# Mass in general relativity

1. Dec 28, 2012

### arindamsinha

I have a broader question on this - is mass at all properly defined in GR context, including rest mass or so-called invariant mass?

Leaving aside concept of mass gain, I have been under the impression that mass itself is defined in different ways in different situations in GR, with caveats, such that there is really no common definition of mass, and that applies even to rest/invariant mass.

Ref: http://en.wikipedia.org/wiki/Mass_in_general_relativity

It is rather surprising. Any thoughts?

2. Dec 28, 2012

### pervect

Staff Emeritus
The wiki is a reasonably accurate summary of the status of mass in GR. GR can define mass in various special circumstances, but there's no universal single good definition for it.

3. Dec 28, 2012

the intrinsic mass is invariant under lorentz transformation. the whole dynamic mass discussion is really confusing but practical sometimes

4. Dec 28, 2012

### Staff: Mentor

If you have small object in GR then you can use a four-vector to describe its energy and momentum and mass (see: http://en.wikipedia.org/wiki/Four-momentum). That mass is invariant.

The problem arises when you want to compute the mass of an extended object. The curvature of spacetime makes it so that there is no unique way to chop it up into small sub-objects and then add them together to get a total mass. Hence the different definitions of mass in GR and all of the caveats associated with each.

5. Dec 28, 2012

### arindamsinha

That's the impression I got.

The point is, what is intrinsic mass in GR? I will come back to that question later in the thread, if some of my thoughts below turn out to be meaningful.

That's a good way of characterizing the problem.

One side question on this: The masses of the Sun, Earth etc. are estimated using Newtonian concepts I believe. Could they be very different in GR? (This is just a curiosity question, not one of the key questions I am trying to address in this post)

---------------------------------
Now I come to some questions that has been bothering me for some time.

Q1) The relativistic energy momentum equation from SR is:
$$E = \frac{mc^2}{\sqrt{1 - v^2/c^2}}$$
I understand that the quantity E/c2 was earlier referred to as 'relativistic mass' but that term is no longer used. However, considering that energy has increased with velocity, why can we not correctly call it 'relativistic mass' using the concept of mass energy equivalence?

If energy increases, isn't it also a mass increase? The gravity of the object does increase, I believe.

Q2) I have been thinking that if the above is the energy momentum equation for SR, would it be correct to extend (with some simplification of course) it in GR context, to have the following corresponding equation?
$$E = \frac{mc^2}{\sqrt{1 - v^2/c^2}\sqrt{1 - 2GM/Rc^2}}$$
I have not seen this anywhere, which is why I want to verify if this is even correct and makes sense. If it does, then I will have some further follow up questions

6. Dec 28, 2012

### pervect

Staff Emeritus
There is something called the "energy at infinity" which is a conserved constant of motion for an orbiting particle. However, it's not equal to your expression.

from http://www.fourmilab.ch/gravitation/orbits/

Schwarzschild coordinates are used throughout. $\tau$ is proper time, it's easy to confuse unfortunatly with r, the Schwarzschild radial coordinate.

~E = potential energy at infinity per unit rest mass

$$\~ E^2 = \left( \frac{dr}{d\tau} \right)^2 + \~ V^2(r,\tau) = \left( \frac{dr}{d\tau} \right)^2 + \left( 1-2M/r\right) \left( 1 + \~ L^2/r^2\right) =$$
$$\~ E^2 = \left( \frac{dr}{d\tau} \right)^2 + \left( 1-2M/r\right) \left( 1 + \left(r\frac{d\phi}{d\tau} \right) ^2\right)$$

You might or might not be able to factor that into a function that depends on v and r, with the proper choice of what you mean by "v".

