Black Holes, Dark Matter, and Dark Energy!

What did Albert Einstein foresee about it? 

Albert Einstein during a lecture in Vienna in 1921. Picture: Ferdinand Schmutzer
When Albert Einstein died in 1955, he had revolutionized our understanding of space and time. So what did he foresee about such concepts as Black HolesDark Matter and Dark Energy? The following lines are written with reference made to the Nobel Prizes in Physics awarded in 20202019 and 2011.
Sir Roger Penrose is awarded one half of the 2020 Nobel Prize in Physics for his theoretical work on Black Holes. He proved that their existence is a realistic expectation from Einstein’s theory of gravitation. Also Sir Isaac Newton’s theory of universal gravitation had earlier been used for discussing the possible existence of invisible super massive objects where the escape velocity would be higher than the velocity of light in vacuum. Einstein did never believe in such objects as a physical reality.

Reinhard Genzel and Andrea Ghez share the second half of the 2020 Nobel Prize for the discovery that an invisible and extremely heavy object controls the star orbits near the center of our own galaxy, the Milky Way. The only logical interpretation is that the object is a supermassive Black Hole.

It is a good choice that Penrose shares the Nobel Prize with the other two physicists. He made his pioneering contribution in 1965 and has up to his present age of 89 years actively contributed to what we know today about the universe. It has been done in collaboration with many others, including the famous cosmologist Stephen Hawking who passed away two years ago.

The concept of a Black Hole at the center of our galaxy as now supported by the 2020 Nobel Prize in Physics is easier to discuss than the Dark Matter concept for explaining the strength of the cohesive gravitational force inside the galaxy and the Dark Energy concept for explaining the accelerated cosmic expansion. While supported by the Nobel Prizes in Physics awarded 2019 och 2011, respectively, their nature is considered mysterious although Dark Energy is commonly related to the cosmological constant proposed by Einstein. In half a dozen texts published from 2009 up to 2019 by the Swedish Mathematical Society, all known interpretations of the two “dark” concepts were challenged by me.

Einstein was awarded the 1921 Nobel Prize in Physics for his contribution to quantum physics. He had until his death a focus on developing a unified field theory for the electromagnetic and gravitational forces so it seems to me that his famous skill of making thought experiments should have made him predict the existence of Dark Matter as accumulations of elementary mass particles providing the gravitational force in analogy with the elementary charge particles providing the electromagnetic force. In the above-mentioned texts, this prediction results from a proposed “Wave Theory of Time” (WTT) linking the strength ratio between the electromagnetic and gravitational forces to a rest mass ratio of their respective elementary particles and to a repetition frequency ratio related to their respective so-called matter waves.

WTT uses a continuous wave model where a scale of time itself varies with a period in the order of 10 exp 20 seconds determining the gravitational force described by Einstein’s curved spacetime and where another scale of time varies with a period in the order of 1/(10 exp 20) seconds determining the electromagnetic force. This explanatory model makes it possible to see the velocity of the cosmic expansion either as increasing with respect to time counted by the atomic clocks which rely on the stability of the elementary charge particles or as decreasing with respect to time counted by an alternative clock which here is named the Einstein clock and which relies on the stability of light wave propagation in vacuum as used in Einstein’s thought experiments.

It was his strong belief in a static universe which in 1917 motivated Einstein’s proposal of a cosmological constant representing a repulsive force that counteracts the cohesive gravitational force. The discovery in the late 1920s that the universe is expanding made him regret this proposal as his biggest blunder. However, the discovery in 1998 that the velocity of the expansion of the universe is increasing made cosmologists quickly state that Einstein was right in believing in a force that counteracts the cohesive gravitational force and that is represented by the cosmological constant. The latter has a magnitude that is linked to the increasing velocity of the cosmic expansion as measured with respect to time counted by the atomic clocks. However, the velocity of the cosmic expansion is not increasing if measured with respect to time counted by the Einstein clock. Then the magnitude of the cosmological constant vanishes and the Dark Energy mystery disappears.

