Landau-Lifshitz pseudotensor

In physics, the law of energy conservation is a foundational principle. While we can easily account for the energy of matter and fields like electromagnetism, a profound puzzle emerges when we ask: where is the energy of the gravitational field itself? This question, simple on its surface, reveals a deep conflict between our intuition about energy and the central tenet of General Relativity—the Equivalence Principle. The very nature of gravity as spacetime curvature prevents its energy from being neatly localized in a single spot, creating a significant knowledge gap in how to formulate a total, conserved energy for a system that includes gravity.

This article navigates this complex and fascinating problem. You will discover the elegant solution physicists devised to keep the universe's books balanced. The first chapter, "Principles and Mechanisms," delves into why gravity resists a simple energy description and introduces the Landau-Lifshitz pseudotensor, a clever mathematical tool designed to work around this issue. Following this, the "Applications and Interdisciplinary Connections" chapter demonstrates the immense power of this concept, showing how this "bookkeeping trick" is used to weigh black holes, calculate the power of gravitational waves, and even probe the energy budget of the entire cosmos.

Let’s begin our journey with a question that seems, on the surface, almost childishly simple: when you lift a bowling ball off the floor, you give it potential energy. Where does that energy go? In Newtonian physics, we’d say it’s “stored in the gravitational field.” It’s a beautifully simple picture. When you drop the ball, the field gives the energy back, converting it into the kinetic energy of motion.

In the world of modern physics, energy, momentum, and pressure are all bundled together into a magnificent and comprehensive object called the ​​stress-energy tensor​​, typically denoted $T^{\mu\nu}$. For any physical system—be it a cloud of dust, a vat of water, or the electromagnetic field of a sunbeam—this tensor tells you everything about the distribution and flow of its energy and momentum. In the flat spacetime of Special Relativity, the law of energy conservation is a crisp, clean statement: the divergence of this tensor is zero, $\partial_\nu T^{\mu\nu} = 0$. This simple equation, through the magic of Gauss's theorem, guarantees that the total energy and momentum of an isolated system are perfectly conserved.

So, what happens when we try to apply this to gravity itself? If the electromagnetic field has energy, surely the gravitational field must as well. After all, a pair of merging black holes radiates gravitational waves that can travel across the universe and jiggle a detector here on Earth—that jiggling takes energy!. It seems obvious that we should be able to write down a stress-energy tensor for the gravitational field, add it to the tensor for matter, and get a total conservation law. But here, Einstein’s theory throws us a curveball, a profound and beautiful puzzle that strikes at the very heart of gravity.

The central pillar of General Relativity is the ​​Equivalence Principle​​. One of its most dramatic consequences is the idea that gravity, as we perceive it, is not a force in the conventional sense. It's a manifestation of the curvature of spacetime. The principle tells us that at any given point in spacetime, you can always choose a special coordinate system—a freely-falling frame, like an elevator with its cables cut—where the effects of gravity locally vanish. Inside this falling elevator, dropped objects float. A laser beam travels in a straight line. For all intents and purposes, you are in an inertial frame, just like one floating in deep space, far from any stars or planets.

Now, let's play a game. Suppose we had a true, physical "tensor" that represented the energy of the gravitational field, let's call it $t^{\mu\nu}_{\text{grav}}$. As a physicist inside the freely-falling elevator, you would perform an experiment to measure the local energy density of the gravitational field. Since gravity has vanished in your frame, your measurement must yield zero. So, at your location, $t^{\mu\nu}_{\text{grav}} = 0$.

But here's the rub: a fundamental property of tensors is their objectivity. If the components of a tensor are all zero in one coordinate system, they must be zero in all coordinate systems. This means that if the gravitational energy tensor is zero for the person in the falling elevator, it must also be zero for their colleague standing on the surface of the Earth, who is most certainly feeling the effects of gravity. If this argument holds for any point, it means the gravitational energy tensor must be zero everywhere! This is a catastrophic conclusion, flying in the face of the evidence from gravitational waves. The energy is real, so our assumption must be wrong. The only way out is to conclude that whatever we use to describe the energy of the gravitational field, it cannot be a tensor. It must be something more slippery, something that can vanish in one reference frame but not another.

This is not just a mathematical subtlety; it’s a profound statement about the nature of gravity. You cannot bottle gravitational energy at a point. It's not "located" in the same way the energy of an electric field is.

