One of the most commonly cited objections to substance dualism is that it violates the principle of energy conservation (PEC). After carefully looking at this energy-conservation (EC) objection, I argue that it fails when one carefully looks at the role of PEC in current physics. The arguments I present, however, also undercut the EC objection against several other philosophical views it has been used against, such as the agent causation view of libertarian free will. Unlike previous responses to this objection, my arguments will be based on a careful examination of the nature of energy and the status of PEC in contemporary physics.
The principle of energy conservation (PEC) is often raised as a major, if not fatal, objection to substance dualism, as can be seen from the following quotations from some of the leading philosophers of mind:
“…no physical energy or mass is associated with them [influences from immaterial mind to brain]. How, then, do they get to make a difference to what happens in the brain cells they must affect, if the mind is to have any influence over the body? A fundamental principle of physics is that any change in the trajectory of any physical entity is an acceleration requiring the expenditure of energy, and where is this energy to come from? It is this principle of the conservation of energy that accounts for the physical impossibility of ‘perpetual motion machines,’ and the same principle is apparently violated by dualism. This confrontation between quite standard physics and dualism has been endlessly discussed since Descartes’s own day, and is widely regarded as the inescapable and fatal flaw with dualism. . . . . Just as would be expected, ingenious technical exemptions based on sophisticated readings of the relevant physics have been explored and expounded, but without attracting many conversions” (Daniel Dennett, 1991, p. 35, italics mine.)
Similarly, according to Owen Flanagan,
If Descartes is right that a nonphysical mind can cause the body to move, for example, when we decide to go to a concert, then physical energy must increase in and around our body, since we get up and go to the concert. In order, however, for physical energy to increase in any system, it has to have been transferred from some other physical system. But the mind, according to Descartes, is not a physical system and therefore it does not have any energy to transfer. The mind cannot account for the fact that our body ends up at the concert. . . .We could maintain that the principle of the conservation of energy holds, but that every time a mind introduces new energy into the world – thanks to some mysterious capacity it has – an equal amount of energy departs from the physical universe – thanks to some perfectly orchestrated mysterious capacity the universe has. Unfortunately, such an assumption is totally unwarranted except as a way of saving Cartesian dualism, and, therefore utterly begs the question” (1991, p. 21).
Finally, Jerry Fodor remarks that
“The chief drawback of dualism is its failure to account adequately for mental causation. . . . How can the nonphysical give rise to the physical without violating the laws of the conservation of mass, of energy and of momentum?” (1994, p.25)
The above quotations each focus on the violation of energy conservation as a key problem confronting substance dualism. This paper will primarily attempt to address this energy conservation (EC) objection by carefully looking at the physics behind the PEC, something that surprisingly has not been done before and is much needed given the pervasiveness of the EC objection in the literature. Although this paper will only be concerned with the EC objection to interactionistic dualism, this objection is of more general interest since it can be raised against most views in which the human person is claimed to causally influence the brain in a way that cause physical events that cannot be explained by current physics. Thus, it is also relevant to certain sorts of property dualism or agency views of libertarian free will. The paper will conclude that the EC objection has little merit, primarily because it is based on a faulty view of the scope of PEC in current physics.
None of the authors above ever provides a precisely stated version of the EC objection. In attempting to formulate a precise version of the objection, one will immediately notice that it is similar to the objection to interactionistic dualism based on the so-called causal closure principle, discussed by Stewart Goetz in chapter ___. According to one version of this principle, “every physical effect has its chance fully determined by physical events alone” (Lowe, 2000, p. 574). It is difficult to see, however, how this and related versions of the principle do not simply beg the question against interactionistic dualism by assuming what this form of dualism denies, a point argued in detail by E. J. Lowe (2000).
