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kaleidoscope
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I know you need to use some spacetime geometry to solve some conservation equations but what is the simplest way you derive the following equation about energy and momentum:
E2 = (pc)2 + (mc2)2
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kaleidoscope said:I know you need to use some spacetime geometry to solve some conservation equations but what is the simplest way you derive the following equation about energy and momentum:
E2 = (pc)2 + (mc2)2
kaleidoscope said:I know you need to use some spacetime geometry to solve some conservation equations but what is the simplest way you derive the following equation about energy and momentum:
E2 = (pc)2 + (mc2)2
Petr Mugver said:[tex](d\tau)^2=dx^\mu dx_\mu[/tex]
is the proper time of the particle, that you can write
.
Altabeh said:Just a minor error: Following the Riemannian line-element for a general spacetime, this must have been
[tex](cd\tau)^2=dx^\mu dx_\mu[/tex]
because the LHS is just the square of the differential of proper length, [tex]ds^2.[/tex]
AB
bcrowell said:I agree with Petr that the answer to this really depends on what you're willing to assume.
Suppose you want these things: (1) A four-vector exists that is a generalization of the momentum three-vector from Newtonian mechanics. (2) The relationship between the relativistic and Newtonian versions satisfies the correspondence principle. (3) The quantity is additive (because we hope to have a conservation law). Then I think it's quite difficult to come up with any other definition for the momentum four-vector than the standard one, and then the [itex]E^2=p^2+m^2[/itex] identity follows immediately. But this isn't anything like a proof of uniqueness or self-consistency.
Petr's derivation is likewise very natural, but it depends on the assumption of a certain form for the action, and there's no guarantee that the results it outputs obey the correspondence principle or result in a conservation law. Those properties have to be checked mathematically and experimentally.
DaTario said:Is there a derivation which only takes into account the LT's and Newton's Second Law ?
Best wishes
DaTario
jtbell said:Another way: assume the equations for energy and momentum,
[tex]E = \frac{mc^2}{\sqrt{1 - v^2/c^2}}[/tex]
[tex]p = \frac{mv}{\sqrt{1 - v^2/c^2}}[/tex]
and solve them together algebraically to eliminate v.
The formula for deriving E2= (pc)2 + (mc2)2 is based on Einstein's famous equation, E=mc2, which states that energy (E) is equal to mass (m) multiplied by the speed of light (c) squared. In this equation, p represents momentum, which is the product of an object's mass and velocity.
The "2" in E2 represents the square of the terms (pc) and (mc2). This is because energy, momentum, and mass are all conserved quantities, meaning that they cannot be created or destroyed, but can be transferred or converted into one another. Squaring these terms allows for a more accurate calculation of the total energy.
The units of measurement for the individual terms in E2= (pc)2 + (mc2)2 are as follows: E is measured in joules (J), p is measured in kilogram meters per second (kg*m/s), and m is measured in kilograms (kg). When the terms are squared, the units become J2, (kg*m/s)2, and kg2, respectively. Therefore, the overall units for E2 are J2 + (kg*m/s)2 + kg2.
E1= (pc)2 is a simplified form of the equation, where the mass (m) is assumed to be at rest and therefore has no kinetic energy. This is commonly used in classical mechanics. However, in E2= (pc)2 + (mc2)2, both momentum (p) and mass (m) are taken into account, making it a more accurate and comprehensive equation that is used in modern physics.
E2= (pc)2 + (mc2)2 is used in various fields of physics, such as particle physics, nuclear physics, and astrophysics. It is used to calculate the energy of particles, such as in the Large Hadron Collider, and to understand the behavior of matter in extreme conditions, such as in nuclear reactions and black holes. It is also used in medical applications, such as in positron emission tomography (PET) scans, which use the equation to detect and measure the energy emitted by radioactive substances in the body.