- The author considers a Klein-Gordon scalar field in a two-dimensional Rindler space-time background.
- They define a two-dimensional action for the scalar field and derive from it the Klein-Gordon equation of motion in this background.
- The equation can be solved exactly using imaginary time. The solution is oscillatory with an angular frequency that corresponds to an integral number, suggesting quantization of the scalar field frequencies in this space-time.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
Methods to determine pressure drop in an evaporator or a condenserTony Yen
This articles aims to explain how one can relatively easily calculate the pressure drop within a condenser or an evaporator, where two-phase flow occurs and the Navier-Stokes equation becomes very tedious.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
Methods to determine pressure drop in an evaporator or a condenserTony Yen
This articles aims to explain how one can relatively easily calculate the pressure drop within a condenser or an evaporator, where two-phase flow occurs and the Navier-Stokes equation becomes very tedious.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...
The klein gordon field in two-dimensional rindler space-time 200920ver-display
1. The Klein-Gordon field in two-dimensional Rindler space-time
Ferdinand Joseph P. Roa
Independent physics researcher
rogueliknayan@yahoo.com
Abstract
The Klein-Gordon scalar in the background of two-dimensional Rindler space-time is considered in this exercise. In an
informal way without resorting to methods of dimensional reduction, a two-dimensional action for the Klein-Gordon
scalar is written with the said background and obtaining from this action the equation of motion for the scalar field. The
equation of motion is solvable exactly in this two-dimensional space-time using imaginary time. In imaginary time, the
solution is oscillatory with a given frequency that corresponds to an integral number.
Keywords: Coordinate Singularity, Series Solution
1 Introduction
The Schwarzschild metric[1]
𝑑𝑆2
= − 𝜂𝑑𝑡2
+ 𝜀 𝑑𝑟2
+ 𝑟2 (𝑑𝜃2
+ 𝑠𝑖𝑛2
𝜃 𝑑𝜙2 )
𝜂 = 𝜀−1
= 1 −
2𝐺𝑀 𝑞
𝑟
(1)
as expressed in the standard coordinates has a (coordinate) singularity[2] at 𝑟 𝐻 = 2𝐺𝑀 𝑞. (Note that in this
entire document we take the square of the speed of light as unity, 𝑐2
= 1 and whenever consistency in
units needed, we insert its appropriate value.) This can be seen crudely from the fact that 𝜀 = ∞ at 𝑟 =
𝑟 𝐻. This coordinate value of 𝑟 at which one piece of the metric is singular defines a horizon[2, 3] that puts
bounds to (1), confining it in a portion of space-time where this metric in that form is sensible. That is, in
rough language say (1) is for all those regions of space-time where 𝑟 > 𝑟 𝐻 and dipping below 𝑟 𝐻 can no
longer be covered by the given metric as expressed in that form.
Figure 1: This is the space-time graph on rt-plane.
𝑑𝑡
𝑑𝑟
= ±
1
1 −
2𝐺𝑀 𝑞
𝑟
(2)
On a space-time graph where one can draw a light-cone bounded by the intersecting lines whose
slopes are given by (2), it can be superficially shown that the region at 𝑟 > 𝑟 𝐻 is not causally connected to
that at 𝑟 < 𝑟 𝐻 . This is so since asymptotically the light-cone closes as 𝑟 𝐻 is approached from the right. As
the light-cone closes there can be no way of connecting a time-like particle’s past to its supposed future
along a time-like path that is enclosed by the light-cone. So any coordinate observer won’t be able to
construct a causal connection between the past and the future for a time-like particle falling into that region
𝑟 < 𝑟 𝐻.
However, such singularity is only a coordinate one specific to the form (1) since expressing the
same metric in suitable coordinates will remove the said coordinate singularity.
