Hawking-Unruh Temperature. PHYS 612: Advanced Topics in Quantum Field Theory. Spring Taught by George Siopsis. Written by Charles Hughes

Transcription

1 Hawking-Unruh Temperature PHYS 612: Advanced Topics in Quantum Field Theory Spring 2018 Taught by George Siopsis Written by Charles Hughes

2 Table of Contents 0) Abstract 1) Introduction to Rindler Coordinates 1.1) Reminder of Minkowski Space 1.2) Reference Frames: Inertial vs. Accelerated 1.3) Minkowski Polar Coordinates 1.4) Rindler Coordinates 2) Massless Klein-Gordon Fields 2.1) Massless Klein-Gordon Fields in Minkowski Space 2.2) Massless Klein-Gordon Fields in Rindler Space 3) Vacuum Expectation Value of the Number Operator for Klein-Gordon Fields Space Space 3.1) Vacuum Expectation Value of the Number Operator of Klein-Gordon Fields in Minkowski 3.2) Vacuum Expectation Value of the Number Operator of Klein-Gordon Fields in Rindler 4) Unruh Radiation 5) Conclusion 4.1) Blackbody Spectrum

3 0) Abstract This term paper is written in fulfillment of the requirements for the Spring 2018 section of PHYS 612: Advanced Topics in Quantum Field Theory as outlined in: The chosen topic for this paper is the Hawking-Unruh Temperature. A brief description of this phenomenon is as follows. In Quantum Field Theory (QFT), free fields in Minkowski space are expanded in terms of the annihilation (a (k)) and creation (a(k)) operators and the corresponding normal modes. These free fields are the solution of the Klein-Gordon (K-G) equation. The vacuum expectation value of the number operator for these non-interacting fields ( <0 a (k)a(k) 0> ) in Minkowski space is 0. This is, however, not generally the case for non-interacting scalar fields in other space-time geometries. The coordinates of hyperbolically accelerated reference frames, Rindler coordinates, are a great example of a geometry which results in unusual QFT consequences for scalar fields. In this space time geometry, solutions to the K-G equations are expanded in two sets of normal modes and two sets of creation (b (1) (k), b (2) (k)) and annihilation (b (1) (k), b (2) (k)) operators. This is necessary to extend the coverage of the normal mode solutions to the K-G equation in Rindler space to the entire space. Using the Bogoliubov transformations (or a clever argument made by Unruh detailed in this paper), one can find relations between the b (r) (k) (r = 1, 2) modes and the a(k) modes. This allows one to find the vacuum expectation value of the Rindler number operator on the Minkowski vacuum ( 0>). Suprisinlgy this is not 0. In fact its form is exactly that of the Planck distribution that Max Planck found as a solution to the Ultra-Violet Catastrophe in the early 20 th century. This is the frequency (or wavelength) distribution of a black-body. This leads to an apparent contradiction. How does the inertial (Minkowski) observer see nothing and the accelerated (Rindler) observer see a thermal spectrum? The resolution to this paradox is to realize that to achieve a constant acceleration something must be supplying energy to the accelerated observer. The inertial observer will observe that the accelerated body is therefore EMITTING radiation. If the accelerated body is a charged particle then this is the known effect of bremsstrahlung (braking radiation). The source of acceleration is supplying the energy for this emitted radiation. Familiar sources of acceleration include the controlled burning of compressed gasoline/air mixture in a cylindrical piston, the release of the chemical energy stored in rocket fuel by burning it and directing its exhaust through a narrow opening, and the use of powerful electromagnets to direct charged particles in circular trajectories. Regardless of the source of radiation, the accelerated observer is oblivious to the details of what is going on outside their reference frame. They only see the net effect of the radiation in their reference frame. That is, the source of energy enabling their acceleration is also supplying the energy necessary for radiation emission and they are absorbing some of that emission (thermal bath).