Does the EC objection beg the question in the same way by assuming that all physical interactions must obey PEC? As C. J. Ducasse has stated, conservation of energy would be an obstacle to interactionistic dualism “only if it were known to be a universal fact” (1951, p. 241), something Ducasse claims scientists do not know. More precisely, the EC objection assumes that interactions between physical things and non-physical things (e.g., the soul) is relevantly similar to interactions between physical things. Thus, if interactions between physical things obey PEC, then it follows that interactions between non-physical things also do. No reason is ever offered, however, for the assumption that they are relevantly similar. Given the radical difference between physical things and non-physical things, however, such an assumption requires support. Of course, they could argue that assuming that PEC does not apply to all interactions involving physical systems leads to a less simple account of the world, hence providing some reason to reject interactionistic dualism. [2]
Another problem with the EC objection, however, one that undercuts the very basis of the objection, is that PEC does not apply to all known physical interactions, as demonstrated in the next section on general relativity below. Further, in quantum mechanics (see section IV), there are law-like correlations between events without energy or momentum exchange, thus providing a physical precedent for a mind-body dualist to assert that mental events are correlated with brain events in a law-like way without any exchange of energy or momentum.
General relativity is the theory developed by Albert Einstein to account for gravity, and is now accepted by almost all physicists. In general relativity, the energy of a body is considered a single component of a sixteen-component mathematical quantity called a tensor, the first component (T00) of which gives the energy density with respect to a given frame of reference. To explain this notion, it will be helpful to review some of the problems with defining the energy of a system as an intrinsic property of a system within classical mechanics. In classical mechanics, the total energy of an object is equal to the sum of the internal energy of a body and its kinetic energy, 1/2mv2. The latter quantity, however, will depend on the frame of reference from which the velocity of the object is measured. In special relativity, the frames of reference of interest are those that are moving at some uniform (non-accelerating) speed relative to each other. These are called inertial frames of reference. A train moving at a constant velocity with respect to the ground will constitute one such frame of reference, with the ground being another.[3] Now suppose we calculate the total kinetic energy of a ball at rest with respect to a moving train car – e.g., a ball that stays at the same position on the floor from the viewpoint of the passengers in the car. Observers inside the train will calculate its kinetic energy as zero. On the other hand, since for observers stationed on the ground the train and ball will be moving with some velocity v, they will measure its kinetic energy as 1/2mv2, where m is the mass of the ball. Thus, the energy of the ball will depend on the frame of reference – the train or the ground – from which one is measuring the energy of the ball. This means that unless there is a preferred frame of reference, we cannot speak of the energy of the ball as being an intrinsic, non-relational property of it.
A central idea behind both the special theory of relativity and the general theory of relativity is that the laws of physics should be formulated in terms of quantities that are independent of one’s frame of reference. Special relativity does this for the class of inertial frames, whereas general relativity does this for the class of all frames of reference, even ones that are accelerating. In the case of special relativity, the frame independent quantity that substitutes for energy is a four component entity, called the energy-momentum vector, whose first component is the energy of the object in question, and whose remaining components are the momentum of the object in the frame of reference in question. In different frames of reference, the momentum and energy components of this vector will take on different values, and thus the values of its components will vary from frame to frame. Nonetheless, there is a well-defined mathematical sense in which this four-component quantity itself remains the same, and hence can be considered a frame-independent, intrinsic quantity characterizing the object.
In general relativity, the frame-independent quantities are what mathematicians call tensors. As mentioned above, the value of energy is first component of a sixteen-component tensor, called the stress-energy tensor. Like the energy-momentum vector of special relativity, however, even though the components – such as the value of the energy – of this tensor vary from frame to frame, there is a well-defined mathematical sense in which the tensor remains the same. Hence, in general relativity the stress-energy tensor that can be considered an intrinsic property of an object or system of particles and fields, but its energy cannot. Thus, the corresponding conservation of energy law in general relativity is that of the conservation of stress-energy. When one only considers a single frame of reference, however, this conservation law implies energy conservation, even though the total energy should not be considered an intrinsic quantity of a system since it is frame dependent. As we will emphasize below, however, the stress-energy that this conservation law applies to does not include the gravitational field itself, which in turn gives rise to a problem for formulating energy (or more precisely, stress-energy) conservation in general relativity. Only non-gravitational fields and material particles contribute to the stress-energy tensor.
