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Hawking Radiation

Modified by Ilja Schmelzer 1997.
Original by John Baez 1994.

Hawking Radiation

In 1975 Hawking published a shocking result: if one takes quantum theory into account, it seems that black holes are not quite black!  Instead, they should glow slightly with "Hawking radiation", consisting of photons, neutrinos, and to a lesser extent all sorts of massive particles. This has never been observed, since the only black holes we have evidence for are those with lots of hot gas falling into them, whose radiation would completely swamp this tiny effect. Indeed, if the mass of a black hole is M solar masses, Hawking predicted it should glow like a black body of temperature

                     6 × 10-8/M kelvin,

so only for very small black holes would this radiation be significant. Still, the effect is theoretically very interesting, and folks working on understanding how quantum theory and gravity fit together have spent a lot of energy trying to understand it and its consequences. The most drastic consequence is that a black hole, left alone and unfed, should radiate away its mass, slowly at first but then faster and faster as it shrinks, finally dying in a blaze of glory like a hydrogen bomb. But the total lifetime of a black hole of Msolar masses works out to be

                     1071 M3 second

so don't wait around for a big one to give up the ghost. (People have looked for the death of small ones that could have formed in the big bang, but they haven't seen any.)

How does this work?  Well, you'll find Hawking radiation explained this way in a lot of "pop-science" treatments:

Virtual particle pairs are constantly being created near the horizon of the black hole, as they are everywhere. Normally, they are created as a particle-antiparticle pair and they quickly annihilate each other. But near the horizon of a black hole, it's possible for one to fall in before the annihilation can happen, in which case the other one escapes as Hawking radiation.

In fact this argument also does not correspond in any clear way to the actual computation.  Or at least I've never seen how the standard computation can be transmuted into one involving virtual particles sneaking over the horizon, and in the last talk I was at on this it was emphasized that nobody has ever worked out a "local" description of Hawking radiation in terms of stuff like this happening at the horizon. I'd gladly be corrected by any experts out there... Note: I wouldn't be surprised if this heuristic picture turned out to be accurate, but I don't see how you get that picture from the usual computation.

The usual computation involves Bogoliubov transformations. The idea is that when you quantize (say) the electromagnetic field you take solutions of the classical equations (Maxwell's equations) and write them as a linear combination of positive-frequency and negative-frequency parts. Roughly speaking, one gives you particles and the other gives you antiparticles.  More subtly, this splitting is implicit in the very definition of the vacuum of the quantum version of the theory!  In other words, if you do the splitting one way, and I do the splitting another way, our notion of which state is the vacuum may disagree!

This should not be utterly shocking, just pretty darn shocking. The vacuum, after all, can be thought of as the state of least energy. If we are using really different co-ordinate systems, we'll have really different notions of time, hence really different notions of energy—since energy is defined in quantum theory to be the operator such that time evolution is given by exp(-itH). So on the one hand, it's quite conceivable that we'll have different notions of positive and negative frequency solutions in classical field theory—a solution that's a linear combination of those with time dependence exp(-iωt) is called positive or negative frequency depending on the sign of ω—but of course this depends on a choice of time co-ordinate t. And on the other hand, it's quite conceivable that we'll have different notions of the lowest-energy state.

Now when we are in good old flat Minkowski spacetime, a la special relativity, there are a bunch of "inertial frames" differing by Lorentz transformations. These give different time co-ordinates, but one can check that the difference is never so bad that different co-ordinates give different notions of positive or negative frequency solutions of Maxwell's equations.  Nor will different people using these co-ordinate systems ever disagree about what's the lowest-energy state.  So all inertial observers agree about what's a particle, what's an antiparticle, and what's the vacuum.

But in curved space/time there aren't these "best" co-ordinate systems, the inertial ones. So even very reasonable different choices of co-ordinates can give disagreements about particles vs antiparticles, or what's the vacuum. These disagreements don't mean that "everything is relative", because there are nice formulas for how to translate between the descriptions in different co-ordinate systems.  These are Bogoliubov transformations.

So if there is a black hole around...

on the one hand we can split solutions of Maxwell's equations into positive frequency in the most blithering obvious way that someone far from the black hole and far in the future would do it...

and on the other hand we can split solutions of Maxwell's equations into positive frequency in the most blithering obvious way that someone far in the past, before the collapse into a black hole has happened would do it.

