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Perspective Chapter: Slowing Down the “Internal Clocks” of Atoms: A Novel Way to Increase Time Resolution in Time-Resolved Experiments through Relativistic Time Dilation

Written By

Hazem Daoud

Reviewed: January 28th, 2022 Published: March 24th, 2022

DOI: 10.5772/intechopen.102931

IntechOpen
Recent Advances in Chemical Kinetics Edited by Muhammad Akhyar Farrukh

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Recent Advances in Chemical Kinetics [Working Title]

Dr. Muhammad Akhyar Farrukh

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Abstract

Traditional time-resolved studies typically rely on a pump laser beam that triggers a reaction dynamic in an atom or molecule and is subsequently probed by a probe pulse of photons, electrons or neutrons. This traditional method is reliant on advancements in creating ever shorter probe and pump pulses. The shorter the pulses the higher is the time resolution. In this chapter we would like to present a novel idea that has the potential to achieve 2–3 orders of magnitude higher time resolutions than is possible with laser and electron compression technology. The proposed novel method is to slow down the ‘internal clock’ of the sample. This can be achieved by accelerating the sample to relativistic speeds, which can be realized in particle accelerators such as cyclotrons and synchrotrons.

Keywords

  • ultrafast science
  • femtosecond
  • attosecond
  • spectroscopy
  • electron diffraction
  • molecular dynamics
  • special relativity

1. Introduction

Until a few decades ago capturing molecular dynamics was in the realm of Gedanken- or thought experiments [1, 2]. Chemists know about the reactants and the final products of chemical reactions, but how the molecules and atoms rearrange themselves to produce the reaction products had always remained in the realm of imagination. This is due to the technical difficulties in making these measurements. In solids, chemical reactions occur at the speed of sound (1000 m/s) and atomic bond lengths are on the order of 1 Å, which means that the time resolution required is on the order of femtoseconds [3, 4].

Recent advances in laser technology have made it possible to produce laser pulses that are femtoseconds and even attoseconds in duration [5]. This has enabled rapid developments in the field of ultrafast science. Typically, a short laser pulse initiates a photo-induced reaction dynamic in a molecular sample, which is then probed by a probe pulse. Probe pulses can be short X-ray pulses [6] in X-ray free electron lasers (XFELs) [7] or compressed electron pulses [8] in tabletop experiments [3, 9]. Additionaly, laser pulses can be used in table-top spectroscopy experiments, which temporally probe molecules and atoms but not with the same spatial resolution as X-rays [10, 11]. XFELs are multibillion dollar facilities that are very costly to operate and entail very complex engineering [12]. However, electron beams are generated in table-top experiments. The electrons are usually accelerated via a DC electric field for a short distance in order to avoid rapid expansion due to Coulomb forces between them [13], or they are accelerated via a DC field, then compressed via an RF field [14, 15]. There are also designs where acceleration and compression take place through the same RF field [16]. Furthermore, utilizing relativistic electron sources can also improve brightness and time resolution since they greatly reduce pulse broadening effects [17, 18, 19]. In all cases a short probe pulse is produced. Probe pulses capture molecular dynamics and produce a diffraction pattern. By varying the time delay between pump and probe pulses, the molecular dynamics at different time points can be captured and a’molecular movie’ can be generated. As a result, the time resolution is mainly limited by the technological ability to produce ever-shorter laser and electron pulses [20], both for triggering a photo-induced dynamic rapidly and for imaging it. In other approaches, the probe pulse is dissected to increase time resolution. In the case of electrons, streak cameras [21] that spatially separate a long electron beam into smaller pieces have been developed, enabling higher time resolutions [22]. Another proposed method, known as optical gating [23], uses ultrashort laser pulses to dissect the electron beam and achieve a higher time resolution than was originally possible based on the length and speed of the beam.

For the sake of making this chapter as self-encompassing as possible, we will start with a review of special relativity (SR) and the concepts of time dilation and length contraction. This shall make for a smoother understanding of the core ideas of the proposed novel experiment.

