Bondi mass-loss formula

In the universe described by Einstein's general relativity, violent events like the collision of black holes shake the very fabric of spacetime, radiating away colossal amounts of energy as gravitational waves. But if energy escapes, then according to $E=mc^2$, the system's mass must decrease. This raises a fundamental question: how do we define and track the mass of a system that is constantly changing? The static concept of mass is insufficient for these dynamic cosmic cataclysms. The Bondi mass-loss formula provides the elegant and powerful answer to this puzzle. This article delves into this cornerstone of gravitational physics. First, in "Principles and Mechanisms," we will explore the core concepts of Bondi mass, the "news function," and the distant boundary of spacetime where radiation is measured. Following that, "Applications and Interdisciplinary Connections" will reveal how this theoretical tool is used to interpret gravitational wave signals, account for the energy of black hole mergers, and test the limits of Einstein's theory itself.

Imagine you are trying to weigh a spinning, sputtering firecracker. The task seems impossible. Not only is it moving, but it's actively throwing parts of itself away as sparks and smoke. How can you define its "mass" when it's constantly changing? This is precisely the dilemma physicists faced when thinking about dynamic, violent events in the cosmos, like the collision of two black holes. Einstein's theory tells us that these events must shake the very fabric of spacetime, sending out ripples of gravitational energy. And since $E=mc^2$, if energy is being radiated away, the mass of the system must be decreasing.

But how do we write down a law for this? The mass of what, exactly? And how does it relate to the radiation being emitted? The beautiful framework developed by Hermann Bondi, M. G. J. van der Burg, A. W. K. Metzner, and Rainer K. Sachs provides the answer. It’s a story about finding the right place to stand, figuring out what to listen for, and writing down the universal law that connects them.

Our intuitive notion of mass is something static and unchanging—the "amount of stuff" in an object. For a stable, isolated system like our solar system (to a good approximation), this works fine. General relativity gives this a precise name: the ​​ADM mass​​ ($M_{ADM}$), named after Arnowitt, Deser, and Misner. You can think of it as the total gravitational charge of a system as seen from infinitely far away. It's a conserved quantity; it represents the total, unchanging energy content of the entire spacetime.

But what about our firecracker—or a pair of merging black holes? Such systems are radiating. A single, unchanging number can't capture the physics of energy being lost over time. We need a concept of mass that can evolve. This is the ​​Bondi mass​​, denoted $m_B$. Unlike the ADM mass, which is like a statement of your total wealth locked in a vault, the Bondi mass is like the balance in your current account. It can decrease as you spend money—or as the system radiates energy.

So, how do these two masses relate? For a system that starts out static, becomes dynamic, and then settles down again, the ADM mass is equal to the initial Bondi mass before any radiation has occurred. The ADM mass represents the "initial deposit" in the spacetime bank account. The Bondi mass, $m_B(u)$, tracks the balance at any given moment, and the difference between the initial and final Bondi mass is precisely the total energy radiated away.

If energy is escaping, where does it go? Like ripples from a stone dropped in a pond, gravitational waves travel outwards at the speed of light. To measure the total energy lost, we must build a "dam" that catches all the escaping ripples. In spacetime, the ultimate destination for all outgoing light (and gravitational waves) is a place physicists call ​​future null infinity​​, denoted $\mathcal{I}^{+}$. It’s not a physical place you can travel to in a spaceship, but rather a conceptual boundary of our spacetime, the asymptotic "celestial sphere" where all light rays from an event eventually arrive.

This is the key insight of the Bondi formalism: don't try to weigh the messy, evolving source directly. Instead, go to the far-away boundary of spacetime and measure what escapes. To do this, we need a clock that's suited for this perspective. This is the ​​retarded time​​, $u = t - r/c$. It simply means that if an event happens at the source at time $t=0$, an observer at a huge distance $r$ will only see it at time $t = r/c$. The retarded time $u$ labels the wavefronts as they arrive at infinity, allowing the distant observer to reconstruct a history of the source's activity.