That'd be the heuristic explanation I'd give that most closely corresponds to the usual computation. There are additional things to say about the fact that the guy far in the future and far away from the black hole can't see what's in the hole, so he has incomplete information about the state, so he sees a state with entropy, in fact a thermal state. (Here I'm assuming the black hole was NOT eternal, so the guy way back in the past didn't have the black hole to contend with.  Apparently Hawking's original computation dealt with this case, but people subsequently watered down his explanation by assuming the black hole was there eternally, to simplify the math. This is what the guy at the talk said... I'd only seen the watered-down version!)

Now in fact when you do a Bogoliubov transformation to the vacuum you get a state in which there are pairs of particles and antiparticles, so this is possibly the link between the math and the heuristic explanation. Hopefully whoever made up the usual heuristic explanation understood the link better than I do!

  • Robert M. Wald, General Relativity, Sections 14.2–14.4, University of Chicago Press, Chicago, 1984. (A good precise introduction to the subject.)
  • Stephen W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975), 199–220. (The original paper.)

[source: ]

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Story and Facts of CPI, a modern technological Company with strong ethical commitments



CPI Antenna Systems Division (CPI ASD) Is a multinational manufacturer of high-performance, highly engineered antenna systems for satellite communications, telemetry, radar, electronic warfare and high-frequency (HF) applications. CPI ASD's customers include international broadcasters and Fortune 500 companies, satellite operators and telecommunications carriers, and military and government organizations. 

Operating across all radio frequency (RF) bands, CPI ASD's products are produced through the use of CAD, CAE, large-scale graphics and computer-controlled machining that delivers extreme precision, supported by in-house microwave and electronics assembly and test laboratories. CPI ASD's in-house service organization provides worldwide installation, logistics and repair support with assistance from local CPI offices.

CPI Antenna Systems Division represents a merger of two leading organizations: CPI ASC Signal Division, with a 50-year heritage of engineering creativity and excellence, and CPI Malibu Division, a recognized leader in antenna design, development and production since 1975.



Russell and Sigurd Varian (photographs by Ansel Adams)

As the former Electron Device Business of Varian Associates, Inc., Communications & Power Industries LLC (CPI) has a long heritage of technological and product excellence. In 1948, Russell H. Varian and Sigurd F. Varian, the historical founders of CPI’s business, founded Varian Associates and introduced the klystron as its first commercial product. Varian Associates was incorporated in California on April 20, 1948, with the stated purpose to “conduct general research in the fields of physical science of every kind or nature.” At the time, Varian Associates had $22,000 of capital and six full-time employees, receiving technical and business assistance from several members of the faculty at nearby Stanford University. (Read more about the early history of Varian Associates.) In 1953, the company established its headquarters in Palo Alto, Calif., in what has become the Stanford Industrial Park. (Read more about Russell and Sigurd Varian and the invention of the klystron.)

Today, CPI is a global manufacturer of electronic components and subsystems focused primarily on communications and defense markets. The company develops, manufactures and globally distributes innovative and reliable technology solutions used in the generation, amplification, transmission and reception of microwave signals for commercial and military applications. CPI serves customers in the communications, defense, medical, industrial and scientific markets.

The company is organized into five operating divisions:

  • Microwave Power Products Division in Palo Alto, Calif., which now includes the Eimac operations, also in Palo Alto, Calif., and the Econco operations in Woodland, Calif.;
  • Beverly Microwave Division in Beverly, Mass.;
  • Satcom & Medical Products Division, which includes the operations of the former Satcom Division in Georgetown, Ontario, Canada and San Jose, Calif. and the operations of the former Communications & Medical Products Division in Georgetown, Ontario, Canada;
  • Radant Technologies Division in Stow and Hudson, Mass.; and
  • Antenna Systems Division, which includes the operations of the former Malibu Division in Camarillo, Calif., the former ASC Signal Division in Plano, Texas and Whitby, Ontario, Canada, and Orbital Systems in Irving, Texas.


Original headquarters of Varian Associates in San Carlos, California, circa 1949

In its early years, Varian Associates’ products included vacuum electron tubes, an expanding instruments product line, an embryonic accelerator activity and a venture into geophysical instruments. In the mid-1950s, the company invented an all-electronic vacuum pump for manufacturing vacuum electron tubes. The company soon realized that the vacuum pump had applications far beyond tube processing and launched additional lines of business.

In the early 1950s, the Canadian government asked Varian Associates to create a local source for the production of microwave vacuum electron devices. The company established Varian Associates of Canada Limited in 1954 in Georgetown, Ontario, Canada. This business has since become the Satcom & Medical Products Division of CPI, a leader in uplink amplifier products and systems for satellite communications and in high-frequency medical X-ray generators, integrated system controls, digital imaging solutions, and compact power supplies for physical vapor deposition applications.