The Earth provided a wrong reference for discovering the nature of the planet orbits in the solar system, and atomic clocks provide the wrong reference for discovering the nature of the cosmic expansion. In a thought experiment where the Einstein clock is the one and only available reference for describing an expansion of space in a universe with elementary mass particles but without the elementary charge particles, WTT shows that the physical reality of Dark Energy is an unnecessary assumption. What the scientific authorities tell us is about 70% of the total energy in the universe could be the biggest bubble ever.   

The Universe and Human Stupidity

Albert Einstein improved greatly our understanding of the universe. He remarked once (Source: Frederick S. Perls): “Only two things are infinite, the universe and human stupidity, and I’m not sure about the former.”’

It seems to me that he improved our understanding also of human stupidity by proposing a repulsive force that would just balance the effect of gravity in intergalactic space. After the discovery that the universe is expanding, he called his proposal “the biggest blunder of my life” (Source: George Gamow).

In spite of Isaac Newton’s theory of gravity as an omnipresent contracting force, a static view of the universe had become generally adopted among scientists. When he proposed a perfectly balanced universe and rejected early criticism from Alexander Friedmann, Albert Einstein confirmed the truth of Schopenhauer’s remark that “There is no opinion however absurd, which men will not readily embrace as soon as they can be brought to the conviction that it is generally adopted”.

Isaac Newton. Source:
Arthur Schopenhauer. Portrait by Ludwig Sigismund Ruhl. Source:

An apparent acceleration in the expansion of the universe was discovered in 1998. Einstein’s repulsive force became then quickly embraced by scientists. It was for them the simplest explanation of data in line with the idea about an initial “push” overcoming gravity and called “the Big Bang“. While another explanation of the apparent acceleration is better in view of Occam’s Razor, it has been ignored long enough. Here is a brief description.

Occam's Razor (William of Occam). Source:
How the Big Bang may have happened... Source:
Stephen Hawking boarded a modified Boeing 727 at the Kennedy Space Center in Florida in 2007, which then performed parabolic arcs where the aircraft is in free fall, allowing the passengers to experience moments of weightlessness. Source:

Before the 1998 discovery, a continuous wave perspective on time was proposed by me. It predicted the apparent acceleration of the cosmic expansion, and a paper was sent to Nature and other publishers. Rejected as too speculative and rewritten, the resulting “Wave Theory of Time” (WTT) became published with quite a delay first by the Swedish Mathematical Society (SMS) in the autumn of the International Year of  Astronomy 2009 and later by the STAR society at the old Observatory in Stockholm. Both SMS and STAR published also a text written by me as a tribute to Stephen Hawking after his death 2017.03.14 precisely on Pi Day.

Nicolas Copernic (Nicolaus Copernicus1473-1543). Source:

According to WTT, the Big Bang beginning of the on-going expansion of the universe is merely a zero-crossing point of a continuous wave in which subsequent half-cycles have opposite directions of a varying time measure. Besides predicting the apparent acceleration of the cosmic expansion and the magnitude of the cosmological constant so that the generally adopted assumption of Einstein´s repulsive force becomes unnecessary, WTT makes the so-called big number coincidences derivable. The continuous wave perspective on time is as crucial for these achievements as the Copernican perspective on space was for Newton’s explanation of the solar system.

Dark energy. Source:

It is of interest here to note that a cosmological model in which the direction of time is changed at an expansion peak was once proposed by Stephen Hawking who referred later to that as a big mistake. In the light of history, Einstein’s remark about human stupidity is applicable to authorities in science as well as to independent researchers like me who challenge the scientific community. And it can one day be seen as applicable to those who now believe in the physical existence of “dark energy” as the origin of a force that counteracts gravity and as a dominating phenomenon in the universe. 

Leo Tolstoy. Source:

Leo Tolstoy made this remark:

“I know that most men — not only those considered clever but even those who are very clever and capable of understanding most difficult scientific, mathematical, or philosophic, problems — can seldom discern even the simplest and most obvious truth if it obliges them to admit the falsity of conclusions they have formed, perhaps with much difficulty — conclusions of which they are proud, which they have taught to others, and on which they have built their lives.”

The following lines are enlightening. They originate from the Royal Swedish Academy of Sciences (Source: The Nobel Prize Organization).