Even though the total energy is not an intrinsic property of a physical system, for convenience the term “energy” will often be used below, with the understanding that energy is merely a component of a quantity expressing the sixteen component stress-energy tensor. All claims regarding energy made below, however, can be translated into frame-independent claims by substituting the term “stress-energy” for the word “energy” in the corresponding sentences below.
Finally, we come to the issue of exactly how to define PEC in physics. In practice, the most general statement of PEC in physics is what could be called the boundary version of PEC, or BPEC for short. According to BPEC, from the perspective of any given frame of reference, the rate of change of total energy in a closed region of space is equal to the total rate of energy flowing through the spatial boundary of the region. Unlike more popular statements of PEC, BPEC neither makes reference to energy as a quantity that cannot be created or destroyed, nor to the idea of a causally isolated system. Thus, for instance, if 1000 joules of energy per second (that is 1000 watts) is flowing into a region of one cubic meter, and 300 joules per second is flowing out, then BPEC requires that the amount of energy in the region increase by 700 joules per second. As an example, one could imagine a leaky one cubic meter that has an energy loss of 300 watts at 2000 C and has a 1000 watt heating element. If one turns up the temperature above 2000 C, BPEC dictates that the total rate of increase of the heat energy in the oven would be 700 watts – the difference between the heat energy coming in from the element and the amount leaking out.
Now that we understand energy and energy conservation in modern science, we are ready to explicate the problem posed by General Relativity (GR). GR presents a major problem for the EC objection. The problem is that no local concept of stress-energy (and hence energy or momentum) can be defined for the gravitational field in GR. Consequently, BPEC does not typically apply in GR since one can neither define the total gravitational energy in a region of space nor the rate at which gravitational energy flows in or out of the region. This implies that although gravitational fields and waves clearly causally influence material objects, their influence cannot be understood in terms of movement of energy through space. As physicist Robert Wald notes,
In general relativity there exists no meaningful local expression for gravitational stress-energy and thus there is no meaningful local energy conservation law which leads to a statement of energy conservation. (1984, p. 70, note 6.)
The reason that local energy cannot be defined for gravitational fields is that no tensor can be defined in GR to represent the gravitational energy in a region of space-time. As mentioned above, all physical quantities in GR are represented by quantities that are in a well-defined mathematical sense invariant with respect to any frame of reference, whether moving with a uniform velocity or accelerating. This is called the condition of general covariance, and it is central to the formulation of GR. Since tensors are defined in such a way as to be invariant with respect to a change of coordinates (though their expression in terms of components is not), expressing a quantity in tensoral form guarantees its invariance. The problem for gravitational energy (and gravitational momentum) is that no physically plausible tensor, nor any other frame invariant quantity, can be found for the stress-energy of a gravitational field. Further, Wald notes, given the fundamental physical principles behind GR, “it seems highly unlikely that a generally applicable prescription exists for obtaining a physically meaningful local expression for gravitational energy...” (1984, P. 286). The only way of obtaining a local expression for gravitational energy would be to add additional structure to space-time. (1984, p. 286). As Wald points out, however, “such additional structure would be completely counter to the spirit of general relativity, which views the spacetime metric as fully describing all aspects of spacetime structure and the gravitational field.” (1984, p. 286).