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2. Special relativity: a quick review

2.1 A brief history

Towards the end of the nineteenth century there was one major inconsistency plaguing the structure of theoretical physics. Newton’s1 equations described very well the mechanics of moving objects ranging from tiny objects on earth to the orbits of planets in space. Maxwell2 had successfully completed the theoretical framework of electromagnetism in 1865, a monumental task that crowned the gradual understanding of electromagnetism throughout the eighteenth and nineteenth centuries through the work of scientists such as Coulomb3, Ampère4 and Faraday5. Despite the enormous success of both Newton’s and Maxwell’s frameworks in describing mechanical objects and light, respectively, they were inconsistent with each other. Maxwell’s equations predicted a constant speed of light and Newton’s equations suggest that objects moving at different speeds should measure the speed of light to be different. On a more rigorous level, Newton’s equations are invariant under Galilean6 transformations while Maxwell’s equations are invariant under Lorentz7 transformations. So, each set of equations suggested a different symmetry present in nature. Moreover, in 1887, on the experimental side, Michelson8 and Morley9 tried to detect a difference in the speed of light for observers moving at different speeds but their results were decidedly negative. There were some attempts to resolve this consistency but it was Einstein,10 who, in 1905, successfully presented the correct solution through a radical theory that would change our understanding of nature forever.

To resolve the inconsistency Einstein suggested that:

  1. The laws of physics are invariant in all inertial frames of reference.

  2. The speed of light in vacuum is constant in all frames of reference.

Simply put, he suggested that Newton’s equations were fundamentally wrong and he replaced them with the equations of SR that were invariant under Lorentz transformations. The implications were dramatic: different observers experience different rates of time, lengths can shrink or elongate and many other peculiar effects take place as objects approach the speed of light. Newton’s equations were only a very good approximation as long as objects moved slowly compared to the speed of light.

2.2 Mathematical framework

There are many ways to approach SR from a mathematical point of view. In this chapter we will present the mathematical framework of SR in a simple manner.

2.2.1 Galilean transformations

Limiting our consideration to one spatial dimension for simplicity, the most general way one can transform between two coordinate systems Oand O, where Ois moving with speed vin the positive x-direction compared to O, is the following:

xt=abefxtE1

Assuming that space is homogeneous and noticing that the principle of relativity requires that Omoves at speed -vcompared to O, we are restricted to linear transformations. A deeper mathematical analysis of this is outside the scope of this chapter. Taking all this into account the forward and backward transformations are calculated to be:

xt=aav1a2avaxtE2
xt=aava21avaxtE3

The Galilean transformations coincide with our everyday intuition (a = 1). Velocities are additive, acceleration is invariant and time is the same for all observers.

x=xvtE4
t=tE5

2.2.2 Lorentz transformations

Going back to the general form of the transformations, we have:

x=axbtE6
x=ax+btE7

By setting x=0, we can calculate the relative velocity of Owith respect to O

V=b/avE8

and similarly, by setting x=0, we calculate

V=b/a=vE9

Assuming the speed of light is constant in all inertial frames of reference, we consider a light signal when the origins coincide (t=t, x=x). The propagation of the light signal in both frames is:

x=ctE10
x=ctE11

Substituting Eqs. (10) and (11) into Eqs. (6) and (7) yields:

ct=acbtE12
ct=ac+btE13

Substituting Eq. (13) into Eq. (12) and using Eq. (8) yields:

a=11v2/c2γE14
b=av=γvE15

Substituting Eqs. (14) and (15) into Eqs. (6) and (7) yields:

x=γxvtE16
x=γx+vtE17

By substituting Eq. (16) into Eq. (17) the transformation of time is obtained:

t=γtvxc2E18
t=γt+vxc2E19

2.2.3 Time dilation

Consider an observer in the frame of reference O at the origin, so x=0with a clock that has a period Δt=t2t1. For the observer in Othat period is much longer, namely

Δt=γΔt.E20

Hence, the clock is ticking slower for the moving observer.

2.2.4 Length contraction

Similar to the peculiar effect of time dilation, a moving observer experiences a contraction in length along the direction of movement. Consider an observer in the frame of reference O that measures a moving ruler to be of length Δx. This measurement happens instantly at one point in time such that t2=t1. This is an important detail as simultaneous times in one frame are not simultaneous in the other one. For the observer in Othat length transforms to

Δx=γΔx.E21

Now, by symmetry, i.e., a ruler at rest in O, that is measured to be of length Loin O, is measured to be of length

Lo=1γΔLoE22

in O. Alternatively, we could have considered an object at rest in Oat the beginning but this would have added an extra step in the derivation as we would have considered the transformations both in space and time. Either way we arrive at the same result; moving objects experience length contraction along the direction of motion.