Basically, the good news is that there is an answer in this case that I think may be similar to what you're looking for. The bad news is that you're not going to find it by pursuing quasi-Newtonian thinking. You really need to learn GR to learn GR. And you really need to learn SR before you can learn GR,

7. Dec 28, 2012

### arindamsinha

OK, that's good. I agree with that last bit of advice. I have known that for a long time, but am not anywhere near that level of knowledge yet. In the meantime I am trying to understand a few things a bit better by asking certain questions, which may sometimes be a bit naive.

I threw in the equation to try to convey my meaning rather than trying to ask whether it is mathematically correct. It saved me a thousand words perhaps. It seems to have served its purpose, since you have got my meaning.

I think it appears that the invariant mass in SR is not necessarily invariant in GR terms, because the gravitational potential does get involved. In other words, a particular amount of matter may have slightly different 'invariant masses' in different gravitational potentials. Would you say that this is correct?

In that case, my follow up question is this: Is even 'invariant mass' an intrinsic property of matter, or is it itself entirely determined by gravity/spacetime curvature?

8. Jan 5, 2013

### arindamsinha

I was really interested in getting some insights on this... the main questions being around whether 'invariant mass' is an intrinsic property of matter, or whether it is entirely of gravitic origin (or anything else) in some way?

My reason for asking - SR defines something called 'invariant mass', while GR does not seem to have a clear definition of mass at all, unless some 'qualifications' are made... GR being the more general theory encompassing SR, this seems a very unsatisfactory situation...

I am not necessarily looking for any 'proven' answers, just a free discussion on the thoughts of people on this...

9. Jan 5, 2013

### Staff: Mentor

The invariant mass in SR remains invariant in GR for small objects. It is only for large objects where curvature becomes important.

10. Jan 5, 2013

### PAllen

With the obvious caveat implied in earlier post that if you have a bunch of small of small objects, you can't treat the collection of them without running into the general issues of mass in GR.

As to how different physicists feel about the mass situation in GR: it varies (some are willing to accept that such ambiguities are a feature of reality - there are reasons to believe in much broader frameworks than just GR; others believe it represents a deficiency in GR, with no better alternative at present). The basic problem was recognized almost immediately on publication of the theory. Understanding of what can and cannot be achieved is ongoing, on two fronts:

- what are the most general conditions under which global definitions are valid with desired conservation properties (i.e., how much more general than asymptotic flatness can be made to work)?

- how well can quasi-local definitions of mass, momentum, and angular momentum be made to work with desired properties. This was a hot area for a while, but I haven't seen much in the way of new results in the last 10 years. There are proofs that achieving 'all desired properties' is impossible.

Last edited: Jan 5, 2013
11. Jan 5, 2013

### Staff: Mentor

You are correct. I probably should have said "a small object" instead of "small objects". That would have been more clear.

12. Jan 6, 2013

### arindamsinha

Most interesting.
(a) What are the basic differences between the two points of view you mention?
(b) Can you give me some references which provide some detailed insights on these?

Again, very interesting. Is this in some way related to the 'uncertainty principle'? Or is it because we cannot practically hope to gain all detailed knowledge of the Universe to make this possible? Or some other reason altogether?

I get the sense of what you are saying, and understood that from your previous post also. What exactly is the difficulty with this in GR, if you can put it in layman terms?

To amplify based on the previous responses - if we can chop a big mass up into small bits which have invariant masses - why would the sum of them not give us the total invariant mass of the extended object? What comes in the way? Again, in layman terms if possible.

13. Jan 6, 2013

### Staff: Mentor

The issue is parallel transport. In curved spacetimes parallel transporting a vector along different paths gives different results. See figure b in section 3.1.2 here: http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.1 [Broken]

If you have two objects in SR each object has an energy and a momentum which can be combined into a vector called the four-momentum or the energy-momentum four-vector. Then, to find the four-momentum of the system of the two objects you simply add the two vectors. In flat spacetime you can do that since all paths give the same result. But in GR when you try to do the same thing you get different answers depending on which path you take to move the vectors together.

Last edited by a moderator: May 6, 2017