A meaningful expression for the gravitational stress-energy – and hence the total energy – of an isolated region of space-time can be obtained, however, in the highly special case in which the region of space-time is asymptotically flat (that is, flat at spatial infinity for suitably defined hypersurfaces of constant time). An example would be a star surrounded by empty space in a universe with a flat space-time. No such systems exist within our universe (although many star systems can be approximately described in this way for predictive purposes). Further, as philosopher Carl Hoefer points out (p. 194), since our universe is not asymptotically flat, strictly speaking energy is not even conserved for our universe taken as a whole.[4]
Consequently, although in one sense PEC is not violated in GR– since this would require that the total energy be well defined – PEC typically does not apply. Consequently, in systems interacting with a gravitational wave, no conserved quantity that has the right characteristics can be defined. Gravitational fields, however, clearly have real physical effects on matter, even though from within the framework of GR these effects do not obey PEC. One specific consequence of this is that in the presence of a gravitational wave the total non-gravitational energy in an enclosed region of space could decrease or increase, without any net physically definable energy flowing across the boundary of the region. For instance, since gravitational waves exert tidal forces on matter, the waves will cause an increase in the energy content of matter. Yet technically one cannot calculate the gravitational energy transferred from gravitational waves to some object since this would require that the energy of the waves be defined. At best, in the highly special case mentioned above, one could estimate the amount of energy flowing out of a region of space that was asymptotically flat – such as the region surrounding a lone star. One is therefore often simply left with acknowledging a change in non-gravitational energy within a closed region without being able to attribute this change to a transfer of energy from another source or region of space. This leads Hoefer to suggest that for the case of gravity wave detectors, “energy gain in a gravity wave detector could be thought of as genuine gain, without having to say that the energy existed somewhere beforehand.” (P. 196).[5]
This non-conservation of energy in GR is exploited in contemporary cosmology. For example, as the universe expands, the waves of each photon are stretched and hence the wavelengths of the photons become longer and longer, a phenomena known as the cosmic redshift. Since the energy of a photon is inversely proportional to its wavelength, photons with longer wavelengths have less energy. Finally, since the vast majority of photons in the universe – those compositing the cosmic microwave background radiation – are not significantly absorbed by matter, the total number of these photons remains almost constant except for an almost insignificant contribution from starlight. Yet each is losing energy, and the energy is neither going into matter nor anywhere else. For example, in a spatially flat universe, which ours might be, it is not going into the curvature of space. Thus, it seems as though the total energy of the universe is decreasing. As P. J. E. Peebles states, however, “the resolution of this apparent paradox is that . . . there is not a general global energy conservation law in general relativity theory” (1993, p. 139). This non-conservation of energy is also exploited by inflationary cosmology, in which the inflaton field acts as a “reservoir of unlimited energy, which can supply as much as is required to inflate a given region to any required size at constant energy density” (Peacock, 1999, p. 26). In inflationary cosmology, this allows for the entire mass-energy of the universe to come out of a minuscule region of pre-space (e.g., less than 10-30 centimeters in diameter) with a minuscule total energy. Some popular treatments – such as that of Alan Guth (1997, pp. 9-12, 170-174) – try to claim that the total energy of the universe remains constant in the process since the gravitational field produced by the matter in the universe contributes a negative total energy, but this could not be correct since the total energy of the universe is undefined in GR. Rather, as other textbooks recognize (e.g., see Börner, 1988, p. 298), inflationary cosmology exploits the non-conservation of energy in GR.
Given the non-conservation of energy in GR, what should one think of PEC? Should one expect that a future successor to GR will re-establish the universality of PEC? The answer is probably no. As Hoefer points out, the tenure of PEC as a well-established idea “was arguably fairly short (limited to part of the nineteenth and early twentieth centuries), and at all times fraught with difficulties.” (2000, P. 188). Further, Hoefer notes, Newtonian gravitational theory had difficulties with PEC and Einstein was explicitly aware of the problems with PEC as early as 1916. (Pp. 189 - 191). This contrasts sharply with the statements of most advocates of the EC objection who claim that PEC is one of the most, if not the most, well-established principles in physics, as in the quotations at the beginning of this paper. Yet, despite the evidence to the contrary, even in most texts on GR there is “the universal, almost desperate desire to make it seem as though there is such a principle [PEC] at the heart of the theory [of general relativity].” (Hoefer, 200, P. 195).
