A train moving at a constant velocity with respect to the ground will constitute one such frame of reference, with the ground being another. Observers inside the train will calculate its kinetic energy as zero. 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. 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. Finally, we come to the issue of exactly how to define PEC in physics. 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 joules of energy per second that is watts is flowing into a region of one cubic meter, and joules per second is flowing out, then BPEC requires that the amount of energy in the region increase by joules per second. As an example, one could imagine a leaky one cubic meter that has an energy loss of watts at 0 C and has a watt heating element.
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If one turns up the temperature above 0 C, BPEC dictates that the total rate of increase of the heat energy in the oven would be 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. 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. The only way of obtaining a local expression for gravitational energy would be to add additional structure to space-time. 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. 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 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.
Why conservation fails
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. In inflationary cosmology, this allows for the entire mass-energy of the universe to come out of a minuscule region of pre-space e.
 General Relativity, Mental Causation, and Energy Conservation
Some popular treatments — such as that of Alan Guth , pp. Rather, as other textbooks recognize e.
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The answer is probably no. 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. The non-conservation of energy in GR opens up another response a dualist could give to the EC objection. The mind, like the gravitational field, could cause a real change in the energy of the brain without PEC applying to the interaction.
When Energy Conservation Seems to Fail: The Prediction of the Neutrino
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.
Interaction without Energy Exchange in Physics. Underlying the EC objection is the idea that causal interaction requires an exchange of energy.
Many people do not like the light given off by fluorescent fixtures and don't use them even though they use less electricity. Those who bought energy-saving diesel cars in the s spent more time in the shop than they did on the road.
A more subtle explanation of consumer reluctance is the normal process by which any new technology diffuses through the population. Economists who study cost-saving technologies have concluded that even in those cases in which the cost savings are very certain, the rate at which a population adopts cost-saving investments follows a bell-shaped curve. Some consumers adopt a technology initially, the mass middle follows, but others change their behavior very slowly.
But what about the hard cases? What should government do if people do not adopt cheap, energy-saving technologies that have no obvious performance flaws relative to traditional alternatives many years after introduction? Markets are valuable because they convey information through price signals, let individuals make decisions based on those price signals, and then require them to live with the consequences.
If people spend more money than they need to, that's OK because they suffer the consequences. We would not give the time of day to proposals to require people to purchase razors at K-Mart because they are cheaper or to attend college because the return on a college education far exceeds that on other investments. We should treat proposals to require or subsidize energy-conservation investments with similar skepticism. If global warming is a problem and it isn't , then create carbon dioxide emission rights and let individuals decide how to respond.
Long project development cycle and high financing cost : ESCOs have faced continuous challenges because of the fact that typical project development time for an ESCO project is between months depending on the maturity of the customer and the energy services market. Further, many projects are aborted for one reason or the other after significant investment of time and effort has been made by the ESCOs to educate the customer on the subtle nuances and details related to ESCO projects.
In many situations, where there are attractive energy efficiency projects with excellent ROI, the banks are not willing to extend a loan without burdensome collateral or other onerous terms and conditions that the customer is unwilling to accept. Extremely narrow margin of error in critical facilities : While critical facilities e.
The number of ESCOs who can be trusted to execute a critical project within the allocated shutdown time while satisfying all other performance parameters is quite small thereby limiting the potentially accessible market for a traditional ESCO.