This document contains comments about the document "Mass in general relativity" in Wikipedia
• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

The article starts with the following sentence.
The concept of mass in general relativity (GR) is more complex than the concept of mass in special relativity
The concept of mass in SR is also complex, because how is it calculated?
In fact, general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances.
In SR you also have similar problems, because what exactly is a rest-mass and why does mass change as a function of speed?

### 1. Review of mass in special relativity

In special relativity, the invariant mass or rest mass (hereafter simply "mass") of an isolated system can be defined in terms of the energy and momentum of the system by the relativistic energy–momentum equation:
m = sqrt{E^2 - (p*c)^2} / c^2
Why do they use the word "system"?
A system, IMO, assumes more than one object. But how can they be all at rest?
How is the Energy (and momemtum) calculated? This are serious problems in our understanding.

### Reflection 1 "Gravitation by MTW" 1973

In Paragraph \$19.2 the "Measurement of the Mass and angular Momentum" is dicussed.
The values of a sytem's mass and angular momentum can be measured by probing the imprint they leave in its external gravitational field
IMO it is impossible in practice to directly measure the gravitational field. The only thing you can measure are how different objects influence each other.
Of all the tools one might use to probe, the simplest is a tset particle in a gravitationally bound orbit,
That means we in practice are studying the Sun and one small planet.
To determine the source's mass M one need only apply Kepler's third law. (perhaps better called "Keplers 1-2-3 law")
That means in a more general sense we use Newton's Law.
The same subject is also discussed at the pages 450,457 and 636ff. See Subject index under: Mass-energy

In paragraph \$19.3 the "Mass and Angular Momentum of fully relativistic sources" is discussed At page 452 we read:

Two types of nonlinearities turn out far from the source: (1) non linearities in the static Newtonian part of the metric, which generate metric corrections etc etc
therby putting the metric in the form ds^2 = function of (M,r,dt^2,Sk,delta jk dxjdxk) (19.13)
The mass of the sun is measured in practice by studying the orbits of the planets in its external gravitational field, a procedure equivalent to reading the Mass M of the line element (19.13) rather than evaluating the volume integral "integral T00 d3x"
Equation (19.13) is a rather complex equation. See above. Next we read:
one defines the "total mass-energy" M of the sun or any other body to be the constant that appears in the line element (19.13) for its distant external spacetime geometry.
The question is how is this done in reality if more stars are involved.

In paragraph \$20.4 "Why the energy of the gravitational filed cannot be localized" the discussion is about the Earth-moon system and a neutron star.
In paragraph \$20.5 "Conservation laws for total 4-Momemtum and Angular Momemtum" the discussion is about our Galaxy or Solar system.

this force is to be calculated as half the difference between the retarded field and the advanced field caused by that particle. This difference is singularity free
Interesting.

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Created: 1 November 2016

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