The non-conservation of energy in GR opens up another response a dualist could give to the EC objection. A dualist could argue that, like the gravitational field, the notion of energy simply cannot be defined for the mind, and hence one cannot even apply PEC to the mind/body interaction. The mind, like the gravitational field, could cause a real change in the energy of the brain without PEC applying to the interaction. Of course, this leaves open the possibility of a new physical theory being developed that replaces the basic framework GR or of someone finding an ingenious definition of energy that fits within the framework of GR. All one can say for sure is that, within current physics, not all systems can be said to obey PEC.[6] At the very least, this puts the burden of proof on the person offering the EC objection to state why, given that in the best physical theories PEC does not apply to all physical interactions, one should think that it applies to the mind/brain interaction.
IV. Interaction without Energy Exchange in Physics.
Underlying the EC objection is the idea that causal interaction requires an exchange of energy. Even apart from the considerations based on general relativity in the last section, this idea is deeply problematic within contemporary physics. To begin, consider classical physics. Theoretically, within classical physics, one system or entity can have an effect on another without any exchange of energy occurring. In classical physics, this occurs anytime an entity acts on another entity with a force that is perpendicular to the second entity’s direction of motion. The reason is that the work, W, performed on a system – that is, the energy transferred to it – is equal to the force times the distance moved in the direction of the force. Consequently, if the force is perpendicular to the direction of motion of the object, it will influence the motion of the object according to Newton’s law (F = ma), without any work being performed and hence without any energy being transferred. There are many examples of this within physics. An example that philosopher C. D. Broad gives in response to the EC objection, and which is repeated by Keith Campbell (Broad, 1925, chapter 3; Campbell, 1984, pp. 52-53), is that of a bob at the end of a pendulum: the string causally influences the motion of the bob, yet imparts no energy to it since the force the string exerts is always perpendicular to the direction of motion. As long as a “mental force” acting on a particle in the brain is perpendicular to the direction of motion of the particle, therefore, it would causally influence the motion of the particle without any exchange of energy occurring. Such forces, however, do impart momentum to the objects under consideration because of Newton’s Second Law, F = ma. This type of response to the EC objection, therefore, fails when the full conservation law of energy and momentum is considered. The only way to conserve both energy and momentum is to have an equal force in the opposite direction act on another particle, thus causing the two changes in momentum to cancel. Even this scenario, however, would violate the BPEC version of energy-momentum conservation as applied to any region that only contained one of the particles.
Quantum mechanics, however, does provide a good case of interaction (or at least correlation) without either energy or momentum exchange. In quantum mechanics, there are definitely correlations between attributes of particles– and in many realist interpretations, causal interactions -- without energy exchange. Following many authors, we will use the quantum attribute of the spin of a particle to illustrate these correlations. All particles in quantum mechanics have an attribute called spin, which comes in values of 1, ½, and 0. Further, the spin can be measured in an arbitrary direction and will always come in quantized values. For the case of spin ½ particles -- such as protons, neutrons, electrons, and some atoms – this means that the spin will always be measured with a value of either +1/2 or -1/2, no matter what direction one chooses to measure it.
Now consider two particles each with a spin of 1/2-- say two nitrogen (N) atoms -- initially bound together to form a system (such as the nitrogen molecule, N2) with a total spin of zero. Suppose we break these particles apart in a spaceship between Earth and Mars, with one of the particles going to Earth and one to Mars. Call the Earth-bound particle pE and Mars-bound particle pM. Further, suppose there is an observer on each planet that will measure the spin (in some prearranged direction Z) of the particle that arrives on her planet. Quantum mechanics dictates that each observer will either measure her particle as having a spin of +1/2 or -1/2. Further, because of conservation of spin and the fact that they are measuring the spin in the same direction Z, quantum mechanics dictates that if the Earth observer measures pE as +1/2, then the Mars observer will measure pM as -1/2, and vice versa: that is, the measurement results are anti-correlated. Consequently, if our Earth observer measures pE as +1/2, she knows that the Mars observer will measure pM as -1/2.