Royal Swedish Academy of Sciences. Source:

“So what is it that is speeding up the Universe? It is called dark energy and is a challenge for physics, a riddle that no one has managed to solve yet. Several ideas have been proposed. The simplest is to reintroduce Einstein’s cosmological constant, which he once rejected. At that time, he inserted the cosmological constant as an anti-gravitational force to counter the gravitational force of matter and thus create a static Universe. Today, the cosmological constant instead appears to make the expansion of the Universe to accelerate.

The cosmological constant is, of course, constant, and as such does not change over time. So dark energy becomes dominant when matter, and thus its gravity, gets diluted due to the expansion of the Universe over billions of years. According to scientists, that would account for why the cosmological constant entered the scene so late in the history of the Universe, only five to six billion years ago. At about that time, the gravitational force of matter had weakened enough in relation to the cosmological constant. Until then, the expansion of the Universe had been decelerating.

The cosmological constant could have its source in the vacuum, empty space that, according to quantum physics, is never completely empty. Instead, the vacuum is a bubbling quantum soup where virtual particles of matter and antimatter pop in and out of existence and give rise to energy. However, the simplest estimation for the amount of dark energy does not correspond at all to the amount that has been measured in space, which is about 10 exp 120 times larger (1 followed by 120 zeros). This constitutes a gigantic and still unexplained gap between theory and observation – on all the beaches of the world there are no more than 10 exp 20 (1 followed by 20 zeros) grains of sand.”

The Comprehensible Universe

When the great physicist and Nobel laureate Steven Weinberg wrote his book The First Three Minutes (1977), he concluded that “The more the universe seems comprehensible, the more it also seems pointless”. His pessimistic conclusion is shared by too many physicists. Relief can be found in a comment made by the late cosmologist Stephen Hawking: “The human intellectual history is meaningful as a record of how we have come nearer and nearer to an understanding of the order in the universe”. However, the question is whether the history of the universe can be understood as meaningful in itself by having a purpose regardless of human intellectual history. 

The generally accepted one-shot model of the universe seemed pointless also to Hawking so he proposed once that the cosmic expansion will reach a peak followed by time reversal. He felt soon forced to abandon that proposal and said later that it was a big blunder. Albert Einstein said the same about his abandoned static model of the universe and he also said that “The most incomprehensible thing about the world is that it is comprehensible” (Physics and Reality, 1936), While he described time as an illusion in line with the view that the universe seems pointless, its intrinsic nature is worth a better interpretation. My own thoughts on the subject were a decade ago published by The Swedish Mathematical Society, SMS. Generalizing the Einstein-Planck equation, it seemed logical to propose a wave theory of time, WTT. And as the universe is comprehensible by means of mathematics, it seemed equally logical to apply the Turing computational model for the purpose of coming nearer to an understanding of its order.   

The great physicist and Nobel laureate Steven Weinberg
Alan Turing c. 1928 at age 16

SMS has published detailed descriptions of my handheld models for explaining the solar system, the Milky Way, and the apparent accelerated expansion of space-time. Designed to complement earth globes and to present cosmic proportions in an accurate manner, they are intended to present at a glance the message that the nature of the universe is dynamic sustainability and that repetition rules. According to WTT, the macro- as well as micro-scale phenomena of the universe can be traced back to repetition periods of fundamental waves described by a generalized version of the Einstein-Planck equation. WTT describes the nature of dark matter, of dark energy, and of the huge difference in strength between electromagnetic forces and gravitation. And the so-called big number coincidences are made derivable instead of being interpreted by anthropic reasoning.

Albert Einstein
Isaac Newton surrounded by symbols of some of his greatest findings

Einstein described Isaac Newton‘s gravitational force as caused by curved space-time. WTT explains that curvature as due to the continuous-wave nature of time described as a flow with a varying scale of time itself and with zero-crossing points where the direction of the flow is reversed. The simple two-dimensional model is extendable Into a three-dimensional continuous-wave model asserting the many-worlds interpretation of quantum mechanics.

The three-dimensional continuous-wave model of time is meaningful. It makes macro- and micro-scale phenomena describe an ever-changing resilient universe. One day, it will make the present one-shot model of the universe seem as obsolete as the old static model.