4 1) Introduction to Rindler Coordinates The following sections detail the necessary background information needed to understand and derive the Rindler coordinate space and its corresponding metric. 1.1) Reminder of Minkowski Space The advent of special relativity signaled an end to the long-held notions of simultaneity and absolute distance. In special relativity inertial observers in different reference frames will observe that the passage of time in one inertial rest frame is different from that in another. That is, an event which takes place in one observer's rest frame will be observed to have a different duration in another observer's rest frame. The difference, it turns out, is quantified by the relative motion between the inertial observers. This phenomenon is known as time dilation. A similar effect is observed for the distance between events in different reference frame. This phenomenon is known as length contraction. These effects don't become noticeable at everyday scales familiar to humans until the relative motion between reference frames becomes somewhat comparable to the speed of light. However, the use of precision equipment such as laser interferometers and atomic clocks allows experimental physicists to experimentally test these ideas. An example of an experimental test for time dilation is the Hafele- Keating experiment in In this experiment, Caesium atomic clocks were placed on commercial airliners and flown in opposite directions (east and west) around the world. The clocks were then compared with a reference Caesium clock on the ground. The difference between the two sets of clocks was found to be in agreement with the predictions from special relativity (within experimental and theoretical error bars). While length contraction cannot be directly measured for extended objects of any appreciable size on scales familiar to the everyday experience of humans, its consequences can be indirectly observed for several physical phenomena. One example of such a phenomenon is the observation of muons produced from the collisions of cosmic rays with molecules in earth's upper atmosphere. The muon has an observed mean lifetime of around 2.2 x 10-6 s. This means, that even if the muon were traveling at the speed of light, it should decay before it reaches muon detectors at the Earth's surface. However, such detectors observe muons from cosmic ray showers all the time. This has a ready explanation in special relativity; the muon sees the atmosphere as being contracted and thus it does not travel the same distance that Earth based observers measure the atmosphere's thickness to be. It travels a smaller distance. In any case, the consequences of special relativity have at least a century long history of being in agreement with the overwhelming majority of experimental tests. Thus, one can reasonably consider special relativity to be held up to all standards of scientific rigor. It can then be said that special relativity is a predictive model of reality and should be taken seriously. Special relativity replaces the cartesian coordinate system whose distances are defined by the Euclidean metric represented by the 3x3 matrix: g ab = ( 0 1 0) (a,b = 1-3) 0 0 1

5 with the Minkowski metric which is represented by the 4x4 matrix: (μ, ν = 1-4 ) The coordinate space defined by the Minknowski metric is referred to as the Minkowski space. In Galilean relativity distances defined by the Euclidean metric are invariant under Galilean trasformations. In special relativity distances defined by the Minkowski metric: are invariant under Lorentz transformations. These distances are often referred to as "space-time" intervals because of the 4 th time like coordinate introduced in the Minkowski space. Any vector quantity such as displacement, momentum, velocity, or acceleration must have a length according to the Minkowski metric. Minkowski space is often understood visually via the space-time diagram. Since Minkowski space is 4- dimensional it cannot be truly represented by a two-dimensional surface in a way that is easily understood. The space-time diagram thus only considers the case where space is 2 dimensional so that space time can be represented as the projection of a 3-d (2 spatial + 1 time) space: Figure Light Cone Space-Time Diagram

6 Figure shows a typical space time diagram and reveals the light cone structure. Often in spacetime diagrams the units of the time axis (the vertical axis in Figure 1.1.1) and the units of the space axes are chosen such that an object traveling at exactly the speed of light in a vacuum will have a trajectory defined by a 45 degree angle. This forms the light cone surface. Any trajectory inclined at less than 45 degrees from the vertical is defined as time-like, any trajectory inclined at more than 45 degrees from the vertical is defined as space-like and any trajectory at exactly 45 degrees is defined as light-like. For reasons discussed in the next section, the inertial trajectories of massive objects can never be inclined at greater than or equal to 45 degrees. 1.2) Reference Frames: Inertial vs. Accelerated Figure Figure shows the world lines of inertial observers traveling at various speeds for a 2-D space-time. Notice two things. 1) Inertial trajectories are always straight lines and 2) the greater the angle, the greater the speed. In section 1.1, it was mentioned that inertial trajectories can never be inclined more than 45 degrees (assuming the correct choice of units for the axes). This is equivalent to saying that massive objects can never travel at the speed of light in a vacuum ( c ). The reason for this is that it requires an infinite amount of energy to accelerate a massive object to the speed of light. This is shown below in Figure 1.2.2

7 Figure In special relativity, the relativistic kinetic energy is defined as: This means it is undefined at v = c and tends to infinity for v ~ c. Accelerated trajectories are a different but related story. Massive objects follow hyperbolic trajectories in space time. They have an equation of motion: Where x 0 is the hyperbolic intercept with the x or t axis. The trajectory is, as mentioned before, is a hyperbolic equation of the form: This has an acceleration of the form: a = c 2 /x 0. This can be seen from Taylor expanding the trajectoy in the approximation that x 0 >> ct: x ~ (x 0 ct) ~ x 0 ( (ct ) + ) x 0 c 2 x ~ x t x 0

