Komar mass
Encyclopedia
The Komar mass of a system is one of several formal concepts of mass
that are used in general relativity
. The Komar mass can be defined in any stationary spacetime
, which is a spacetime
in which all the metric
can be written so that they are independent of time. Alternatively, a stationary spacetime can be defined as a spacetime which possesses a timelike Killing vector field
.
The following discussion is an expanded and simplified version of the motivational treatment in (Wald, 1984, pg 288).
. Using the Schwarzschild basis, a frame field for the Schwarzschild metric, one can find that the radial acceleration required to hold a test mass stationary at a Schwarzschild coordinate of r is:
Because the metric is static, there is a well-defined meaning to "holding a particle stationary".
Interpreting this acceleration as being due to a "gravitational force", we can then compute the integral of normal acceleration multiplied by area to get a "Gauss law" integral of:
While this approaches a constant as r approaches infinity, it is not a constant independent of r. We are therefore motivated to introduce a correction factor to make the above integral independent of the radius r of the enclosing shell. For the Schwarzschild metric, this correction factor is just
, the "red-shift" or "time dilation" factor at distance r. One may also view this factor as "correcting" the local force to the "force at infinity", the force that an observer at infinity would need to apply through a string to hold the particle stationary. (Wald, 1984).
To proceed further, we will write down a line element for a static metric.
where gtt and the quadratic form are functions only of the spatial coordinates x,y,z and are not functions of time. In spite of our choices of variable names, it should not be assumed that our coordinate system is Cartesian. The fact that none of the metric coefficients are functions of time make the metric stationary: the additional fact that there are no "cross terms" involving both time and space components (such as dx dt) make it static.
Because of the simplifying assumption that some of the metric coefficients are zero, some of our results in this motivational treatment will not be as general as they could be.
In flat space-time, the proper acceleration required to hold station is du/d tau, where u is the 4-velocity of our hovering particle and tau is the proper time. In curved space-time, we must take the covariant derivative. Thus we compute the acceleration vector as:
where ub is a unit time-like vector such that ub ub = -1.
The component of the acceleration vector normal to the surface is
where Nb is a unit vector normal to the surface.
In a Schwarzschild coordinate system, for example, we find that
as expected - we have simply re-derived the previous results presented in a frame-field in a coordinate basis.
We define
so that in our Schwarzschild example 
.
We can, if we desire, derive the accelerations ab and the adjusted "acceleration at infinity" ainfb from a scalar potential Z, though there is not necessarily any particular advantage in doing so. (Wald 1984, pg 158, problem 4)
We will demonstrate that integrating the normal component of the "acceleration at infinity" ainf over a bounding surface will give us a quantity that does not depend on the shape of the enclosing sphere, so that we can calculate the mass enclosed by a sphere by the integral
To make this demonstration, we need to express this surface integral as a volume integral. In flat space-time, we would use Stokes theorem and integrate
over the volume. In curved space-time, this approach needs to be modified slightly.
Using the formulas for electromagnetism in curved space-time
as a guide, we write instead.
where F plays a role similar to the "Faraday tensor", in that
We can then find the value of "gravitational charge", i.e. mass, by evaluating
and integrating it over the volume of our sphere.
An alternate approach would be to use differential forms, but the approach above is computationally more convenient as well as not requiring the reader to understand differential forms.
A lengthly, but straightforward (with computer algebra) calculation from our assumed line element shows us that
Thus we can write
In any vacuum region of space-time, all components of the Ricci tensor must be zero. This demonstrates that enclosing any amount of vacuum will not change our volume integral. It also means that our volume integral will be constant for any enclosing surface, as long as we enclose all of the gravitating mass inside our surface. Because Stokes theorem guarantees that our surface integral is equal to the above volume integral, our surface integral will also be independent of the enclosing surface as long as the surface encloses all of the gravitating mass.
By using Einstein's Field Equations
letting u=v and summing, we can show that R = -8 π T.
This allows us to rewrite our mass formula as a volume integral of the stress-energy tensor.
Note that
replaces 
in our motivational result.
If none of the metric coefficients
are functions of time, 
While it is not necessary to chose coordinates for a stationary space-time such that the metric coefficients are independent of time, it is often convenient.
When we chose such coordinates, the time-like Killing vector for our system
becomes a scalar multiple of a unit coordinate-time vector 
, i.e 
. When this is the case, we can rewrite our formula as
Because
is by definition a unit vector, K is just the length of 
, i.e. K = 
.
Evaluating the "red-shift" factor K based on our knowledge of the components of
, we can see that K = 
.
If we chose our spatial coordinates so that we have a locally Minkowskian metric
we know that
With these coordinate choices, we can write our Komar integral as
While we can't choose a coordinate system to make a curved space-time globally Minkowskian, the above formula provides some insight into the meaning of the Komar mass formula. Essentially, both energy and pressure contribute to the Komar mass. Furthermore, the contribution of local energy and mass to the system mass is multiplied by the local "red shift" factor
The formula for the Komar mass in terms of the metric and its Killing vector is (Wald, 1984, pg 289, formula 11.2.9)
The surface integral above is interpreted as the "natural" integral of a two form over a manifold.
As mentioned previously, if none of the metric coefficients
are functions of time, 
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
that are used in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. The Komar mass can be defined in any stationary spacetime
Stationary spacetime
In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike....
, which is a spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
in which all the metric
Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
can be written so that they are independent of time. Alternatively, a stationary spacetime can be defined as a spacetime which possesses a timelike Killing vector field
Killing vector field
In mathematics, a Killing vector field , named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold...
.
The following discussion is an expanded and simplified version of the motivational treatment in (Wald, 1984, pg 288).