With that knowledge in mind, we are now ready to discuss the proposed experiment.

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3. The proposed experiment

This section presents a method for studying molecular and atomic dynamics using time-resolved diffraction or spectroscopy studies with greater time resolution without relying on laser or electron beam technology advancements [24]. In this method, instead of shortening the probe pulse, the ‘internal clock’ of the sample (charged molecule or ion) is slowed down. This can be accomplished by accelerating the sample to relativistic speeds, which can be realized in particle accelerators, such as cyclotrons and synchrotrons. A sample, which is accelerated to speed vs, undergoes a slowing down of its ‘internal clock’ by a factor of γ, where

γ=1/1vs2/c2E23

relative to the lab frame irrespective of its velocity direction. As a result, the time resolution becomes a function of the sample’s energy rather than being mainly reliant on pump and probe pulse durations. This can easily enable new time resolutions that have never been unlocked before.

3.1 Experimental considerations

To successfully implement any novel method, there are several barriers and challenges to overcome. We will introduce the experimental setup and discuss the experimental challenges and limitations, as well as the physics involved in the proposed setup.

3.1.1 Setup

We propose accelerating the samples in a cyclotron or synchrotron and studying them at a fixed energy Eand, consequently, at a fixed speed vs, which remains constant during data collection. Figure 1 shows a schematic of the experimental setup. During the experiment, samples are accelerated into a chamber in which a pump pulse is directed parallel to a probe pulse and perpendicular to the direction of the sample’s motion. The delay between the pump pulse and probe pulse can be controlled by changing the distance Lbetween the two beams. The resulting time delay τdaccording to the sample’s clock is

Figure 1.

Schematic of the proposed experimental setup: Samples are accelerated to a fixed energy. A pump pulse and a probe pulse are directed parallel to each other and perpendicular to the sample direction of motion. The delay between the pump pulse and probe pulse can be controlled by changing the distance between the two beams.

τd=Lvsγ.E24

As a result, the signal will reflect the changing dynamics according to the ‘internal clock’ of the sample.

3.1.2 Sample suitability

To begin with, this novel method can only be applied to electrically charged ions or molecules so that they can be accelerated to relativistic speeds, and to molecules that are in the gas phase in order to achieve the required energies. To run the experiment for a given time resolution, the higher the mass of the sample, the more energy is needed, so the best candidates for such studies are light, charged molecules, and ions.

The typical number density in gas phase UED [25, 26, 27] and spectroscopy [28] experiments is 1015cm3. Proton bunches at the LHC contain 1.15×1011protons with a proton number density of 1016cm3[29, 30]. There have been many schemes to reduce bunch length, and in fact an order of magnitude shorter bunch length has been produced for the purpose of accelerating electrons with plasma wakefields of proton bunches [31].

It is also necessary for a sample to be stable when subjected to the accelerator conditions. H-anions with a binding energy of 0.75 eV are accelerated regularly to 520 MeV at TRIUMF [32]. Molecules with covalent bonds typically have binding energies of 1 eV or higher. The typical length scale of a covalent bond is 1 Å. Hence, to break a bond, a force of 1600 pN is needed. The forces exerted by typical electric fields (<10 MV/m) and typical magnetic fields (<10 T) are orders of magnitude less than 1600 pN. Furthermore, second ionization energies for atomic ions are typically significantly higher than 1 eV.

In addition to originally being designed to accelerate only protons and positively charged ions, the LHC ring has recently been used to accelerate partially stripped Pb+81ions with one electron to an energy of 6.5 Q TeV [33] as part of the gamma factory proposal [34] to create a new type of high intensity light source. Although originally designed to accelerate protons or positively charged ions only, the LHC ring has recently accelerated partially stripped Pb+81ions with one electron to an energy of 6.5 Q TeV [33], where Qis the ion charge number, as part of the gamma factory proposal [34] to create a new type of high intensity light source. Currently, engineering challenges with regard to collimation are being addressed [35].