When we look out at this celestial sphere at future null infinity, what do we see? If the source back home is perfectly quiet and unchanging, we see nothing new. But if the source is dynamic—say, two neutron stars are spiraling into each other—it sends out "news" in the form of gravitational waves. This "news" is captured by a mathematical object called the ​​news function​​, $N(u, \theta, \phi)$. It tells us, for each moment of retarded time $u$ and in each direction on the sky $(\theta, \phi)$, what new information is arriving about the source's changing gravitational field.

What, physically, is this news? It is the time rate of change of the ​​shear​​ of outgoing light fronts. Imagine a perfectly spherical flash of light emitted from the source. In empty, flat spacetime, it remains a perfect sphere as it expands. But a gravitational wave will distort it, stretching it in one direction and squeezing it in a perpendicular direction. This distortion is called shear. The news function measures how fast this shear pattern is changing. A static, distorted field (like that of a single, non-spinning, slightly oblong potato-shaped star) has a constant shear but zero news. It's only when the distortion itself is changing—the potato is wobbling—that we get "news" and thus radiation.

Crucially, not just any motion will do. Consider a star that is perfectly spherical but pulsates in and out. Its radius changes, its density changes, its surface gravity changes. Surely that's a dynamic event? Yet, as a consequence of Birkhoff's theorem, the spacetime outside remains perfectly static (it's the Schwarzschild geometry), and no gravitational waves are emitted. The news function is identically zero. This is because gravitational radiation has no monopole (spherical) component. It starts at the quadrupole level—you need a changing shape, like a spinning dumbbell, to generate waves. The universe doesn't radiate gravitationally if you just "breathe" in and out spherically.

Now we can put all the pieces together. The Bondi mass, $m_B(u)$, is the energy account balance at time $u$. The news function, $N(u, \theta, \phi)$, is the signal of expenditure arriving at that time. The connection between them is the celebrated ​​Bondi mass-loss formula​​: $\frac{dm_B}{du} = - \frac{1}{4\pi G} \int_{S^2} |N(u, \theta, \phi)|^2 d\Omega$ (Here we've restored the gravitational constant $G$; in many theoretical calculations it's set to 1). This equation is a masterpiece of physical insight. Let's break it down.

First, notice the minus sign. The rate of change of mass is negative (or zero). A system can only lose mass by radiating; it can never gain it. The equation enforces the second law of thermodynamics for gravitational radiation.

Second, the rate of loss depends on the integral of $|N|^2$ over the entire celestial sphere. The term $|N|^2$ can be interpreted as being proportional to the ​​flux​​ of energy—the power per unit area—being carried away by the waves in a specific direction. By integrating over all directions (the solid angle $d\Omega$), we get the total power being radiated at that instant.

But why is it proportional to the square of the news function, $|N|^2$? This is a deep and universal feature of all waves. The energy carried by a wave—be it a wave on a string, a sound wave, or a light wave—is always proportional to the square of its amplitude. A wave with twice the amplitude carries four times the energy. This quadratic dependence ensures two crucial things:

1. ​​Positivity​​: Since $|N|^2$ is always non-negative, the energy flux is always directed outwards. Energy is always lost, never spontaneously gained from empty space.
2. ​​Reality​​: The news function is technically a complex number that encodes the two different polarizations of gravitational waves ($h_+$ and $h_\times$). The quantity $|N|^2$ is always a real, positive number, as a physical energy flux must be.

So, if there is no "news" ($N=0$), the right-hand side is zero, and the Bondi mass is constant. This happens before a merger begins, and it happens after the final object has settled into a quiet, stationary state. For example, when a system settles into a final Schwarzschild black hole, its Bondi mass becomes constant and is exactly equal to the mass parameter $M$ that characterizes the black hole.

With this framework, we can now tell the complete story of a cataclysmic event, like the merger of two black holes, from the perspective of energy conservation.

1. ​​In the distant past ($u \to -\infty$):​​ The black holes are far apart and orbiting slowly. They are effectively in a static state. The news function is zero. The Bondi mass is at its maximum initial value, $m_i$, which is equal to the total ADM mass of the spacetime, $M_{ADM}$.

