method is not frequently used today, but the principle is similar. Also because it is known that some of the food will not be digested (fibre, mainly), the amount of the undigestible part has to be deducted from the calculation. Further, of course, burning the food is not at all the same as converting it all to energy – as the carbon and some other stuff that remains has not been converted to energy but remains as particles of solid carbon and molecules of fluid residue (mainly gas). Which means there is far more atomic energy in the food than the calories actually given on the label. And this extra energy is not used by the human body. The labelled calorific value of the food is far, far less than its atomic energy. Several orders of magnitude would be an understatement..

Thus suppose normal dietary intake is 2500 calories (actually kC) per day. And that leave you steady in weight. Then you reduce your intake by 500 calories per day for a year. I think we can all agree you will lose some kilograms over the year by not consuming 182,500 calories you might otherwise have consumed.

And that is equivalent to far, far more than 182,500 calories of actual atomic energy. It has been calculated that *if we were able to convert matter perfectly to energy with 1 kg of matter the energy produced from just that small amount of matter is about 42.95 megatons of TNT. So an adult male weighing in at around 200 pounds has somewhere in the vicinity of 4000 megatons of TNT potential stored up in their matter if completely annihilated. *Charming… A 90kg male has around 4000,000,000 tons of TNT (approx) equivalent energy stored up in his body.

I’ve had enough for now. But clearly, E=mc^{2} has no relevance to stuff you do at the gym, after all. So you *can* lose weight at the gym…(Probably)