3.2 Theoretical considerations

3.2.1 Energy considerations

At the moment, the Large Hadron Collider at CERN can accelerate protons to energies on the order of 7 TeV [36] and lead ions to 5 TeV collision energies [37], which is enough to boost the resolution of time measurements significantly. As an example, a hydronium molecule (H3O+, rest mass: 3.16×1026kg) accelerated to an energy of 1.8TeV would experience a slowing down of time with a γfactor of 100. Since

E=γmc2,E25

the time resolution scales proportionally to energy, so an energy of 18TeV would result in an astonishing γ=1000. Additionally, time resolution is inversely proportional to mass, so hydrogen ions, for example, would experience an order of magnitude more gain in time resolution than hydronium molecules with the same energy.

The effect of relativistic time dilation on dynamical processes will still be extremely fascinating to observe, even if the particle accelerators cannot be commissioned to perform this experiment in the near future. With current laser technology it is possible to observe changes in differential detection, with and without a perturbation, as small as 104to 108[38, 39] using standard modulation techniques and photon detectors. There has also been major advances in laser based particle accelerators up to field gradients as high as 100GeV/m [40, 41, 42, 43] that will soon enable particle kinetic energies up to 10GeV range or higher. As compared with particle accelerators, this level of relativistic energy would result in only modest time dilation. It would nonetheless constitute a direct measurement of time retardation, which would prove to be an important test case for the development of laser-based particle acceleration with the goal of controlling the time variable directly, asymptotically approaching’stopping’ time. A control of the time variable could open up new avenues, beyond simple imaging, to driving dynamics that are otherwise too rapid to control.

3.2.2 Pump and probe beam dynamics

As in standard ultrafast studies, pulsed pump and probe beams can be used. The time resolution is largely determined by the pulse duration of the pump and probe pulses. In conventional terms, the pulse duration refers to the time during which the full width at half maximum (FWHM) of the pulse crosses the sample. The pump pulse determines the trigger speed, and the probe pulse determines the imaging time resolution.

Besides velocity mismatch [44, 45], which takes place due to the difference in velocity between pump and probe pulses and their different incidence angles, other factors that affect the time resolution are the time of arrival jitter [25] for RF accelerated electron pulses. By the very nature of the experimental geometry, however, a lower resolution due to velocity mismatch or time of arrival jitter is avoided, as both pulses are parallel, so the delay time from time zero is solely determined by the speed of the sample between the pump and probe pulses. According to our proposed setup, the pulse crosses the sample in two directions, and we will thus consider the pulse duration during which the pulse crosses the sample or vice versa, in both directions. To explain the physics we denote the direction, parallel to the direction of propagation of the sample beam, yand denote the perpendicular direction x. For clarification, we will treat the problem from the lab frame of reference as well as from the sample frame of reference.

  1. Lab frame of reference.The pulse duration in the x-direction τxis given by

    τx=lxvp,E26

    where lxindicates the pulse length in the x-direction and vpindicates the speed of the pump/probe pulse. The pulse duration in the y-direction τyis given by

    τy=lyvs,E27

    where lyindicates the pulse length in the y-direction and vsindicates the speed of the sample. Vswill always be close to the speed of light cin the proposed experiment.

  2. Sample frame of reference.With relativistic speeds approaching the speed of light, the effect of length contraction along the direction of sample propagation ybecomes significant. Hence, the length of the pulse in the y-direction is contracted to

ly=lyγ.E28

The pulse duration in the y-direction τyis then given by

τy=lyγvs.E29

The pulse duration in the x-direction τxis given by

τx=γlxvpE30

as vptransforms to

vp=vpγE31

in the sample frame of reference. This happens because of relativistic angle aberration. In the lab frame of reference a pulse that is emitted at 90does not hit the sample perpendicularly in the sample frame of reference. The closer vsis to cthe smaller is the incidence angle between the beam and the sample’s line of motion in the sample frame of reference. When discussing the observable signal relativistic angular aberration will be discussed in more detail.

However, as a quick check, if we assume that the probe beam consists of photons, then vp=c. Utilizing Eq. (41), the longitudinal component of velocity in the sample frame of reference is given by ccosθi=ccosθivc1cosθivc, where θiis the angle in the lab frame of reference and vis the speed of the sample in the lab frame of reference. Plugging in θi=π2, we end up with vas the longitudinal component. The total speed of the photon probe pulse is then given by cγ2+v2=c21v2c2+v2=c, which is expected as the speed of light is constant in all frames of reference.