The seemingly obvious explanation of this is that when the two particles were initially separated on the ship, the process of separation caused each of them to be in some definite state that was anti-correlated with its partner -- e.g., the pm was forced into a +1/2 state while pE was forced into a -1/2 state. This explanation is an example of what is called local causation. To see why this explanation only needs to invoke local causation, first note that it explains why pE was measured as +1/2 by saying that it had a certain attribute, being in a +1/2 state, that caused the measuring apparatus on Earth to register +1/2. This causation is purely local, since once pE hits the apparatus, there is no longer any relevant spatial distance between it and the apparatus. In the same way, it explains using only local causation why the Mars observer apparatus registered -1/2 when measuring the spin of pM. Finally, only local causation is required to explain why the two particles started off in their respective spin states via the mechanism that separated the two particles: when the two particles were bound together on the ship, no relevant spatial distance separated them from the mechanism that split them apart and imparted to them their respective spins, and hence only local causation was involved.
Now for the punch line. A theorem proved by John Bell in 1966, called Bell's theorem ruled out the above explanation and any other explanation of these quantum correlations involving only local causation. Bell showed that if certain experimental results predicted by quantum mechanics occurred, explanations based on local causes could not explain the correlations. Since 1977, these predictions have been verified numerous times. Now within all physical theories energy exchange always involves non-instantaneous and hence local causation, since the packets of energy (or stress-energy) must move through space. Consequently, Bell's theorem rules out any explanation of these correlations by means of energy exchange. Consequently, quantum mechanics requires the existence of correlations that cannot be explained by an exchange of energy.
There have been two main responses to these correlations in the literature: (i) the causal realist response, according to which these correlations are grounded in some instantaneous causal connection between the two particles or in some non-local and thus instantaneously acting common cause; and (ii) the causal anti-realist response, according to which these correlations are not grounded in any further causal facts. If the causal realist response is adopted, the burden is on the advocate of the EC objection to state why she thinks that the causal interaction between the mind and the brain should require an exchange of energy when these quantum interactions do not. If the causal anti-realist interpretation is adopted, then versions of dualism in which there are law-like connections between mental events and physical events without any corresponding causal interaction become much more plausible.
These quantum correlations, therefore, show that positing an interaction (or at least a correlation) between the mind and brain that does not involve an energy exchange, or any other mediating field, has precedent in current physics, thus severely weakening the EC objection. Further, these quantum correlations are not merely some minor “technical exemption” within physics, but pervasive throughout the microscopic world, playing a fundamental role in the operation of nature.[7] Finally, they cast severe doubt even on the suggestion that causation requires an intermediate carrier (e.g., Hoefer p. 196), such as gravitational waves in the case of general relativity, whether or not that carrier involves a transference of energy and momentum. Since quantum mechanics predicts that, for any given frame of reference, one can always find an experimental arrangement in which the quantum correlations are instantaneous, it follows that causal interactions (or at least law-like correlations) do not require an intermediate carrier. Further, since by hypothesis there is no spatial distance between the immaterial mind and the brain, there is no need for such a carrier.
The EC objection against interactionistic dualism fails when one considers the fact that energy conservation is not a universally applicable principle in physics and that quantum mechanics sets a precedent for interaction (or at least law-like correlation) without any sort of energy-momentum exchange, or even any intermediate carrier. Of course, the more general interaction problem for interactionistic dualism still remains, although this is beyond the scope of this paper to address. What this paper has shown is that from the perspective of the role of PEC in modern physics, the pervasive EC objection to dualism has no merit.
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[1] This chapter is a condensed and simplified version of my article, "Modern Physics and the Energy Conservation Objection to Mind-Body Dualism" originally published in The American Philosophical Quarterly, Volume 45, Number 1, January 2008, pp. 31-42. I would like to thank the John Templeton Foundation for support of this work as part of a group grant “The Soul Hypothesis: Developing Non-Supervenience Views of the Mental,” several members of the group (Stewart Goetz, Charles Taliaferro, William Hasker, and Dean Zimmerman) for reading over an earlier draft, Richard Swinburne, Marc Lange, and William Lane Craig and an anonymous referee for providing helpful comments.