Beyond Hawking

Stephen Hawking has been the Albert Einstein of our time. He became recognized as a prominent physicist who popularized the mysteries of the universe. But his thinking resulted, unlike the thoughts of Einstein, neither in mass destruction weapons nor in any products used in our daily life. And the universe seems more mysterious now at his death than when Einstein died. While another such superstar in science is highly unlikely, perhaps instead a skunkworks project team will in near-future get fame from pioneering research at those frontiers on which some light is shed in the following lines.

Albert Einstein

Many physicists believe, like Hawking and Einstein, in the possibility of explaining all the physical forces by means of “a theory of everything”. Considered the holy grail of physics, it would be something to celebrate like the first steps in 1953 on the highest peak of Mount Everest and in 1969 on the surface of the Moon. However, the Nobel laureate Steven Weinberg has expressed a common opinion of the leading authorities: “The more the universe seems comprehensible, the more it also seems pointless”. Einstein had to abandon a static model of the universe and Hawking abandoned a dynamic model with a nice symmetry for an oscillation back and forwards in time. Presently, a pointless universe is proposed as existing either in an infinite series as described by Roger PenrosePaul Steinhardt and others or in a multiverse as described by Andrei Linde.  While time has been described as an illusion in line with the view that the universe seems pointless, a wave interpretation of its flow makes everything less mysterious. Physicists describe elementary charges as having infinite existence if left alone and the same goes for photons. A continuous-wave model is applicable on both these elementary particles so why not try it also on corresponding yet to be discovered elementary particles of gravity? Could that open a door to make all the physical forces described by a unified field theory as sought in vain by Einstein and others? I showed some years ago in an article entitled Beyond Higgs how to motivate an extended use of the Einstein-Planck equation so as to develop a wave theory of time, WTT. It offers a key to interpret the nature of dark matter and of the huge difference in strength between electromagnetic forces and gravitation. And it makes the so-called big number coincidences derivable instead of “explained” by anthropic reasoning.  Like a ring, Einstein’s static model of the universe was limited but endless and definitively not pointless. A rolling ring can be used to describe a continuous wave (CW) that is sinusoidal and that will in the following lines be used to do what has been said to be impossible, namely to describe time itself. In a most simple CW model of a cosmic flow of time tc, a small measure of time dtc varies with regard to a reference measure of time dtr as expressed by the equation dtc/dtr = sin (2π tc/T)  where T is the wave period and dtr is equal to the positive peak value of dtc. If the cosmic expansion with time is chosen to be described by means of a reference flow of time tr as represented by atomic clocks, it shows the positive acceleration expressed by the estimated value of Einstein’s cosmological constant Lambda used today for describing the apparent accelerated expansion of space. That acceleration is predicted by the equality between the ratio dtc/dtr and, as derived from the sine wave function above, the ratio between 2 and the sum of exp(2π tr/T) and exp -(2π tr/T). The cosmic expansion is conceivable, however, without any atom clocks generating the reference time tr. In order to interpret the intrinsic nature of the cosmic expansion, it is, therefore, logical to apply the cosmic flow of time tinstead of the reference time tr making the positive acceleration an illusion and the physical existence of a force counteracting gravity and represented by Lambda simply an unnecessary assumption.  Einstein described the gravitational force as curved spacetime. WTT interprets that curvature as due to the CW nature of time where the period is in the order of 10 exp 20 seconds in the context of a yet to be discovered elementary particle of gravity and 10 exp -20 seconds in the context of the elementary charge controlling the atomic clocks. In a more developed version of WTT, the two-dimensional CW model of time tc becomes a three-dimensional CW model related to the Everett many-worlds interpretation of quantum mechanics asserting the objective reality of the universal wavefunction and challenging the view that the universe is pointless.

A Tribute To Stephen Hawking

He passed away on Pi Day…

Stephen Hawking passed away on March 14th which is dubbed as Pi Day and that happens to be Albert Einstein’s birthday. The International Mathematical Union had recently proposed it to be declared by Unesco as the International Day of Mathematics.