By 1958, Varian Associates had grown to approximately 1,300 employees and generated approximately $20 million in sales.

In 1959, the company helped launch S-F-D Laboratories in Union, N.J. to design and build new classes of magnetrons and other microwave tubes. That same year, Varian Associates acquired Bomac Laboratories, a manufacturer of microwave tubes and components in Massachusetts. Varian Associates acquired LEL, Inc., which specialized in microwave components, in 1964. These three operations were combined into one operation in Beverly, Mass., and have become the Beverly Microwave Division of CPI.


Russell and Sigurd Varian with the V-42 klystron used in ultra high-frequency (UHF) television transmission, circa 1953. This klystron was the first of a series of Varian Associates high-power tubes used for radar and communication systems. (photograph by Ansel Adams)

In the 1960s, Varian Associates expanded its business with research in solid state devices, new forms of printing and the development of devices for commercial communications. The company also acquired the traveling wave tube business of the General Electric Company in Palo Alto during this period, merging it into Varian Associates’ existing local operations.

In 1965, the company acquired Eitel-McCullough, Inc., a Calif.-based manufacturer of a broad line of specialty electron tubes, primarily for various broadcast and industrial markets. Eitel-McCullough was originally founded in 1934 by William Eitel and Jack McCullough to produce more powerful and reliable transmitting vacuum tubes operating at low voltages for the Ham radio community. Its products were quickly adopted for commercial and military transmitters: they were used by the U.S. Navy in radar experiments starting in 1938 and they powered Allied radars during World War II. Eitel-McCullough became the Eimac Division of Varian Associates and eventually became the Eimac operations of CPI. In June 2006, those operations were integrated into the Microwave Power Products Division of CPI in Palo Alto, Calif.

In 1972, Varian Associates purchased Ryka Scientific, which became part of the company’s solid state operations. In 1979, the Solid State Microwave Division consolidated all its activities and moved into a new facility “dedicated to the development and production of microwave solid state devices and components.” This division’s products were used for low-power applications in communications, avionics, radar, instrumentation and electronic warfare applications. CPI later sold these operations in 2002.

In 1979, Varian Associates’ Microwave Components and Subsystems Division consolidated the majority of its activities in a new, 100,000 square foot facility in Santa Clara, Calif. This division, which focused on power amplifiers, power supplies, microwave components and electromagnets for satellite communications, radar and electronic warfare applications, eventually became the Satcom Division of CPI.


Headquarters of Varian Associates in the Stanford Industrial Park in Palo Alto, California, circa 1953. CPI's present-day headquarters remain in the same location.

In 1989, as a result of the energy efficient performance of the Klystrode® inductive output tube (IOT), the Eimac Division of Varian Associates was awarded an Emmy® Award for technological achievement by the Academy of Television Arts and Sciences. The Klystrode IOT used in television transmitters at WCES in Augusta, Ga. was the first of the high-power output devices to be on-air for UHF television.

Over the years, through internal development and acquisitions, Varian Associates grew its Electron Device Business and developed new devices and uses for its products, including applications for radar, electronic warfare, communications, medical, industrial and scientific markets. The Electron Device Business of Varian Associates became a technological leader in microwave and x-ray generating devices and electronic equipment to drive these devices.

In August 1995, Varian Associates sold the Electron Device Business to Leonard Green & Partners, L.P., a private equity fund, and members of management. Together, they formed Communications & Power Industries, Inc. with the mandate to continue to grow the business’ tradition of technology leadership and manufacturing excellence.

In January 2004, affiliates of The Cypress Group, a private equity fund, acquired CPI.

In October 2004, CPI acquired Econco Broadcast Service, Inc., a rebuilder of power grid devices originally founded in 1968 in Woodland, Calif. Econco Broadcast Service had received the first contracts ever awarded by the U.S. government and the U.S. Navy to remanufacture microwave devices, and twice received the U.S. Navy’s Award for Excellence. Econco is currently part of the Microwave Power Products Division of CPI.

In April 2006, CPI International, Inc., the parent company of CPI, held its initial public offering and began trading on The Nasdaq Stock Market under the symbol “CPII.”

In August 2007, CPI acquired Malibu Research Associates, Inc., a leader in the design, manufacture and integration of advanced antenna systems for radar, radar simulators and telemetry systems, as well as for strategically vital data links used in ground, airborne, unmanned aerial vehicles (UAV) and shipboard systems. Malibu Research Associates, founded in 1975 and located in Camarillo, Calif., became the Malibu Division of CPI. The Malibu operations are currently part of the Antenna Systems Division of CPI.

