Can someone help me to better understand energy

[Pattonias]

Ok, I'll read a bit about EM-waves.

 

[jambaugh]

I really appreciate the extended explanation. While I don't immediately know all the references and principles that you have laid out, I can use this as a starting point for more reading.

Another question.

Can and does energy exist apart from matter? And if it does then in what form? Can it attach itself to matter that it comes in contact with?

Gets tricky here qualifying things like this. You can define "matter" as anything which possesses energy (above the vacuum) and even then you can speak of the vacuum itself containing "matter" (like the Dirac sea).

But the best answer I can say is that every system has an energy even a block of empty space. Energy is a property, the value may be zero but that doesn't mean "there is no Energy". You can also in most cases speak of energy in a relative sense. I.e. it is changes in energy that are important not the absolute value.

The cosmological constant basically is a way of absorbing such resetting of the zero value while still giving meaning to the energy affecting space-time curvature. You can "reset" the zero energy (density) plus change the value of the cosmological constant to effect equivalent gravitational effects. (I make it sound simple but its tricky doing this. One also has to play with the definition of inertial mass when working with non-gravitational forces.)

 

[Dale]

I really don't think it is so complicated. Energy is the capacity to do work, work is force times distance. jambaugh is correct that there are all sorts of interesting advanced topics, but none of it changes the fundamental definitions, especially not at the level of the OP.

 

[Phyisab****]

I just want to say that Jambaugh, that was a very good explanation.

 

[stevefaulkner]

Would it be true to say that energy is the only conserved quantity that transforms as a scalar?

 

[Gerenuk]

Would it be true to say that energy is the only conserved quantity that transforms as a scalar?

If you consider frame transformations, which is different from the time evolution transformations, then any product of two vectors is a scalar (rest mass of a particle, ...).

Not sure how this refers to time evolution transformation. I think in general energy, momentum and angular momentum are the only integrals of motion unless you deal with a special problem.

Edit: Oh, I see. You meant something that is both
Then in general only energy should be the conserved scalar. Unless the problem is a special one.

 

[jambaugh]

Would it be true to say that energy is the only conserved quantity that transforms as a scalar?

Not at all. Firstly it is only scalar in a non-relativistic setting. In SR energy is one component of the 4-vector energy-momentum. Electric charge and the other gauge charges are space-time scalars (with some slight qualification on weak isospin due to weak parity issues).

The nice part of the link between conserved quantities and symmetry transformations is that how the quantities transform (whether they are scalar, vector, tensor, ...) with respect to the various groups (rotation, Lorentz, Poincare, ...) is made explicit by how the symmetry transformations themselves transform. Consider that translation in the x direction can be rotated to translation in e.g. the y direction. Hence rotations act not only in concert with translations on objects, but also rotations act on translations. This is called the adjoint action of one transformation on another.

Since time translation is "scalar" i.e. invariant under spatial transformations (rotations and spatial translations) Energy is scalar in this non-relativistic setting. Since spatial translation in a given direction gets rotated by a rotation transformation so does momentum. Momentum is a vector quantity.

In the relativistic setting you can think of energy as "momentum in the time direction" and Lorentz transformations act on the four momenta as a 4-vector.

One final note. You can imagine for some system that you can "tweak" the dynamics to have any arbitrary symmetry or that you break a given symmetry. This does not interfere with the link between measurable quantities (which may or may not be conserved) and transformations (which may or may not be symmetries).

EDIT: In short the link is there without worrying about the quantity being conserved or the transformation being a symmetry :END EDIT

This is made most explicit in the formulation of quantum mechanics where the operators generating the transformations are identified mathematically with the operators corresponding to the observables.
The momenta ARE the generator of translations,
Energy (the Hamiltonian) IS the generator of time evolution,
Angular Momenta ARE the generators of rotations.

It is all very elegant and beautiful. That it also yields an accurate description of how nature behaves is gravy on the biscuit!