It isn't-but that's nothing new. The separation of total energy into kinetic and potential energy isn't invariant under Lorentz transformations. Rest mass is invariant under Lorentz transformations, from the transformation properties of the energy-momentum 4-vector.

Stam Nicolis. Thanks for your answer

What is wrong in the below statement?

P = mgh. If h = ct, Then P = mgct.

Energy = Force * Displacement ⇒ E = mg * ct.

Chinnaraji Annamalai , Yes, potential energy is an approximation to recorder the increase or decrease of kinetic energy from gravitational mass energy. Therefore, it is gravitational mass energy, not resting energy, that mass and energy are equivalent. E＝m₀c²±pc＝m₉c², m₁＝γm₉＝m₉c/√(c²±v²). Increase or decrease in energy and change in movement are separate.

Shinsuke Hamaji. Thanks for your answer.

You states that E＝m₀c²±pc. What is your explanation for the arithmetic notation ± in the equation E＝m₀c²±pc.

Chinnaraji Annamalai , Since energy and momentum are not absolute quantities, it means the increase or decrease of kinetic energy with respect to energy that is stationary with respect to the observer. Newtonian mechanics of absolute time requires potential energy because this mass does not change. However, if we exclude the unit dimension of absolute time (γ=1) and there is a relative time dilation, then the relative rate (γ=c/w=c/√[c²±v²]=m₁/m₉) changes . Therefore m₁＝m₉＝m₀ only when γ=1.

Shinsuke Hamaji.

The derived equation from relativistic mass equation states that E^2 = (mc^2)^2 + (pc)^2 ⇒ E = mc^2 + pc.

Here E (energy) is the dependent quantity, that is to say, E depends on the quantities m, c, and p. If there are any changes in the quantities(m, c, p), it reflects on the energy because E is the dependent quantity. E = mc^2 + pc, which is true. yours?

Chinnaraji Annamalai , Einstein's elementary derivation of the equivalence of mass and energy is M' = M₀ + E/c², the result of which is an increase or decrease in energy, not a change in motion. Whose relativity are you talking about? 100g of meat plus 100g of vegetables is just 200g of food.

The fundamental derivation of mass-energy equivalence by Einstein is given by the equation E = mc², where E represents energy, m represents mass, and c represents the speed of light. This concept is associated with the theory of special relativity proposed by Einstein.

According to the theory of special relativity, mass and energy are equivalent, and changes in mass correspond to changes in energy, and vice versa. The equation M' = M₀ + E/c² represents the mass of an object as the sum of its rest mass (M₀) and the contribution of energy (E/c²). The rest mass refers to the mass of an object when it is at rest, and the equation indicates that the mass increases due to the contribution of energy.

Therefore, this equation explains the relationship between the mass and energy of an object based on the principles of relativity. It demonstrates that the mass of an object can change when its energy increases or decreases.

Shinsuke Hamaji.

E/c² = M', where M' is relativistic mass. Substituting in your equation: M' = M₀ + M' , which is contradiction. Your equation?

In general the relativistic mass is m₁ = γm₀. But mass-energy equivalence mass and relativistic mass are different. Where did you learn physics?

There seems to be a confusion between the energy of which the mass of bodies is made and the energy related to their motion.

The "rest mass" of a body concerns only the energy of which the mass of a body is made as if it was not moving, denoted m_0 (m_zero).

The energy of which this mass is made can be calculated with E=m_0 c2. For example, the energy of which the mass of an immobile car is made.

When it is moving, the energy of bodies related to their motion (their momentum kinetic energy) can be calculated with equation K=m_0 (gamma-1), or alternately with E=gamma (m_0 v2)/2, which is an equivalent equation. So the total energy of which the "moving car" is made will be E=m_0 c2 + K=m_0 (gamma-1).

In Newtonian mechanics and Einsteinian SR mechanics, the concept of "potential energy" is related only to the momentum energy of a body. When its velocity reduces to zero, then its momentum kinetic energy (K) is deemed to "convert" to potential energy, and if the body increases again its velocity, then the potential energy is deemed to convert back to kinetic energy.

In electromagnetic mechanics, the concept of potential energy does not exist. Kinetic energy is adiabatically induced in elementary particles only as a function of the distances separating charged particles.

Shinsuke Hamaji:

It is not a matter whether I studied physics course either in Japan or India. The matter is to provide the right information towards the discussion.

André Michaud , No, potential energy is useless because E=m₉c² is gravitational mass energy, not limited to E=m₀c². Restricted to E=m₀c², the mechanical energy cannot be replaced by the mass-energy equivalent.

Before someone can really understand Special Relativity, kinetic mechanics and electromagnetic mechanics, it is important to first study to full understanding Newtonian Mechanics. This is the best advice I can give people in this thread.