In February 2011, an affiliate of Veritas Capital, a private equity investment firm which invests in companies that provide critical products and services to governments worldwide, acquired CPI for approximately $525 million. CPI International ceased trading on The Nasdaq Stock Market.

In June 2012, CPI acquired the Codan Satcom business from Codan Limited, increasing CPI’s portfolio of solid-state products and subsystems for satellite communications applications to support commercial and government customers. The Codan Satcom business, consisting of operations in Australia and the Locus Microwave operations in Boalsburg, Pa., are currently part of the Satcom & Medical Products Division of CPI.

In June 2013, CPI acquired M C L, Inc. (MCL), a manufacturer of power amplifier products and systems for the satellite communications market. The acquisition, which was integrated into what is now the Satcom & Medical Products Division of CPI, further expanded CPI’s satellite communications product and service offerings.

In October 2013, CPI acquired Radant Technologies, Inc., a manufacturer of advanced composite radomes, reflector antennas and structures for defense aerospace, defense naval and commercial aerospace applications. The acquisition broadened CPI’s portfolio of microwave and radio frequency products for radar, electronic warfare and communications applications. Radant Technologies, founded in 1984 and based in Stow, Mass., became the Radant Technologies Division of CPI.

In September 2015, CPI acquired ASC Signal Corporation, which designs and builds advanced satellite communications, radar and high-frequency antennas and controllers that are used in commercial and government satellite communications, terrestrial communications, imagery and data transmission, and radar and intelligence applications. The addition of ASC Signal’s products and technology broadens CPI’s existing offerings for communications and radar customers to include one of the industry’s broadest portfolios of high-performance, large-aperture fixed and mobile antennas. Headquartered in Plano, Texas and with manufacturing facilities in Whitby, Ontario, Canada, the ASC Signal operations are currently part of the Antenna Systems Division of CPI.

In July 2017, Odyssey Investment Partners, LLC, a private equity investment firm, acquired CPI.

In March 2018, CPI announced that it was acquiring the limited-motion satellite antenna family from Viasat Inc., extending the larger-diameter antenna product line of CPI's Antenna Systems Division to include 6m to 18m multi-band antennas.

In August 2018, CPI acquired the related companies Orbital Systems, Ltd. and Quorum Communications, Inc., merging them into one organization, Orbital Systems LLC, and integrating them into CPI's Antenna Systems Division. Based in Irving, Texas, Orbital designs and manufactures ground-based full-motion antenna systems primarily for communications with earth observation satellites in low earth orbit and telemetry, tracking and control applications, as well as provides satellite microwave receivers, downconverters and other communications products.

Today, CPI continues its long tradition of leading the world in developing and producing microwave devices. In fiscal 2018, CPI employed approximately 2,000 people and generated approximately $492 million in sales.

 Corporate Citizenship

CPI strongly believes in its duty to be a responsible corporate citizen. The company expects its management and employees to live up to the highest standards of ethical conduct and to make a meaningful contribution to the health and well-being of the communities in which it operates. CPI encourages social responsibility in a number of different ways:

Giving Back:

CPI supports numerous non-profit and charitable organizations through a combination of corporate donations, employee donations, fundraising drives, blood drives and other philanthropic efforts. These organizations include:

American Cancer Society, Inc.

The American National Red Cross and Canadian Red Cross

American Society for the Prevention of Cruelty to Animals (ASPCA)

Beverly Bootstraps Community Services, Inc.

Beverly Police Department and Police Relief Association

California Polytechnic State University San Luis Obispo Disability Resource Center

CARE (Cooperative for American Relief Everywhere)

ChildFund International

Elderly Nutrition Program in Yolo County

The Family Giving Tree

Food Bank of Yolo County

Habitat for Humanity International

Juvenile Diabetes Research Foundation International

Marine Toys for Tots Foundation

North Shore Chamber of Commerce Honor Scholar Program

The Salvation Army

Second Harvest Food Bank of Santa Clara and San Mateo Counties

Special Olympics International

Stanford Blood Center

The Terry Fox Foundation

Ticket to Dream Foundation

United Way Worldwide and United Way of Canada

Weekend to End Women’s Cancers

Woodland Healthcare Foundation

World Vision Canada

Protecting the Environment:

CPI encourages the preservation of natural resources and the protection of the environment through a variety of programs and initiatives. The company and its employees continually strive to minimize the environmental impact of their activities on a daily basis. These environmental sustainability efforts include:

  • Ongoing initiatives to:
    • Minimize requirements for natural resources in manufacturing processes,
    • Minimize peak power usage,
    • Reduce the storage and usage of chemicals,
    • Promote recycling of wastes generated by manufacturing and other processes,
    • Reduce the amount of hazardous wastes and emissions generated by manufacturing processes, and
    • Improve waste water treatment plant processes and eliminate water from disposal
    • Support and subsidies for employees using public transit and other commute alternatives, including:
      • Eco Pass (Santa Clara Valley Transit Authority): public transit subsidy
      • Commuter Check: public transit subsidy
      • Vanpooling and carpooling
      • Bike lockers
      • Ride Your Bike to Work Day
    • CPI Econco Division has been a California Department of Resources Recycling and Recovery (CalRecycle) Waste Reduction Awards Program (WRAP) winner every year since 2003

Promoting an Ethical Supply Chain:

CPI believes in the ethical treatment of all human beings, and is strongly opposed to human trafficking, forced servitude, child labor and all other types of slavery. To see CPI’s disclosure on this topic under the California Transparency in Supply Chains Act, click here.

Protecting Our Employees:

CPI believes that its employees are its greatest resource. The company has a number of programs aimed at improving employee wellness. CPI also has robust health and safety programs and devotes significant resources to safety training at each of its major locations.

Assessing Conflict Minerals:

CPI believes in a humane and ethical supply chain. In August 2012, the U.S. Securities and Exchange Commission (SEC) adopted a rule requiring companies to publicly disclose their use of certain “conflict minerals” originating in the Democratic Republic of Congo (DRC) and its neighboring countries. According to the SEC, this disclosure is required due to “concerns that the exploitation and trade of conflict minerals by armed groups is helping to finance conflict in the DRC region and is contributing to an emergency humanitarian crisis.” Under the SEC rule, companies must disclose their use of tantalum, tin, gold or tungsten and identify if the minerals originated in the DRC region if these minerals are necessary to the functionality or production of a product they manufacture.

CPI fully supports the eradication of violence and exploitation from the supply chain and is committed to taking all steps necessary to comply with the rules related to the reporting of conflict minerals. CPI believes in the responsible sourcing of minerals and will make its best effort to promote and ensure a conflict-free supply chain.

Some of CPI’s products may contain tantalum, tin, tungsten or gold. Because the company does not purchase these minerals directly from mines or smelters, CPI has commenced a comprehensive due diligence process to determine the source of the minerals used in its products, including conducting a reasonable country of origin inquiry with its suppliers, using a coordinated, risk-based approach. The supply chain for these minerals is complex and broad, including several hundred suppliers from numerous countries, and tracing the source of these minerals is a lengthy, challenging process. To learn more about CPI’s conflict minerals process, please see the Conflict Minerals Report in the company’s most recent Form SD filing with the Securities and Exchange Commission.


CPI believes that a diverse community makes for a more interesting and rewarding environment, and our experience continues to confirm our belief that it is a good business practice to have personnel policies and practices with a strong commitment to providing an equal opportunity workplace. The company values the contributions of a diverse workforce and is proud to be an Equal Opportunity Employer.

In addition, CPI supports diversity in its supply chain. CPI is a proud customer and supporter of small businesses and businesses owned by military veterans, minorities and women.

 Careers at CPI

We invite you to view our current job opportunities.

Whether you are a seasoned professional or just embarking on your career, CPI is an ideal place to expand your wealth of knowledge and expertise. As a global industry leader, we offer a dynamic and diverse team environment. We value the unique skills and qualities that each employee brings to CPI, and we respect each employee as an integral member of our growing team. Join CPI and build your career with some of the most highly skilled and dedicated individuals you will ever meet.

Thank you for considering CPI as an employer of choice. CPI values the contributions of a diverse workforce and is proud to be an Equal Opportunity Employer. CPI encourages veterans, disabled veterans and disabled individuals to apply for any open position for which they may be qualified.

In compliance with the ADA Amendments Act (ADAA) in the United States, or Accessibility for Ontarians with Disabilities Act (AODA), in Ontario, Canada, if you are an individual with a disability and require reasonable accommodation to complete any part of the application process, or are limited in the ability or unable to access or use this online application process and need an alternative method for applying, you may contact us for assistance:

  • In the United States, at This email address is being protected from spambots. You need JavaScript enabled to view it.
  • In Canada, at This email address is being protected from spambots. You need JavaScript enabled to view it. 

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MilSatMagazine (MSM)

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