For example, from the standard coordinates (𝑡, 𝑟, 𝜃, 𝜙) we can change (1) into
𝑑𝑆2
= − 𝜂𝑑𝑢̃2
+ 2𝑑𝑢̃𝑑𝑟 + 𝑟2 (𝑑𝜃2
+ 𝑠𝑖𝑛2
𝜃 𝑑𝜙2 ) (3)
using the Eddington-Finkelstein coordinate
𝑢̃ = 𝑡 + 𝑟 ∗ (4)
with the Regge-Wheeler coordinate
2. 𝑟 ∗ = 𝑟 + 2𝐺𝑀 𝑞 𝑙𝑛 (
𝑟
2𝐺𝑀 𝑞
− 1)
(5)
Noticeable in (3) is that none of the metric components goes infinite at 𝑟 𝐻 so the singularity
( 𝜀 = ∞ at 𝑟 = 𝑟 𝐻) in (1) is not a case in (3).
In this paper, we deal with the Klein- Gordon scalar as dipped very near the horizon but not
having completely fallen into those regions at 𝑟 < 𝑟 𝐻. We consider that near the horizon we can make the
substitution[2]
𝑥2
8𝐺𝑀 𝑞
= 𝑟 − 2𝐺𝑀 𝑞
(6)
Hence, in approximate form we write (1) as
𝑑𝑆2
≈ −(𝜅𝑥)2
𝑑𝑡2
+ 𝑑𝑥2
+ 𝑟𝜅
2
𝑑Ω2
(7)
𝜅 =
1
4𝐺𝑀 𝑞
where 𝑟𝜅 = 1/2 𝜅 is the approximate radius of a two-sphere 𝑆2
: 𝑟 𝜅
2 𝑑Ω2
and we think of the (3+1)-
dimensional space-time ascribed to metric (7) as a product of a two-dimensional Rindler space-time
𝑑𝑆(𝑅)
2
≈ −(𝜅𝑥)2
𝑑𝑡2
+ 𝑑𝑥2
(8)
and that of the two-sphere. We give to this two dimensional Rindler space-time the set of coordinates
𝑥 𝜇
= {𝑥0
= 𝑡, 𝑥1
= 𝑥} (9.1)
with t as the real time to be transcribed into an imaginary time by
𝑡 → 𝜏 = −𝑖𝑡 (9.2)
2 Two-dimensional action
In this Rindler space-time we just write a two dimensional action for our scalar field
𝑆 𝐶 = ∫ 𝑑2
𝑥 √−𝑔 (
1
2
𝑔 𝜇𝜔
(𝜕 𝜇 𝜑𝑐)(𝜕 𝜔 𝜑𝑐) +
1
2
𝑀2
𝜑 𝐶
2
) (10)
The metric components in this action are those belonging to the two-dimensional Rindler space-time given
by metric form (8). This is rather an informal way without having to derive it from an original 3 + 1
dimensional version that would result into (10) through the process of a dimensional reduction[4] with the
Rindler space-time as background. Anyway, on the way the solution exists for the resulting equation of
motion to be obtained from the given action and this solution is oscillatory as taken in the imaginary time.
Taking the variation of (10) in terms of the variation of our classical scalar 𝜑 𝑐would yield the
equation of motion
1
√−𝑔
𝜕𝜇(√−𝑔 𝑔 𝜇𝜔
𝜕 𝜔 𝜑𝑐) − 𝑀2
𝜑 𝑐 = 0
(11.1)
or with (8) as the said background we have explicitly
𝜕1
2
𝜑𝑐 +
1
𝑥
𝜕1 𝜑𝑐 −
1
𝜅2 𝑥2
𝜕0
2
𝜑𝑐 = 𝑀2
𝜑 𝑐
(11.2)
We take that the solution is variable separable, 𝜑𝑐 = 𝜒(𝑥)𝑇(𝑡)so that (11.2) could be written into
two independent equations
1
𝜒
(𝜕1
2
𝜒 +
1
𝑥
𝜕1 𝜒) −
1
𝜅2 𝑥2
𝜇 𝐸
2
= 𝑀2
(12.1)
and
1
𝑇
𝜕0
2
𝑇 = −
1
𝑇
𝜕𝜏
2
𝑇 = 𝜇 𝐸
2
𝜇 𝐸
2
> 0 (12.2)
Later, the constant 𝜇 𝐸 is to be identified as the angular frequency 𝜔 in the imaginary time (9.2).