Hawking was born on January 8th, 1942. It happened to be the date of the 300th anniversary of Galileo’s death. Following the footsteps of another genius – Sir Isaac Newton – the chair of Lucasian Professor of Mathematics at the University of Cambridge in England has been held by him. He was offered to be knighted and granted the title of “Sir” but declined the honor. Hawking became a winner of many prizes, among them the 2013 Special Breakthrough Prize in Fundamental Physics set up 2012 to be the most lucrative prize ever established in science, and the Crafoord Prize, astronomy’s equivalent to of the Nobel prize that he did not receive.

Like Ludwig Boltzmann, Hawking wished a tombstone adorned with his most famous equation. In combination with Boltzmann’s constant k, Einstein’s speed of light c, Planck’s constant h, and Newton’s gravitation constant G, it connects the black hole entropy S to the surface area A of the black hole event horizon. This equation between these fundamental quantities in physics is a fine example of mathematical beauty. The supreme example is Euler’s equation between the most fundamental numbers in mathematics: Euler’s number e, pi, and the imaginary unit i.

In Hawking’s famous book A Brief History of Time, he admitted having made a big mistake like Einstein who claimed that the biggest mistake of his life was to introduce the cosmological constant. Hawking’s idea was at first that the universe oscillated with a nice symmetry between expanding and contracting phases. He applied a cosmological model in which the contracting phase looked like the time reverse of the expanding phase. However, his colleagues proved soon to him that disorder would continue to increase during the contraction. Perhaps the last word is not said in this matter. His idea of oscillation with a nice symmetry is – like Einstein’s cosmological constant – worth a second chance in a new cosmological model. Hawking was optimistic: “All we need to do is make sure we keep talking… My goal is simple. It is a complete understanding of the universe, why it is as it is and why it exists at all.”

As a young man thinking about marriage, Hawking was told that he could expect to die in less than five years. All kinds of support and a strong will added more than fifty fruitful years to his life. Considered by many to be the Einstein of our time, a speech synthesizer and a wheel-chair contributed to making him known to the millions.

Hawking leaves three children and two remarried ex-wives. During his funeral on March 31st, I have been writing these lines inspired by the view of Stockholm’s Observatory Hill in front of which I have lived and worked since 1985. To the left, the human curiosity is represented by the Old Observatory built 1753, and to the right, a sculpture erected 1939 shows the centaur Chiron raising himself from the ground. Today, it symbolizes in my eyes the triumph of mind over matter and the now buried man’s outstanding life.

What Existed Before Big Bang?

The topic of this article has interested many thinkers. One of them is the world renowned emeritus professor Stephen Hawking at Cambridge University who claims that the concept of time has no meaning before the beginning of the universe. He states that to ask what existed before the Big Bang is like asking what is South of the South Pole and that the universe has created itself as a one-shot product of quantum fluctuations. According to him, there was nothing before the big bang. However, Sir Roger Penrose, emeritus professor at the University of Oxford, winner of the Copley Medal and the Wolf Prize in Physics which he shared with Stephen Hawking, believes that the Big Bang is one of perhaps an endless succession of Big Bang events where quantum fluctuations in the remote future of a previous cosmic aeon has produced the Big Bang of our aeon.

Along with professor Vahe Gurzadyan of the Yerevan State University in Armenia, Penrose states that images of cosmic microwave background radiation received from NASA show imprints that are older than the Big Bang. According to the new theory of “conformal cyclic cosmology” described in his book Cycles of Time, black holes will eventually consume all the matter in the universe. When they have finished, all that will be left in the universe is energy which will trigger the next Big Bang and the new aeon.

Anyway, Big Bang is merely a name for a theory that describes but does not explain the cosmic expansion, like Dark Energy is a name for a theory describing but not explaining the apparent acceleration of the same expansion. Sir Fred Hoyle coined the term “Big Bang” to ridiculate the idea of a sudden beginning for the universe. He developed a theory of such a continuous creation that the density of matter in the expanding universe remains unchanged. While his “steady state theory” is now abandoned, the present Dark Energy hypotheses build on ideas of continuous creation and of unchanged density. Another version of a steady state theory is proposed by C. J. Masreliez at EST Foundation. According to him, time has no beginning and progresses in discrete increments of the expanding spacetime

An eternal succession of discrete increments of time as proposed by Masreliez or of Big Bang events as proposed by Penrose represents thinking of time as moving only in one direction. It makes me remember the endless sequence of turtles in an old story incorporated in Hawking’s book A Brief History of Time: A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You’re very clever, young man, very clever,” said the old lady. “But it’s turtles all the way down!”