Typical pulses are Gaussian temporally and spatially (τxτy) and so the time resolution τreswould be determined by the relativistically shortened duration τy. It would thus be given by

τres=1γτpump2+τprobe2,E32

where τpumpand τprobeare the transit time durations of the sample beam through the pump and probe pulses in the lab frame, respectively. As an example for pump and probe beams with ly=10μm and γ=100, the time resolution would be roughly 470 as. Figure 2 shows a visual representation from both frames of reference along the relevant direction y.

Figure 2.

Lab and sample frames of reference: Along they-direction the length of the pulse is contracted in the sample frame of reference relative to the lab frame of reference.

The lab frame of reference is the same as the sample frame of reference in conventional pump-probe experiments. Signals (e.g., diffraction patterns) always reflect interaction time according to the clock rate of the sample, so our proposed setup exploits the involved relativistic effects that result from the differences between two frames of reference.

3.2.3 Doppler effect and frequency shifts

In order to properly conduct the experiment, it is imperative to understand how the frequency of a laser, x-ray or electron pulse is ‘seen’ by the sample in its own rest frame. Changing the frequency of a laser can cause it to be outside the absorption spectrum of the sample, preventing the intended interaction. The spatial resolution of scattering x-rays and electrons would decrease if they undergo significant redshifts, for example. Furthermore, there is the relativistic effect of time dilation in addition to the classical Doppler effect. Even if the source and receiver are not crossing paths, relativity dictates a frequency shift known as the transverse Doppler effect [46].

If we let θbe the angle between the sample wave vector and the wave vector of the pump/probe particles, as measured in the lab frame of reference, then the frequency that the sample ‘sees’, fs, is given in terms of the frequency in the lab frame of reference, fl, by

fs=flγ1βcosθE33

for photons (see derivation in Appendix A) [47]. For other particles, e.g., electrons, one needs to replace βwith

βe=vsve,E34

where veis the speed of the particles but γ=1/1vs2/c2remains the same. In our proposed setup, where the pump and probe beams are perpendicular to the direction of motion of the samples, we have

fs=flγ.E35

Depending on the angle, there could either be a redshift or a blueshift. For one critical angle θcthere is no frequency shift. For β(βe) 1, as γbecomes larger, θcbecomes smaller, meaning that the two wave vectors are more collinear. Although this would eliminate frequency shifts entirely, it would decrease the time resolution significantly.

3.2.4 The observable signal

The interaction between light and matter can take many different forms (e.g., absorption, scattering, etc.). In this section we present a general scheme for calculating the final observable signal in our proposed experiment.

The main steps are: (1) transforming the incident field from the lab frame of reference to the sample frame of reference by applying the Lorentz transformations; (2) calculating the resultant signal in the sample frame of reference; (3) transforming the signal from the sample frame of reference to the lab frame of reference by applying the Lorentz transformations one more time.

Without loss of generality, for an incident electric field Eiwith angular frequency ωipolarized in the z-direction and incident at angle θi(angle between photon wave vector kiand sample velocity vector vs), the field would be Lorentz transformed to the sample frame of reference to Ei'in the following way: [48, 49].

Ei=ẑEiexpikicosθiy+sinθixexpiωit,E36
Ei=ẑEiexpikicosθiy+sinθixexpiωit,E37

where

Ei=γ1βcosθiEi,E38
ki=ωic,E39
ωi=γ1βcosθiωi,E40
θi=cos1cosθiβ1cosθiβ.E41

For particles other than photons moving with speed vethe angular frequency and angle transform in the following way:

ωi=γ1βecosθiωi,E42
θi=tan1sinθiγcosθiβe.E43

However, as before γ=1/1β2with β=vscremains the same.

Due to the nature of the angular transformations (see derivation in Appendix B), we expect scattering and diffraction angles to be wider than for the static case, and hence, we recommend detectors that cover as much of the 4πsr solid angle as possible.