2. ​​During the merger (finite $u$):​​ The black holes spiral together violently. The spacetime is furiously churned. The news function $N(u, \theta, \phi)$ becomes large and complex, broadcasting the details of this cosmic dance across the sky. For every moment that $N$ is non-zero, the Bondi mass $m_B(u)$ ticks downwards.

3. ​​In the distant future ($u \to +\infty$):​​ A single, spinning black hole remains. It has settled down and is now stationary. All the ripples have subsided. The news function returns to zero. The Bondi mass settles at a final, constant value, $m_f$. This $m_f$ is the mass of the new black hole.

The total mass lost during the event is simply the integral of the mass-loss rate over the entire duration of the event, or more simply, the difference between the initial and final mass: $\Delta m_B = m_f - m_i = -\Delta E_{\text{rad}}$ This $\Delta E_{\text{rad}}$ is the total energy radiated away in gravitational waves. For the first black hole merger ever detected (GW150914), the initial masses were about 29 and 36 solar masses, and the final mass was about 62 solar masses. The "missing" 3 solar masses were converted into a stupendous burst of energy in the form of gravitational waves, all perfectly accounted for by Bondi's elegant formula. It is a profound and beautiful confirmation that mass is not just a static property, but a dynamic quantity that participates in the great energy transactions of the universe.

Having acquainted ourselves with the beautiful machinery behind the Bondi mass-loss formula, we might be tempted to leave it as a pristine piece of theoretical physics. But that would be like forging a magnificent key and never trying to see which doors it unlocks. The true power and beauty of this formula lie not in its elegant derivation, but in its profound ability to connect the abstract geometry of spacetime to the most violent and energetic events in our universe, and even to guide our search for physics beyond Einstein. It is our master tool for energy accounting in a universe governed by gravity.

Imagine you are sitting infinitely far from a dynamic, self-gravitating system—say, two stars dancing around each other. General relativity tells us this dance must disturb the spacetime around it, sending out ripples of gravitational curvature. The Bondi-Sachs formalism gives us a language to describe these ripples as they arrive at our distant perch. The "news function," $N(u, \theta, \phi)$, is the core of this language. It is, in essence, the broadcast signal—the "news"—of what the system is doing at any given moment. The Bondi mass-loss formula, $\frac{dm_B}{du} = - \frac{1}{4\pi G} \oint |N|^2 d\Omega$ is our receiver, telling us the power of this broadcast. A loud broadcast—a large $|N|^2$—means a high rate of energy loss. Silence, where $N=0$, means the system is quiet, and its mass-energy is conserved.

What kind of "news" do we expect? We can start with simple models to build our intuition. If a system that was previously quiet suddenly undergoes a brief, violent convulsion—perhaps a star collapsing or two objects having a near miss—it might emit a short, sharp burst of radiation. We could model the news from such an event as a simple pulse, like a Gaussian curve or even a rectangular blip. By integrating the power over the duration of this pulse, the formula allows us to calculate the total, permanent mass lost by the system. A stronger or longer pulse means more energy is radiated away, leaving the system with a slightly smaller final mass.

On the other hand, a system in a steady, periodic motion, like a binary star system in a stable orbit, would send out a continuous, periodic signal. The news function in this case might look like a sine wave, oscillating with the orbital frequency. The mass-loss formula tells us that the power radiated would also oscillate, peaking twice per orbit (as the oscillating shape passes through its maximum distortion), but never becoming negative. The system steadily leaks energy, causing the orbit to slowly shrink.

This might all seem a bit arbitrary. Can we just invent any news function we like? The answer is a resounding no, and this is where a crucial connection is made. The news function is not an independent entity; it is a direct consequence of the source's physical motion. For most astrophysical systems, the "news" broadcast is dominated by the third time derivative of the source's mass quadrupole moment, $\dddot{Q}_{ij}$. The quadrupole moment is just a measure of how "out-of-round" the mass distribution is. A perfectly spherical, pulsating star has no changing quadrupole moment and hence radiates no gravitational waves. A spinning potato, or two stars orbiting each other, will radiate because their mass distribution is non-spherical and changing in a particular way. It's not the shape itself, nor its speed, nor even its acceleration, but the jerkiness of the changing shape—the rate of change of acceleration—that generates the news. This beautiful link connects the abstract field at infinity to the tangible lump of matter doing the wiggling.