8 This looks like the parabolic trajectories that accelerated objects take in Cartesian coordinates. Consider two observers one inertial and one accelerated. It is true that the Lorentz transformation is only valid for transforming from one inertial frame to another. However, if one considers the comoving frame of the accelerated observer, then this frame is related to the comoving frame of the inertial observer by a Lorentz transformation AT ANY GIVEN INSTANT. One can consider an inertial reference frame that is moving with the instantaneous velocity, v(t = T), of the accelerated reference frame. This reference frame is certainly related to that of the inertial observer via the Lorentz transformation: Where dτ is the proper time in the comoving frame (with accelerated observer's instantaneous velocity) of the accelerated observer and dt is the time in the accelerated observer's comoving frame as measured by the inertial observer. One can derive t as a function of τ in the following way: One can also find the spatial coordinate, x, using the equation of motion and using the identity cosh 2 (x) - sinh 2 (x) = 1 Now one has obtained a parametric equation for hyperbolic trajectories. This is shown below in Figure for various values of a and for a range of tau (for each chosen a).

9 Figure Various Accelerated Trajectories From Figure one can see how the hyperbolic trajectories arise naturally from the coordinate transformation between frames. Using natural units, the asymptotes of these hyperbolic trajectories are at x = ± t. These asymptotes divide Minkowski space into 4 distinct regions. Figure Minkowski space divided into 4 regions by the light cone These regions can be expressed in terms of the light cone variables. x ± = x ± t

10 These light cone variables take on the following signs for each region of Minkowski space shown in Figure Figure Light Cone Variable Signs in Different Regions of Minkowksi Space

11 1.3) Minkowski Polar Coordinates In analogy with Cartesian polar coordinates, the Minkowski coordinates can also be expressed in polar coordinates. Figure From Figure 1.3.1, one can see that η acts like an angle and ρ acts like a "radius". Lines of constant ρ are hyperbolas (accelerated trajectories). The polar Minkowski coordinates are known as the Rindler Coordinates. ρ has to be greater than 0 and eta has to be between to. The metric in Rindler coordinates is then: Where ρ takes the form of the space-like coordinated and eta takes the form of the time-like coordinates.

12 Figure 1.32 Rindler Coordinate Frame 1.4) Rindler Space-Time Coordinates The Rindler coordinates can be parametrized in ξ just as the Minkowski coordinates were parametrized in tau. The metric then becomes: Light cone coordinates arise naturally from this parametrization (as they did before in the Minkowski parametrization in τ): The metric in the light cone coordinates then becomes: This form for the light cone-coordinates is not defined for x +, x - 0. This occurs in regions III and IV for ξ+ and II and IV for ξ-. Thus this coordinate frame is not supported in region IV. This will be an important detail which will be addressed in the proceeding sections.

13 2) Massless Klein-Gordon Fields In QFT, the simplest Lagrangian density that one can write is: This Lagrangian is that of the non-interacting, massive scalar field, phi. Using Noether's theorem, equation of motion for this field is: Which is the Klein-Gordon Equation. This simple Lagrangian can be further simplified by considering the special case of massless scalar fields, m = 0. In this case the Klein-Gordon (K-G) equation becomes: 2.1) Massless Klein Gordon Fields in Minkowski Space The previous cases discussed are for Minkowski space with the metric defined from the positive signature (+, -, -, - ). Solutions to the massless K-G equation are plane waves: Where omega = k because the case under consideration is the massless case. In terms of the light cone variables, one can consider two cases, k > 0 (right moving waves) and k < 0 (left moving waves). Interestingly, one observes that these solutions are bounded: This occurs only in regions I and III. For the left moving waves: (k > 0) This only occurs in regions II and IV. (k < 0) The solution of the massless K-G field is usually expanded in terms of the positive and negative energy modes as well as the creation and annihilation operators:

14 So far, there is nothing interesting going on. Even when one uses the light cone variables, x + and x -, the modes still over the entirety of the MInkowski space (different modes cover different regions but all the modes cover the entire region). 2.2) Massless Klein-Gordon Fields in Rindler space Using the Rindler metric derived in Section 1, the Lagrangian for the massless scalar fields is: This leads to an equation of motion (using Noether's theorem) of: The solutions to this equation are, as in Minkowski space, are plane waves: One can use the light cone coordinates Xi± to split the plane waves into two different kinds, right moving waves (k > 0): and left moving waves (k < 0): Now an interesting property is observed. Xi- is not defined for regions I and III. Xi+ is not defined for regions I and II. That means that neither the left or right moving modes are defined in region IV. This is shown below with Figure Figure Non-Analyticity of Rindler modes in region IV