The formal expression of energy conservation in General Relativity is the equation $\nabla_\mu T^{\mu\nu} = 0$. The Del-like symbol $\nabla$ represents a ​​covariant derivative​​, which is the proper way to take derivatives in a curved spacetime. This equation is perfectly exact. But it doesn't represent a conservation law in the way we're used to. Because of the complexities hidden within the covariant derivative, we can't simply integrate it over a volume to get a conserved total quantity.

Physically, what this equation describes is the local exchange of energy and momentum between matter and the geometry of spacetime. It says that if the energy of matter at some point decreases, the energy of the spacetime geometry must increase by a corresponding amount, and vice-versa. It’s a statement of trade, not a statement of the total amount in the bank.

Is there any situation where we can define a true, conserved total energy? Yes, but only in very special circumstances. If a spacetime possesses a continuous symmetry—for example, if it is unchanging in time—then it has what mathematicians call a ​​Killing vector​​. Such a vector provides a mathematical hook to construct a genuinely conserved quantity from the matter stress-energy tensor, $T^{\mu\nu}$. For a static star, we can define its total energy perfectly. But what about a dynamic, evolving system like the binary black hole merger? Its spacetime is rippling and changing; it has no time-translation symmetry. It has no Killing vector, and thus no straightforward way to define a conserved total energy.

This is where physicists, in the grand tradition of finding clever workarounds, resort to a beautiful piece of mathematical trickery. The idea, pioneered by physicists like Lev Landau and Evgeny Lifshitz, is this: let's invent a new quantity, which we will call the gravitational stress-energy ​​pseudotensor​​, $t^{\mu\nu}_{LL}$. The "pseudo" is the crucial word here. This object is not a tensor. It's a mathematical construct, a sort of scaffold built from the metric tensor and its derivatives. We design it with one specific goal in mind: to create a new, combined quantity whose ordinary divergence vanishes. That is, we arrange things so that: $\partial_\nu \left( \sqrt{-g} (T^{\mu\nu} + t^{\mu\nu}_{LL}) \right) = 0$ By getting back to an equation with an ordinary derivative, $\partial_\nu$, we can once again use Gauss's theorem. This allows us to define a total energy-momentum four-vector, $P^\mu$, for an isolated system by integrating over all of space. The problem of global energy conservation seems to be solved!

This solution comes at a price. Because $t^{\mu\nu}_{LL}$ is a pseudotensor, its value at a specific point depends on the coordinate system you use to calculate it. This is exactly the "out" we needed to resolve the falling elevator paradox. For the observer in free fall, the local components of the pseudotensor can indeed be made to vanish. For the observer on the ground, they will calculate a non-zero value.

The coordinate dependence can lead to some frankly bizarre results if you're not careful. Consider flat, empty Minkowski spacetime. There's no gravity, no matter, nothing. The energy should be zero everywhere, right? But if you describe this same flat spacetime using a rotating coordinate system, the Landau-Lifshitz pseudotensor will tell you there is a non-zero, and in fact negative, energy density!. This "energy" is entirely an artifact of the non-inertial frame; it's the phantom force you feel on a merry-go-round, dressed up in the language of General Relativity. This shows we must be extremely cautious about interpreting the local values of a pseudotensor as physical energy density.

So, if it’s so strange and frame-dependent, what good is it? The magic happens when we stop looking at it locally and instead use it for its intended purpose: to calculate global properties of an entire, isolated system. When we integrate the energy component, $\sqrt{-g}(T^{00} + t^{00}_{LL})$, over all of space, the weird, coordinate-dependent artifacts at infinity cancel out or vanish, leaving a single, finite, and physically meaningful number: the total energy of the system.

This total energy, known as the ADM mass (after Arnowitt, Deser, and Misner), is robust. If you calculate the total energy of a black hole solution, the formalism dutifully returns its mass, $M$,. If you calculate the energy of a mere ripple in your coordinate system—a "pure gauge" perturbation—it correctly gives you zero, confirming that coordinate tricks don't contain real energy. The conservation law itself is an identity, it holds true even in peculiar coordinate systems for flat spacetime. The bookkeeping, though strange, is consistent.

Most beautifully, this "pseudo" idea connects directly to real, measurable physics. When we analyze a weak gravitational field in the Newtonian limit, the Landau-Lifshitz energy density $t_{LL}^{00}$ takes on a very familiar form: $t_{LL}^{00} = -\frac{1}{8\pi G} (\nabla\Phi)^2$ where $\Phi$ is the good old Newtonian gravitational potential. This is strikingly analogous to the formula for the energy density of an electric field, which is proportional to $E^2$. It’s a sign that our strange pseudotensor formalism has deep connections to our physical intuition.