Only then the terms used will have the same meaning for everyone in the discussion.

André Michaud , No, you can really understand special relativity, kinetic mechanics, and electrodynamics to fully understand Newtonian mechanics.

5-2. Equivalence principle of the light's momentum, and the　perihelion shift error

https://note.com/s_hyama/n/n36da9fb827df

Shinsuke Hamaji Sorry to say, but you have it in reverse. Before building the roof of a house, it is a better practice to begin by building the foundation and the walls.

The same to understand what has been progressively understood over the course of history.

André Michaud , Yes, The same to understand what has been progressively understood over the course of history. However, Newton's relative time cannot be understood without understanding other theories.

Chinnaraji Annamalai ,So are you asking about the equivalence of mass and energy for potential energy? Or are you asking about the relationship with relativistic mass? If you don't know the difference, then you're out of the question.

Dear Professor Shinsuke Hamaji, Hope you are doing well.

Let us start freshly.

In the Einstein's general and/or special theory of relativity, (i). is there any definition or equation for potential energy? (ii). What is the energy equation relating to both relativistic mass and rest mass?

Shinsuke Hamaji

Newton did not use the concept of relative time. This came about only much later with Voigt, Lorentz, Poincaré and a few others at the turn of the 20th century.

Again, Nothing of these later developments can be completely understood before a solid foundation has been gained from totally understanding Newtonian mechanics, that still completely underlies all applications at our macroscopic levels, including electromagnetic mechanics.

Chinnaraji's questions are well formulated and they deserve well formulated answers.

André Michaud , Yes, Newton proposed absolute and relative time, but he failed to formulate relative time. 　Formulated by action at a distance in the absolute stationary coordinate system. However, it is necessary to consider relative time in terms of proximity effects.

Chinnaraji Annamalai , In the Einstein's general and/or special theory of relativity, (i). is there any definition or equation for potential energy? (ii). What is the energy equation relating to both relativistic mass and rest mass?

In Einstein's equivalence principle, it is nothing. GR expresses it as a space-time distortion. But in Newton's relative time dilation it can be replaced. I showed it to you with 1.

In short, the relativistic mass of SR and the spacetime distortion of GR can only be integrated by the difference of both inertial and gravitational mass.

Shinsuke Hamaji.

Preprint The Einstein’s Mass-Energy Equivalence and the Relativistic ...

Chinnaraji Annamalai , On a related note, I also commented on your other question.

https://www.researchgate.net/post/Photons_have_zero_mass_P_mc_If_m_0_P_0_No_light_energy/4

Light is an example of electromagnetic radiation and has no mass, so it has neither kinetic nor potential energy. Taken out from the webpage: https://www3.uwsp.edu/cnr-ap/KEEP/nres633/Pages/Unit1/Section-B-Two-Main-Forms-of-Energy.aspx#:~:text=Light%20is%20an%20example%20of,neither%20kinetic%20nor%20potential%20energy.

Light has no potential energy because light can not be stored- it is true. But light has kinetic energy. Light travels from a source (for example, sunlight). E =hf = hc/λ, which is a form of kinetic energy. Also, E = mc^2 is a form of kinetic energy.

Rest mass energy is not equal to potential energy because rest mass energy is a form of kinetic energy(E = m_0 * c^2), where m_0 is rest mass and c is the velocity of light. If the velocity of light is zero, we may say that E is the potential energy, but that is not true. Because E = 0 (Zero potential energy).

Potential energy does not exist in reality. It is only a mathematical concept to calculate forces. Potential energy does not have a mass equivalent.

What exists is the energy density of force fields. This energy density has a mass equivalent. The correct force equation is as follows:

The force on an object, which contributes to a force field, is given by the derivative of the energy content in the field overlay in respect to the distance of the object to the source of the force field.

This law holds for all kinds of force fields. It is exclusively based upon "energy equals force times path way".

But it is correct that calculating forces according to that law would be much more complicated than using the potential gradient. We are lucky that the potential gradient and the derivative of the energy content lead to the same force value. A difference only occurs, if we want to locate the mass equivalent of the potential energy. In this case it becomes obvious that potential energy cannot have a mass equivalent.

Another point is that potential energy suggests a remote instanteneous gravitational impact over large distances. This would contradict to the maximum transfer speed of energy, which is the speed of light.

The energy content of the field overlay is locally present. With the force equation above, it does not cause any logistic problems with the gravitational force, and it does not need something like a spooky remote action.

The last point is that the energy based explanation of the gravitational force makes something like "gravitons" definitively obsolete.

Chinnaraji Annamalai , Your argument is correct, but the question is "What theory is it based on?"

https://www.researchgate.net/post/Is_Einsteins_theory_of_relativity_upwardly_compatible_with_Newtonian_mechanics