On the top line is given what people like Galileo knew. Its called Gravity. It says as you go away from center of gravity your energy increases as the force of attraction you experience goes on decreasing with the increase in square of r, r is separation of you from the center of the gravity creating object such as earth. So Newton could have differentiated it. (why Galileo isn’t ascribed to derivatives and calculus? he definitely had to understand it to see gravity forces? thats the kind of questions historians better be talking about) Now if I differentiated the top what I get is the 2nd line. (I took derivatives, which are ratio of small changes in different variables) So I get on 2nd line on Left hand a fractional change in energy dE/E (eg did energy change 1% or 5 % ? .. )
So deceleration is equivalent to a gravity field. Which is exactly what principle of equivalence of Einstein is. ‘In osme paperwork” I have proved this equivalence from classical mechanics. [only from Newtonian formula for gravity].
This means many special and general relativity ideas are actually also classical mechnaics ideas but Einstein extended them and made them more general, exact and advanced. eg Einstein studied them in more situations than were studied before. Time dilation, equivalence principle and gravitational dilation are all simple classical mechnaics ideas. All the details I have written in recent articles in in this website check last 1-3 months under Relativity/Physics tag.
What is beer?
Beer is what you want to drink when a beautiful girl is sitting not very far from you and yet you cannot tell her you would like to spend some time with her in privacy and everyone else sitting with you are also thinking so, so they are also drinking it.
All energy slows down clocks …, not just gravitational energy and there is nothing Einsteinian about it. Gravity just slows down in a particular way than any other forms of energy and Einstein gave its exact form. I will derive time dilation of gravity from Non-Einsteinian case.
You may read the linked article — Time Dilation, before or after reading the current one.
It’s a beauty of physics that not only the most successful laws of physics viz quantum mechanics and theory of relativity are super-set of the canonical laws of physics which is classical mechanics and over-binding on the latter but they are also a good extension of the latter. They are a modified version of the latter, hence they must all fall back onto the classical laws in their requisite limits.
In some situations this fact is amply reflected in the equations or ideas itself. I have hit upon two of these, while thinking of the same, another one is perhaps well known. The one that is known well enough is the so-called principle of relativity which is a set of transformations relations, between quantities in one frame-of-reference to another.
But this is merely a different form of transformation laws, the principle of relativity being one known even in Newtonian mechanics. In the latter, the principle of relativity of ordinary classical mechanics, it lacks the characteristics of certain kind, which are only envisaged by Einstein.
I am going to show why the two other such similarities of relativistic nature of our world known to be special relativistic properties are actually Newtonian properties. The only difference that crops up is perhaps in the newer ideas of Einstein, perhaps because I haven’t reviewed the constancy of speed of light.
The first one is time dilation. Time dilation is strictly speaking not an Einsteinian property. It’s a Newtonian property. And this I described in much detail in my last article “Time dilation”. Here one does not even need any equations explicitly, to see the idea.
Why is time dilation a Newtonian property? Because as I explained in that article “time dilation is a result of energy disappearance or unavailability”. Period. And there is no special Einsteinian thinking here. If there is no energy there is no “running of time”. If there is less of energy there is slower running of time and if there is more energy there is faster running of time.