And what about the gravitational waves that inspired our initial question? While the instantaneous energy density at a point is ill-defined, we can average the pseudotensor over a region of spacetime several wavelengths across. This process smooths out the coordinate-dependent jitters and gives a well-defined, positive energy density for the wave. For a simple plane wave, the average energy density is proportional to the square of its frequency and the square of its amplitude. This energy is real. It is the energy that travels from a violent cosmic cataclysm and, after a journey of millions of years, causes a mirror in an L-shaped detector on Earth to move by a distance smaller than the width of a proton.

The story of the Landau-Lifshitz pseudotensor is a perfect encapsulation of the spirit of theoretical physics. It begins with a deep paradox born from a beautiful principle. It proceeds via a clever, almost cheeky, mathematical trick that seems to break the rules. And it ends by providing a powerful, consistent, and indispensable tool for calculating real physical quantities that we can actually go out and measure. It tells us that while we may not be able to point to a spot and say "the gravitational energy is right here," we can confidently say how much total energy a star contains, and how much energy is being carried away by the waves of spacetime itself.

In our previous discussion, we confronted a rather tricky question that lies at the heart of general relativity: if gravity is the curvature of spacetime, where does the energy of the gravitational field reside? We found that you can't point to a spot and say, "the energy is right there." The equivalence principle forbids it. To navigate this, we introduced a clever, if slightly peculiar, beast: the Landau-Lifshitz pseudotensor, $t^{\mu\nu}_{LL}$. You might think of it as a sophisticated accounting tool. While not a true tensor itself, it allows us to do the books for the total energy and momentum of a system—matter plus gravity—and wonderfully, the final balance is always physically meaningful and conserved.

Now, having met this mathematical creature, you are right to ask: What is it good for? What can it do? This is where the real fun begins. It turns out that this abstract bookkeeping device is a master key that unlocks profound insights into some of the most fascinating objects and phenomena in our universe. It allows us to weigh black holes, to measure the power of ripples in spacetime, and even to talk about the energy budget of the cosmos itself. Let’s embark on a journey to see how this one idea ties together the universe in the most beautiful and unexpected ways.

Perhaps the most straightforward question you can ask is: how much does a black hole weigh? We have the Schwarzschild solution, which describes a non-rotating black hole with a parameter $M$, which we call "mass." But is that its total mass-energy, including the energy of its own intense gravitational field? The Landau-Lifshitz formalism provides the answer. By performing a calculation at a great distance from the black hole—effectively "weighing" it from afar—we find that the total energy is precisely $E = Mc^2$. This is a-ha moment number one. The formalism works! It confirms that the parameter $M$ we write down in the metric is, in fact, the total mass-energy that another object would feel.

What if the black hole has an electric charge, as in the Reissner-Nordström solution? This spacetime contains energy in the electric field surrounding the black hole. Does gravity weigh this energy too? Again, we put the pseudotensor to work. We calculate the total energy and find, once more, that it is simply $E = Mc^2$. This is a beautiful statement about the unity of physics. The energy locked in the electric field doesn't appear as a separate term; it contributes to the total mass $M$. Gravity, through its curvature, packages all forms of energy—rest mass, kinetic energy, field energy—into a single, total gravitational mass.

The power of this tool extends beyond simple energy. For a rotating Kerr black hole, which has a parameter '$a$' related to its spin, the formalism can be used to calculate the total angular momentum. And what does it find? The total angular momentum is $J = Ma$. Again, the abstract parameter in the mathematical solution is given a concrete, physical meaning. This is the pattern: the Landau-Lifshitz pseudotensor acts as a bridge between the abstract geometry of spacetime and the conserved physical quantities we know and love—energy, momentum, and angular momentum.

But this isn't just about exotic black holes. The formalism beautifully connects back to the physics we first learn. Consider a simple star. Its own gravity pulls it together, creating immense pressure. This self-interaction, this "binding energy," must be part of the star's total mass-energy budget. Using the weak-field approximation of the Landau-Lifshitz pseudotensor, we can calculate this gravitational binding energy. We find it is a negative quantity, proportional to $-\frac{GM_0^2}{R}$. Of course, it must be negative! Energy must be released to form the star from dispersed dust; you would have to add energy to pull the star apart against its own gravity. The fact that the total mass of the Sun is slightly less than the sum of the masses of all its individual protons and neutrons is a direct consequence of this negative binding energy, and the LL formalism gets it just right. It shows how Einstein's theory contains Newton's within it, but in a richer, deeper way.