15 To get complete coverage of the Rindler modes, they need to be extended to region IV. This is accomplished by reversing the sign of the coordinates (η, ξ) -> (-η, -ξ). This interchanges regions I and IV and interchanges II and III. This means ξ- is not defined for regions IV and II and ξ+ is not defined for regions IV and III (so now region I is not covered for the reversed coordinates). Now there are two sets of modes: The general solution to the K-G equation in Rindler space can be expanded in terms of these modes: Now one has to relate these modes to the Minkowski modes found in the previous section. 3) Vacuum Expectation Value of the Number Operator of Klein-Gordon Fields As described in the previous section, the K-G Fields are expanded in terms of the normal mode solutions to the corresponding equation. They are also expanded in terms of the creation and annihilation operators. The operator in Minkowski space act on the Minkowski vacuum like: The operators in Rindler space act on the Rindler vacuum like: It is desirable to relate the Rindler creation and annihilation operators to those of the Minkowski creation and annihilation operators. Ordinarily this is accomplished via the Bogoliubov transformations. However, this is an arduous process and William G. Unruh as a more elegant way. Using the following factor: Unruh demonstrated that one could extend the support for one set of Rindler modes into the entire coordinate space (as opposed to using two sets for full coverage which was detailed in the previous section):

16 The (2)* modes contain an ambiguous factor (-) iω. Now e -iπ is equal to 1. So one just adds a phase angle to the expression for (2)* This is the equivalent of a branch cut in the lower half of the x- plane. A similar process is done for the (2) and (1)* modes by placing a branch cut in the upper half of the x+ plane. One combines the (1), (2)*, (1)*, and (2) to obtain a full set of positive energy modes: 3.1) Vacuum Expectation Value of the Number Operator of Klein-Gordon Fields in Minkowski Space In Minkowski space the number operator for the Klein-Gordon field is: The vacuum expectation value of the Number Operator is 0: 3.2) Vacuum Expectation Value of the Number Operator of Klein-Gordon Fields in Rindler Space The Minkowski observer can expand their K-G field in terms of the modes found in the previous section:

17 One can express the Rindler operators b (r) in terms of the new Minkowski operators C (r). Then one can match terms: Figure Term Matching Between the K-G fields of the Minkowski and Rindler Observers After term matching one obtains: Using the commutation relation below, one can normalize the C-modes in the following way:

18 The Rindler modes are then expressed in terms of the Minkowski modes: Now, one can calculate the vacuum expectation value of the number operator in the Rindler space for the Minkowski vacuum 0>. The number operator in Rindler space is then: Consider an accelerated observer in region I (where b (2) modes vanish): Interestingly enough, this is not 0 as in Minkowski coordinates.

19 4) Unruh Radiation The non-zero value of the number operator in Rindler coordinates has an interesting physical interpretation. Its form takes the exact same form as the Planck distribution for the black-body spectrum: If one matches the factor in the exponential then one obtains the temperature (after adding in the appropriate physical constants): This is known as the Hawking-Unruh Temperature or simply the Unruh Temperature. 4.1) Blackbody Spectrum From the previous section, one sees that the temperature depends linearly upon the acceleration the non-inertial observer is undergoing. Figure shows the blackbody curves in spectral radiance vs. wavelength of light emitted for various accelerations: Figure Blackbody Curves for the Unruh Radiation a Non-Inertial Observer Sees

20 One can see that even the faintest temperature thermal bath (below 1 K) requires an acceleration of 4.9 x m/s 2. This kind of acceleration is not achievable in any terrestrial laboratory. More clever methods of detecting Unruh radiation must be sought after. 5) Conclusions Why do the inertial and accelerated observer disagree on the vacuum expectation value of their number operators? The number operators are acting on the same vacuum state, should they not agree on the expectation value? The resolution to this seeming contradiction is that in order for the uniformly accelerated observer to keep accelerating, something must be supplying energy to the observer. This source of energy is the source of the Unruh radiation. In fact, the inertial observer will see that the Rindler observer is emitting radiation. If the Rindler observer is an electrically charged particle, then this is known effect of bremsstrahlung radiation. The Rindler observer, however, is unaware of these details and only observes the net effect. Some of the radiation coming from the source of acceleration will be absorbed by the Rindler observer and seen as Unruh radiation.

21 Sources 1) Siopsis, George. PHYS 611 Notes: 2) Alsing, Paul M. & Milonni, Peter W. Simplified derivation of the Hawking-Unruh temperature for an accelerated observer in vacuum. American Journal of Physics 3) McDonald, Kirk T. Hawking Radiation of a Uniformly Accelerated Charge.

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27 Figure 2 Various Accelerated Word-Lines in Minkowski Space

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29 Fig. 1 Rindler Coordinates for 0.5 Acceleration

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40 Figure 3 Different Black Body Curves for Unruh Radiation for an observer with various accelerations