3. 3 The solution
The differential equation (12.1) is satisfied by a series solution of the following form
𝜒(𝑚) =
1
√ 𝑥
∑
1
𝑥 𝑛
(𝑎 𝑛 exp(𝑀𝑥) + 𝑏 𝑛 exp(−𝑀𝑥) )
𝑚
𝑛=0
(13)
This series solution corresponds to an integral number 𝑚 and the series stops at the 𝑚𝑡ℎ term. (As
a cautionary let us not confuse 𝑚 with 𝑀. The latter is the mass of our scalar field in units of per length.)
The (𝑚 + 1)𝑡ℎ term and all other higher terms vanish as the 𝑎 𝑚+1 and 𝑏 𝑚+1 coefficients are terminated.
That is, 𝑎 𝑚+1 = 𝑏 𝑚+1 = 0. Each coefficient 𝑎 𝑛 is given by this recursion formula
𝑎 𝑛 =
(2𝑛 − 1)2
− (2𝜅−1
𝜇 𝐸)2
8𝑀𝑛
𝑎 𝑛−1
(14.1)
and each 𝑏 𝑛 by
𝑏 𝑛 = −
(2𝑛 − 1)2
− (2𝜅−1
𝜇 𝐸)2
8𝑀𝑛
𝑏 𝑛−1
(14.2)
These formulas are defined for all 𝑛 ≥ 1 and with these the vanishing of those 𝑚 + 1 coefficients would
imply that
𝜇 𝐸(𝑙) = (𝑙 +
1
2
)𝜅 (14.3)
𝑙 = 0, 1, 2, 3, … , 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 (14.4)
𝜅 = (4𝐺𝑀 𝑞)
−1
(14.5)
(There in (14.3) we have relabeled 𝑚 as 𝑙 and this includes zero as one of its parameter values.
With the inclusion of zero, the lowest vanishing (𝑚 + 1) coefficients given 𝑚 = 0 would be 𝑎1 = 0 and
𝑏1 = 0 so that all other higher terms with their corresponding coefficients vanish. Then in this particular
case, the series solution only has the 0th terms with their coefficients 𝑎0 and 𝑏0 that are non-zero.)
Given (14.3), we can write the differential equation (12.1) as
𝜕1
2
𝜒(𝑙) +
1
𝑥
𝜕1 𝜒(𝑙) −
1
4𝑥2
(2𝑙 + 1)2
𝜒(𝑙) = 𝑀2
𝜒(𝑙)
(14.6)
whose solution is in the series form (13).
Put simply, the differential equation (12.2) has as solution the following function of the imaginary
time 𝜏
𝑇(𝜏) = 𝐴𝑐𝑜𝑠𝜔(𝑙) 𝜏 + 𝐵𝑠𝑖𝑛𝜔(𝑙) 𝜏
(14.7)
Here we have identified the separation constant 𝜇 𝐸 as the angular frequency 𝜔 in the solution above. That
is, 𝜇 𝐸(𝑙) = 𝜔(𝑙), and given (14.3), we find that this oscillatory solution has an angular frequency that
corresponds to an integral number 𝑚.
We can choose to set 𝑎0 = 𝑏0 so that 𝑏 𝑛 = (−1) 𝑛
𝑎 𝑛 and at 𝑚 = 0 , implying 𝑎1 = 0 and
𝑏1 = 0, we have
𝜒(0) =
2𝑎0 𝑐𝑜𝑠ℎ𝑀𝑥
√ 𝑥
(15.1)
with 𝜔(0) = 𝜅/2 .