The versatility of this method even allows us to probe the weird world of theoretical physics. For speculative objects like a traversable wormhole, one might ask what its total mass is. For certain models, the formalism reveals a stunning answer: zero. This strange result doesn't mean the tool is broken; it tells us something profound about the kind of bizarre, "exotic" matter with negative energy density that would be required to prop such a thing open.

So far, we have been "weighing" objects that are more or less sitting still. But the real power of gravity reveals itself when things are in motion. Accelerating masses produce gravitational waves—ripples in the very fabric of spacetime. A passing gravitational wave will stretch and squeeze you. Surely, something that can do work must carry energy!

But how much? Here, the Landau-Lifshitz pseudotensor shines. It allows us to calculate the energy flux—the amount of energy per unit area per unit time—carried by a gravitational wave, just as we can for a light wave. It also allows us to calculate the "stress" or momentum flux carried by the wave, which implies that gravitational waves exert a form of radiation pressure. The idea that a ripple of spacetime itself carries energy and momentum is one of the most remarkable predictions of general relativity, and this formalism makes it quantitative.

Where does this radiated energy come from? It must come from the source. The key insight, derivable from the LL formalism and encapsulated in the famous quadrupole formula, is that a system radiates gravitational waves if its mass distribution changes in a specific, "lumpy" way—technically, if its quadrupole moment has a non-zero third time derivative.

Imagine a perfectly uniform, rigid, spherical ball spinning. Because of its perfect symmetry, it does not radiate gravitational waves. Now, imagine a lumpy, triaxial potato-shaped object (an ellipsoid) that is forced to rotate. As it spins, its "lumpiness" changes orientation, its quadrupole moment varies in time, and it radiates energy away in the form of gravitational waves. The amount of power it radiates is directly proportional to its asymmetry. It is a beautiful consequence that only systems that are "imperfect" in their rotation can broadcast their presence across the cosmos with gravitational waves.

This brings us to the most spectacular confirmation of this entire picture: binary systems. Consider two massive objects, like neutron stars or black holes, orbiting each other. As they whirl around their common center of mass, their mass distribution is constantly changing. They form a perfect cosmic generator of gravitational waves. By radiating this energy, the system loses energy. And what is the consequence of a binary system losing orbital energy? The two bodies spiral closer together, orbiting faster and faster, radiating even more powerfully. This orbital decay was first observed with breathtaking precision in the Hulse-Taylor binary pulsar system, providing the first indirect (and Nobel Prize-winning) evidence for gravitational waves and a stunning verification of the energy-loss predictions that stem from the Landau-Lifshitz formalism. When LIGO and Virgo later directly detected the waves from merging black holes, the circle was complete. The energy carried by those ripples, calculated using these very principles, matched perfectly with the energy lost by the black holes as they merged.

We have applied our tool to stars and black holes, and to the waves they generate. Now, let's think bigger. What about the biggest system there is: the universe itself? We live in an expanding universe, described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. The distance between galaxies is growing. The very geometry of space is changing with time. Is there an energy associated with this cosmic expansion?

This seems like a question on a completely different scale, yet our trusty pseudotensor can be applied. In the context of the FLRW cosmology, one can calculate the components of the Landau-Lifshitz pseudotensor for the "gravitational field" of the expanding universe. From this, one can define an effective pressure of the gravitational field, $P_g$. The result is nothing short of profound. The effective pressure of gravity turns out to be negative, and its value depends on the average density $\rho$ and pressure $P$ of the matter and energy filling the universe: $P_g = -\frac{5}{6}\rho - \frac{1}{2}P$.

Let that sink in. In the grand arena of cosmology, the "energy of the expanding space" acts as if it has a negative pressure. It is an active participant in the dynamics of the cosmos. This insight, that components of the gravitational field itself can be associated with properties like negative pressure, resonates deeply with some of the biggest mysteries in modern cosmology, such as the nature of dark energy, the mysterious component driving the accelerated expansion of the universe, which is also characterized by a large negative pressure.

From the simple question of a black hole's mass to the energetics of the entire universe, the Landau-Lifshitz pseudotensor provides a consistent, albeit subtle, language for discussing the energy of gravity. It is a testament to the unifying power of a great physical theory, showing how a single, coherent idea can connect the physics of a star, the ripples from a distant collision, and the very evolution of the cosmos.