[2] Sometimes the purported universal applicability of PEC to all known domains is supported by metaphysical reasons. Indeed, historically, metaphysical motivations – particularly those based on the idea of “conservation of natural forces”-- played a major role in motivating the PEC in the nineteenth century (See Harman, 1982, 33-35, 58-64.). Further, more recently, several authors have suggested an account of causation in terms of energy flow (e.g., Quine, 1973, pp. 4-8, Hart, 1988, p. 62, and Fair, 1979), thus suggesting that PEC would apply to all interactions. since they all involve an exchange of energy.
[3] For illustrative purposes, we neglect the motion and curvature of the Earth itself, which makes all these frames non-inertial.
[4] Insofar as they discuss the issue, general relativity texts agree that there is no adequate definition of local energy in GR: for example, Wald (1984, p. 70, note 6; p. 286-287), Misner, Thorne, Wheeler, (1973, pp. 457-470), and Penrose (1989, p. 220; 2004, p. 467). Penrose, however, has attempted to retain some global conception of the conservation of energy by claiming that GR shows that gravitational energy is non-local. (1989, Pp. 220 - 221). Even this move, however, would require that the total energy of the universe is well-defined, which it is not. Further, if Penrose is right, the total mass-energy of the brain is not entirely confined to any local region, but at least in part involves the entire universe. Such a non-local conception of the mass-energy of the brain, however, makes it difficult even to formulate the EC objection, thus undermining almost all of its purported force.
To practically deal with this problem of the lack of any definition of local energy in GR, in practice physicists often define a pseudo-tensor that they identify with the stress-energy of the gravitational field for purposes of doing calculations. For example, such a pseudo-tensor can be defined for the weak field limit of general relativity and the predictions based on this conform to experiment for at least one binary star system. (See, for example, Carroll, 2004, p. 315 and Penrose, 2004, pp. 467-68.) Such pseudo-tensors, however, cannot be treated as providing the real stress-energy of a system since they are not frame invariant. The pseudo-tensors typically used, for instance, imply that in some frames of reference, flat space-time has gravitational energy, even though by definition flat space-time contains no gravitational fields!
[5] Interactions with gravitational waves might seem even to violate energy conservation for non-gravitating fields, since if one measured the rate of change of the energy of those fields in some region of space, the rate of change would not be equal to the total non-gravitational energy coming into that region. Thus, BPEC is violated for non-gravitational fields. The resolution to this difficulty is that in curved space-time, BPEC does not even apply when we only consider non-gravitational fields. The only sense in which energy conservation holds for these fields is that the divergence of the stress-energy tensor for a system must equal zero: LCT = 0, where L represents the four-divergence and T the stress-energy tensor. In flat space-time – that is, space-time in which no gravitational fields are present – the BPEC follows from LCT = 0 by Gauss’s theorem. (Wald, pp. 62-63, eqs. 4.2.11 and 4.2.18). Consequently, it is legitimate to think of divergence equation LCT = 0 as equivalent in flat space-time to energy conservation. Accordingly, in texts on general relativity this equation is typically presented as the way of summarizing the law of energy and momentum conservation in non-quantum physics. (See, for example, Carroll (pp. 35 - 36, eqs. 1.115 and 1.120). In the curved space-time of general relativity, however, there is no way of deriving BPEC from this divergence equation. Hence, even though this divergence equation is assumed to hold for non-gravitational fields and particles, energy conservation does not hold. (Wald, pp. 69-70).
[6] It should be noted that since energy and momentum are united in relativity, if no adequate stress-energy tensor can be found for the gravitational field, then the momentum of a gravitational field or wave cannot be defined. (See, for example, Misner, Thorne, Wheeler, 1973, 463-469.) This means that there cannot be an applicable principle of momentum conservation in general relativity, although this is often not discussed.
[7] This is true whether or not one adopts a realist interpretation of quantum mechanics. Even if one thinks that quantum mechanics is merely a useful instrument of prediction, these correlations occur between the results of measuring apparatuses, and Bell’s theorem rules out any explanation of them in terms of local causation.