Going back to the recursion relations (14.1) and (14.2), we must take note that these fail when the scalar
field is massless since these are singular at 𝑀 = 0. In the massless case we may consider 𝑎0 = 𝑏0, so as a
consequence, 𝑏 𝑛 = (−1) 𝑛
𝑎 𝑛 and
𝜒(𝑚) =
2𝑎0
√ 𝑥
+
2
√ 𝑥
∑
𝑎 𝑛
𝑥 𝑛
𝑚
𝑛 =2
(15.2)
So in the massless case we apply (14.6) with 𝑀 = 0 to (15.2) to get a form of constraint on the coefficients
𝑎 𝑛 and this is given by
∑ [2 (𝑛 +
1
2
)
2
− 2 (𝑚 +
1
2
)
2
]
𝑚
𝑛 =0
𝑎 𝑛 𝑥−𝑛−5/2
= 0
(15.3)
4. From this constraint we can choose only one coefficient 𝑎𝑙 to be non-zero, while all other coefficients
𝑎 𝑛 ≠ 𝑙 to be zero. Example, if we choose 𝑎0 to be the only non-zero, then 𝑚 = 0, and all other coefficients
𝑎 𝑛 ≠ 0 to be zero. For every non-zero coefficient 𝑎𝑙, there corresponds an angular frequency in the form of
(14.3). Eventually then, following this condition in the constraint, series (15.2) would only be made up of a
term with the non-zero coefficient,
𝜒(𝑚) =
2𝑎 𝑚
√ 𝑥
1
𝑥 𝑚
(15.4)
This satisfies the differential equation in the same form (14.6) for a given integral value of 𝑚 with 𝑀 = 0.
4 Conclusions
Taking the Klein-Gordon field as a classical scalar, we have shown that its two-dimensional equation of
motion in Rindler space-time has a series solution that can terminate at a certain term. As a consequence of
this termination the angular frequency (14.3) with the identification 𝜇 𝐸(𝑙) = 𝜔(𝑙), given (9.2) seems to
have values that correspond to integral values of 𝑙. This is seemingly suggestive that the classical scalar
field can already appear quantized in terms of its angular frequency or can have a spatial mode given by
(13) that corresponds to an integral value of 𝑚. We also have a curious result as manifest in (14.3) that in
the imaginary time, the scalar field can oscillate at frequencies that are odd multiples of the surface gravity
over four pi, 𝑓(𝑙) = (2𝑙 + 1)𝜅/4𝜋, where surface gravity is 𝜅 = 𝑐3
/4𝐺𝑀 𝑞.
Some details of spatial solution
Two-dimensional scalar action defined in the background of two-dimensional Rindler space-time
We have arbitrarily defined a two-dimensional scalar action (10), given the fundamental line element (8) of
the two-dimensional Rindler space-time. This action we explicitly write as
𝑆 𝐶 = ∫ 𝑑𝑡 ∫ 𝑑𝑥
1
2
(−
1
𝜅𝑥
(𝜕𝑡 𝜑𝑐)2
+ 𝜅𝑥 (𝜕 𝑥 𝜑𝑐)2
+ 𝑀2
𝜅𝑥 𝜑 𝐶
2
)
(16)
As earlier stated this Rindler space-time is endowed with a set of coordinates (9.1). Thus, the classical
scalar can have two degrees of freedom and its motion is along 𝑥0
= 𝑡 (the time direction) and one spatial
direction 𝑥1
= 𝑥.
We will no longer (or perhaps in the much later portion of this draft) present here the details of varying this
action in terms of the variation of the classical scalar to arrive at the equation of motion (11.1) or as
explicitly given by (11.2).
The spatial series solution
In tackling the differential equation (11.2), we assumed a variable separable solution in the form of
𝜑 𝑐 = 𝜒(𝑥)𝑇(𝑡) so decomposing (11.2) into equations (12.1) and (12.2). Most of our effort here is to work
on the details involved in solving (12.1).