Armed with this understanding, we can turn our attention to the real universe. And there, the applications are breathtaking. The most powerful events since the Big Bang are the collisions of black holes and neutron stars. When two black holes, with masses many times that of our sun, spiral into each other and merge, they churn spacetime with unimaginable violence. Numerical relativists solve Einstein's equations on supercomputers to predict the "news" from such an event. The resulting signal is a complex "chirp," starting slow and low, then rapidly rising in frequency and amplitude until the final, cataclysmic merger. Applying the Bondi mass-loss formula to this computed news function reveals something astonishing: in the final fractions of a second, an amount of mass equivalent to several suns can be converted purely into the energy of gravitational waves. This is the energy that our detectors like LIGO, Virgo, and KAGRA "hear," and the formula gives us the theoretical key to interpret its staggering power.

The formula also teaches us subtleties. Consider a stationary black hole. What happens if we shine a beam of light—a stream of "null dust"—directly into it? The black hole's mass will certainly increase as it swallows the energy. Does it radiate? The Bondi mass-loss formula gives a clear and perhaps surprising answer: no. Because the energy is purely infalling, it generates no outgoing news at future null infinity. An observer far away sees no gravitational wave broadcast. The Bondi mass, which measures the total energy of the isolated system including its radiation field, remains constant during this process. This highlights a critical point: the Bondi mass is not just the mass sitting at the center; it's the total energy of the system as perceived from the outside. Only energy that actively propagates out to infinity as "news" can reduce it.

Perhaps the most profound applications of a physical principle come when we use it to probe its own limits and explore the unknown. The Bondi mass-loss framework is a perfect tool for this.

General relativity makes very specific predictions. One is that the dominant form of gravitational radiation is quadrupolar. It explicitly forbids dipole radiation from a self-gravitating system. But what if there are other, hidden fields in the universe, as some alternative theories of gravity propose? For instance, a hypothetical scalar field could couple to matter, causing a binary system to radiate dipole scalar waves, something forbidden for pure gravity. Physicists can construct a model for such a scenario, calculate the power that would be radiated in these hypothetical waves, and predict the effect it would have on the binary's orbit. By then observing real binary pulsars with incredible precision, astronomers can check if their orbits decay in the way predicted by GR alone, or if there is an extra energy loss matching the dipole formula. So far, Einstein's theory has passed every test with flying colors, but the Bondi framework provides the very language we use to ask these deep questions.

The formalism also forces us to confront deep theoretical questions about the nature of radiation itself. What happens long after a gravitational wave burst has passed? Does spacetime simply return to its original state? The mathematical structure of the news function suggests this might not be the case. The long-term behavior of the news, such as a "power-law tail" that decays slowly over time, can lead to a permanent change in the spacetime geometry, a phenomenon known as the gravitational memory effect. Whether a system radiates a finite or infinite amount of total energy over all time can depend on how slowly this tail wags. The Bondi formula becomes a tool for exploring the permanent scars that gravitational events can leave on the fabric of spacetime.

Finally, we must acknowledge the ground on which this entire beautiful edifice is built: the assumption of an "asymptotically flat" spacetime. This is a mathematical idealization of an isolated system in an otherwise empty universe. But our universe is not empty; we now know it is filled with a mysterious dark energy, which acts like a positive cosmological constant, $\Lambda$. This seemingly small change has a drastic consequence for the large-scale structure of spacetime. In a universe with $\Lambda > 0$, the "future null infinity" ($\mathcal{I}^+$) where Bondi and Sachs set up their framework is no longer a null surface—it becomes a spacelike surface. This completely changes the rules of the game. The concept of a universal retarded time parameterizing outgoing null rays breaks down. We can no longer stand at infinity and unambiguously collect "news" arriving via light rays. This doesn't mean energy isn't radiated, but it means our entire framework for defining and measuring it must be rethought. How do we define gravitational energy in a cosmological setting? This is one of the great unsolved problems in theoretical physics, standing at the crossroads of general relativity, gravitational waves, and cosmology. The Bondi mass-loss formula, in revealing its own limitations, points the way toward the next frontier.