Momentum transfer due to light radiation pressure would have a more negligible effect on changing the ion beam path than in conventional ultrafast gas electron diffraction. Forces perpendicular to the ion beam path would cause acceleration according to atransverse=Ftransverseγm. Forces along the ion beam path would cause acceleration according to alongitudinal=Flongitudinalγ3m. Hence, acceleration due to transverse forces is reduced by a factor of 1γand acceleration due to longitudinal forces is reduced by a factor of 1γ3as compared to conventional non-relativistic ultrafast gas electron diffraction. This is due to the well-known concept of transverse and longitudinal masses in special relativity.

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4. Conclusions

In this chapter, we proposed a novel method for time-resolved studies that relies on taking advantage of relativistic effects rather than on advances in laser technology or electron beam compression. It was shown that by using currently available technology, this method could improve time resolution by 2or 3orders of magnitude. This has the potential of opening up a whole new domain of ultrafast dynamics that was previously unattainable. With ever more powerful accelerators being proposed, there will be great potential to achieve truly remarkable time resolutions that far exceed the status quo. We hope this would lead to new avenues of collaboration between the particle physics community and the ultrafast science community, which will maximize the research potential of particle accelerator facilities worldwide.

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Acknowledgments

Hazem Daoud would like to thank Prof. R.J. Dwayne Miller for his support, and Prof. Pierre Savaria and Prof. David Bailey for fruitful discussions.

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We will derive the relativistic frequency shift formula in two steps. First we will consider a classical example to derive the classical Doppler shift then we will consider the relativistic effects involved in the second step.

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A.1 Classical Doppler shift

Consider a resting source emitting a photon with frequency f0and period T0collinear with an observer moving with speed uin the same direction as the emitted photon. We denote the frequency observed as fS. The observed wavelength λSis given by

λS=cuT0.E44

Hence,

cfS=cuf0,E45

and so,

f0=1βfS,E46

where β=uc.

A.2 Relativistic Doppler shift

  1. The time interval between consecutive crests in the source frame of reference is T0.

  2. Due to relativistic time dilation, the time interval between consecutive crests in the observer frame of reference is γT0.

  3. The time interval between the reception of consecutive crests in the source frame of reference is T01βcosθ, where θis the angle between the observer’s wave vector and the wave vector of the photon.

  4. The time interval between the reception of consecutive crests in the observer frame of reference is γT01βcosθ. A calculation analogous to the classical Doppler shift yields f0=γ1βcosθfS

Consider a probe beam emitted from a source at an angle θand speed uwith respect to the wave vector of an observer moving with speed vin the frame of reference of the source. Considering a 2-d plane containing the wave vectors of the probe beam and the observer, we can define the components as follows:

ux=ucosθE47
uy=usinθE48

In the observer frame of reference (indicated with a prime), using the relativistic velocity transformations, yields:

ux=ucosθv1vc2ucosθE49
uy=1v2c2usinθ1vc2ucosθE50

Hence,

tanθ=uxuy=1v2c2usinθucosθv.E51

The probe beam speed in the observer frame of reference is

u=ux2+uy2=u2u2v2sinθ2c22ucosθv+v21/21vc2ucosθ.ωE52

So,

cosθ=uxu=ucosθvu2u2v2sinθ2c22ucosθv+v21/2E53

If the probe beam consists of photons, then u=cand this simplifies further to

cosθ=cosθvc1vccosθ.E54

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Notes

  • Sir Isaac Newton: English mathematician, physicist, engineer, philosopher, astronomer, theologian and author (1642–1726).
  • James Clerk Maxwell: Scottish mathematician and physicist (1831–1879).
  • Charles-Augustin de Coulomb: French engineer and physicist (1736–1806).
  • André-Marie Ampère: French physicist and mathematician (1775–1836).
  • Michael Faraday: English physicist (1791–1867).
  • Galileo Galilei: Italian astronomer, physicist, engineer, philosopher, and mathematician (1564–1642).
  • Hendrik Antoon Lorentz: Dutch physicist (1853–1928).
  • Albert Abraham Michelson: German-born American physicist (1852–1931).
  • Edward Williams Morley: American physicist (1838–1923).
  • Albert Einstein: German physicist (1862–1943).

Written By

Hazem Daoud

Reviewed: January 28th, 2022 Published: March 24th, 2022