Given the variable separable solution, we were able to write the spatial part of (11.2) as (12.1) and re-
writing that here as
𝑥2
𝜕1
2
𝜒 + 𝑥 𝜕1 𝜒 − (𝜇 𝐸 𝜅−1)2
𝜒 = 𝑀2
𝑥2
𝜒
(17.1)
In considering the solution in series form as given by (13), we have actually decomposed 𝜒 into two
separate components
𝜒(𝑚) = 𝜒1 + 𝜒2 = 𝜒1(𝑚) + 𝜒2(𝑚)
(17.2)
(Again, we must note that 𝑚 here is an integral number and must not be confused with mass 𝑀 of the
scalar field.) We plug (17.2) into (17.1) and collect like terms we then write two separate differential
equations from (17.1).
𝑥2
𝜕1
2
𝜒1 + 𝑥 𝜕1 𝜒1 − (𝜇 𝐸 𝜅−1)2
𝜒1 = 𝑀2
𝑥2
𝜒1
𝜒1 = 𝜒1(𝑚)
(17.3)
5. where
𝜒1 = 𝜒1(𝑚) = 𝑄1(𝑚)(𝑥) 𝑒 𝑀𝑥
=
1
√ 𝑥
∑
1
𝑥 𝑛
𝑎 𝑛 exp(𝑀𝑥)
𝑚
𝑛=0
(17.4)
while the other part is given by
𝑥2
𝜕1
2
𝜒2 + 𝑥 𝜕1 𝜒2 − (𝜇 𝐸 𝜅−1)2
𝜒2 = 𝑀2
𝑥2
𝜒2
𝜒2 = 𝜒2(𝑚)
(17.5)
to which belongs the other solution
𝜒2 = 𝜒2(𝑚) = 𝑄2(𝑚)(𝑥) 𝑒− 𝑀𝑥
=
1
√ 𝑥
∑
1
𝑥 𝑛
𝑏 𝑛 exp(−𝑀𝑥)
𝑚
𝑛=0
(17.6)
We proceed from (17.3), given (17.4) to get the following differential equation for 𝑄1(𝑚).
𝑥2
𝑄 ′′1(𝑚) + 2𝑀𝑥2
𝑄 ′1(𝑚) + 𝑥𝑄 ′1(𝑚) + 𝑀𝑥𝑄1(𝑚) − (𝜇 𝐸 𝜅−1)2
𝑄1(𝑚) = 0
𝜇 𝐸𝐹 = 𝜇 𝐸 𝜅−1
(17.7)
The form of 𝑄1(𝑚) is already evident in (17.4) and substituting this solution in (17.3) would give us the
recurrence relations between the coefficients 𝑎 𝑛.
Proceeding, we write
∑ [(𝑛 +
1
2
)
2
− 𝜇 𝐸𝐹
2
] 𝑎 𝑛 𝑥−(𝑛+ 1/2)
𝑚
𝑛=0
− ∑ 2𝑀𝑛
𝑚
𝑛 = 0
𝑎 𝑛 𝑥
−(𝑛−
1
2
)
= 0
(17.8)
In the first major summation, we make the shift 𝑛 → (𝑛 − 1) so that (17.8) can be re-written into the
following form
∑ [((𝑛 −
1
2
)
2
− 𝜇 𝐸𝐹
2
) 𝑎 𝑛−1 − 2𝑀𝑛𝑎 𝑛] 𝑥−(𝑛 − 1/2)
𝑚
𝑛=1
= 0
(17.9)
For this to be satisfied for all coefficients we must have the recurrence relations between adjacent
successive coefficients and such relations are already given by (14.1). The series solution (17.4) can be
terminated so that it will consist only of series of terms up to the 𝑙𝑡ℎ place with 𝑚 relabeled as 𝑙. This can
be done by setting the 𝑎 𝑛 = 𝑙 +1 coefficients to zero.