Global sustainability seems to depend on whether we move fast enough from linear to circular thinking. A sustainable model of the universe should be circular as in The Wave Theory of Time (WTT). Predicting the apparent acceleration of the cosmic expansion, WTT was proposed by me for publication in Nature before that discovery in 1998 but became first published by the Swedish Mathematical Society in the autumn of The International Year of Astronomy 2009. It describes the beginning of the cosmic expansion as merely a zero-crossing point in a continuous wave of time. The Planck-Einstein equation is assumed to connect a very low-frequency continuous wave to the rest mass of an elementary particle for gravitation and the rest mass of the elementary charge to a very high-frequency continuous wave. It is the huge ratio between these frequencies that determines the difference in strength between gravity and the electromagnetic force. 

Hawking proposed once that the flow of time changes direction at a future peak of the cosmic expansion. He regretted that proposal later and called it a big blunder. According to WTT, time flows in opposite directions for subsequent half periods of the above-mentioned continuous waves. Our present direction of time is positive by definition so before the Big Bang existed “Negative Time”. That is my short answer!

The Swedish Mathematical Society published in May 2017 an article with equations found in the above-mentioned text published in 2009. The following lines are an extract of that article translated into English with reference made to the graph below. They describe the WTT circular model of the universe and of time flowing in alternating directions.  

It is a generally accepted view that about 14 thousand million years ago the universe had a start 1 of expansion called the Big Bang, that the expansion is related to a variation in the metric of space and has started from nothing, and that this expansion has a positive acceleration at the present point of time 2 as shown by the graph rotated 90 degrees clockwise. Some scientists predict a prolonged phase of accelerated expansion 3 until all the elements of the universe break apart. Other scientists propose that quite soon the positive acceleration could cease and turn negative resulting in a phase of contraction 4.        

The expansion and contraction phases 3 and 4 are marked with arrows between which a mathematical model is shown. The model describes the shift from positive to negative acceleration by means of exponential expressions using the natural base e raised to the power of x and to the power of -x. The contraction results in an end 5 as shown by the graph rotated another 90 degrees clockwise. A hasty interpretation would be that the birth of the universe in the Big Bang is followed by its death with everything disappearing in a Big Crunch.

It is a fact, however, that the above-mentioned mathematical model is derivable from a most fundamental mathematical model of repetitive phenomena, namely the sine wave model. Rotating the graph another 90 degrees clockwise, the shape of the sine wave is shown by the dashed line. If the variation in the metric of space is expressed as a ratio between a measure dtc of cosmic time tc and a measure dtr of reference time tr where dtr is proportional to the second defined by atomic reference clocks and dtc is proportional to the metric of space as determined by the constant speed c of light, then the equation dtc/dtr = sin (2π tc/T) makes x in the above-mentioned exponential expressions equal to 2π tr/T where T is the repetition period of the sine wave. 

In the graph, the end 5 is followed by a new start in a phase that is shifted 180 degrees with respect to the start 1 and that results in new expansion and contraction phases. The two mathematical models have a common peak 6. According to the equation above, the measure dtc alternates between positive and negative values so the cosmic time tc flows in alternating directions inside the dashed line. Curved spacetime describes gravitation and the primary cause behind the curvature can be interpreted by describing simply the cosmic time tc as flowing in a closed loop. Its repetition period T determines the mass of omnipresent elementary particles of gravitation which provide an interpretation of Dark Matter. As to the positive acceleration in the expansion of space, it is here derived from the sine wave model without any assumption of Dark Energy as dominating in the universe and causing a stronger force than gravitation. That assumption can one day be viewed as the cause of the greatest bubble in the history of science. 

A more detailed answer than given above to the question in the headline is that there were Dark Matter and Negative Time before the Big Bang.