Motivation
Consider the Schwarzschild metricSchwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
. Using the Schwarzschild basis, a frame field for the Schwarzschild metric, one can find that the radial acceleration required to hold a test mass stationary at a Schwarzschild coordinate of r is:
Because the metric is static, there is a well-defined meaning to "holding a particle stationary".
Interpreting this acceleration as being due to a "gravitational force", we can then compute the integral of normal acceleration multiplied by area to get a "Gauss law" integral of:
While this approaches a constant as r approaches infinity, it is not a constant independent of r. We are therefore motivated to introduce a correction factor to make the above integral independent of the radius r of the enclosing shell. For the Schwarzschild metric, this correction factor is just
, the "red-shift" or "time dilation" factor at distance r. One may also view this factor as "correcting" the local force to the "force at infinity", the force that an observer at infinity would need to apply through a string to hold the particle stationary. (Wald, 1984).To proceed further, we will write down a line element for a static metric.
where gtt and the quadratic form are functions only of the spatial coordinates x,y,z and are not functions of time. In spite of our choices of variable names, it should not be assumed that our coordinate system is Cartesian. The fact that none of the metric coefficients are functions of time make the metric stationary: the additional fact that there are no "cross terms" involving both time and space components (such as dx dt) make it static.
Because of the simplifying assumption that some of the metric coefficients are zero, some of our results in this motivational treatment will not be as general as they could be.
In flat space-time, the proper acceleration required to hold station is du/d tau, where u is the 4-velocity of our hovering particle and tau is the proper time. In curved space-time, we must take the covariant derivative. Thus we compute the acceleration vector as:
where ub is a unit time-like vector such that ub ub = -1.
The component of the acceleration vector normal to the surface is
where Nb is a unit vector normal to the surface.In a Schwarzschild coordinate system, for example, we find that
as expected - we have simply re-derived the previous results presented in a frame-field in a coordinate basis.
We define
so that in our Schwarzschild example .We can, if we desire, derive the accelerations ab and the adjusted "acceleration at infinity" ainfb from a scalar potential Z, though there is not necessarily any particular advantage in doing so. (Wald 1984, pg 158, problem 4)
We will demonstrate that integrating the normal component of the "acceleration at infinity" ainf over a bounding surface will give us a quantity that does not depend on the shape of the enclosing sphere, so that we can calculate the mass enclosed by a sphere by the integral
To make this demonstration, we need to express this surface integral as a volume integral. In flat space-time, we would use Stokes theorem and integrate
over the volume. In curved space-time, this approach needs to be modified slightly.Using the formulas for electromagnetism in curved space-time
Maxwell's equations in curved spacetime
In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system...
as a guide, we write instead.
where F plays a role similar to the "Faraday tensor", in that
We can then find the value of "gravitational charge", i.e. mass, by evaluating and integrating it over the volume of our sphere.An alternate approach would be to use differential forms, but the approach above is computationally more convenient as well as not requiring the reader to understand differential forms.
A lengthly, but straightforward (with computer algebra) calculation from our assumed line element shows us that
Thus we can write
In any vacuum region of space-time, all components of the Ricci tensor must be zero. This demonstrates that enclosing any amount of vacuum will not change our volume integral. It also means that our volume integral will be constant for any enclosing surface, as long as we enclose all of the gravitating mass inside our surface. Because Stokes theorem guarantees that our surface integral is equal to the above volume integral, our surface integral will also be independent of the enclosing surface as long as the surface encloses all of the gravitating mass.
By using Einstein's Field Equations
letting u=v and summing, we can show that R = -8 π T.
This allows us to rewrite our mass formula as a volume integral of the stress-energy tensor.
- where V is the volume being integrated over
- Tab is the Stress-energy tensorStress-energy tensorThe stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
- ua is a unit time-like vector such that ua ua = -1
Komar mass as volume integral - general stationary metric
To make the formula for Komar mass work for a general stationary metric, regardless of the choice of coordinates, it must be modified slightly. We will present the applicable result from (Wald, 1984 eq 11.2.10 ) without a formal proof.- where V is the volume being integrated over
- Tab is the Stress-energy tensorStress-energy tensorThe stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
- ua is a unit time-like vector such that ua ua = -1
is a Killing vector, which expresses the time-translation symmetry of any stationary metric. The Killing vector is normalized so that it has a unit length at infinity, i.e. so that 
at infinity.
Note that
replaces in our motivational result.If none of the metric coefficients
are functions of time, While it is not necessary to chose coordinates for a stationary space-time such that the metric coefficients are independent of time, it is often convenient.
When we chose such coordinates, the time-like Killing vector for our system
becomes a scalar multiple of a unit coordinate-time vector , i.e . When this is the case, we can rewrite our formula asBecause
is by definition a unit vector, K is just the length of , i.e. K = .Evaluating the "red-shift" factor K based on our knowledge of the components of
, we can see that K = .If we chose our spatial coordinates so that we have a locally Minkowskian metric
we know thatWith these coordinate choices, we can write our Komar integral as
While we can't choose a coordinate system to make a curved space-time globally Minkowskian, the above formula provides some insight into the meaning of the Komar mass formula. Essentially, both energy and pressure contribute to the Komar mass. Furthermore, the contribution of local energy and mass to the system mass is multiplied by the local "red shift" factor
Komar mass as surface integral - general stationary metric
We also wish to give the general result for expressing the Komar mass as a surface integral.The formula for the Komar mass in terms of the metric and its Killing vector is (Wald, 1984, pg 289, formula 11.2.9)
- where
are the Levi-civita Levi-Civita symbolThe Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
symbols
is the Killing vector of our stationary metric, normalized so that 
at infinity.
The surface integral above is interpreted as the "natural" integral of a two form over a manifold.
As mentioned previously, if none of the metric coefficients
are functions of time,