𝑎 𝑛 = 𝑙 +1 =
(2𝑙 + 1)2
− (2𝜅−1
𝜇 𝐸)2
8𝑀(𝑙 + 1)
𝑎𝑙 = 0
(17.10)
As earlier stated in the conclusion the consequence of having a terminated series solution is that the
frequencies are odd multiples of the surface gravity over four pi
𝑓(𝑙) = (2𝑙 + 1)
𝜅
4𝜋
(17.11)
It should already be clear from (12.2) that 𝜇 𝐸 is the angular frequency 𝜔(𝑙) in the imaginary time.
Given the vanishing coefficient (17.10), we can slightly re-write (17.7) as
𝑥2
𝑄 ′′1(𝑙) + 2𝑀𝑥2
𝑄 ′1(𝑙) + 𝑥𝑄 ′1(𝑙) + 𝑀𝑥𝑄1(𝑙) =
(2𝑙 + 1)2
4
𝑄1(𝑙)
(17.12)
with
7. 𝑥2
𝑄1(2)
′
= −
1
2
𝑎 0√ 𝑥 +
9
2𝑀
𝑎 0
√ 𝑥
−
15
2𝑀2
𝑎 0
√𝑥3
(20.4)
𝑥2
𝑄1(2)
′′
=
3
4
𝑎 0
√ 𝑥
−
45
4𝑀
𝑎 0
√𝑥3
+
105
4𝑀2
𝑎 0
√𝑥5
(20.5)
For this case the differential equation to be satisfied is given by
𝑥2
𝑄 ′′1(2) + 2𝑀𝑥2
𝑄 ′1(2) + 𝑥𝑄 ′1(2) + 𝑀𝑥𝑄1(2) =
25
4
𝑄1(2)
(20.6)
So far our tests for 𝑄1(𝑙) in the integral values 𝑙 = 0, 1, 2 prove that (17.12) holds true for all these cited
positive integral values of 𝑙. For the moment let us assert that these tests are sufficient enough to claim that
(17.12) is satisfied in all 𝑙 ∈ ℤ ∶ 𝑙 > 0 .
Going much further we will now deal with the other part of the solution that is given by (17.6) that must
satisfy (17.5) to yield the differential equation for 𝑄2(𝑚). Upon substitution of (17.6) into (17.5) we would
obtain the said differential equation for 𝑄2(𝑚)
𝑥2
𝑄 ′′2(𝑚) − 2𝑀𝑥2
𝑄 ′2(𝑚) + 𝑥𝑄 ′2(𝑚) − 𝑀𝑥𝑄2(𝑚) = (𝜇 𝐸 𝜅−1)2
𝑄2(𝑚)
𝜇 𝐸𝐹 = 𝜇 𝐸 𝜅−1
(21.1)
We read off the series form of 𝑄2(𝑚) from (17.6) and plug this in (21.1) to arrive at an expression that will
put a constraint on the coefficients involved.
∑ [(𝑛 +
1
2
)
2
− 𝜇 𝐸𝐹
2
] 𝑏 𝑛 𝑥−(𝑛+ 1/2)
𝑚
𝑛=0
+ ∑ 2𝑀𝑛
𝑚
𝑛 = 0
𝑏 𝑛 𝑥
−(𝑛−
1
2
)
= 0
(21.2)
Then making the shift 𝑛 → (𝑛 − 1) in the first summation from the left to re-write (21.2) into the
following form
∑ [((𝑛 −
1
2
)
2
− 𝜇 𝐸𝐹
2
) 𝑏 𝑛−1 + 2𝑀𝑛𝑏 𝑛] 𝑥−(𝑛 − 1/2)
𝑚
𝑛=1
= 0
(21.3)
From this we get the recurrence relation between successive 𝑏 𝑛 coefficients and this is already given by
(14.2). Likewise, we can also terminate (17.6) up to the 𝑙𝑡ℎ place with 𝑚 relabeled as 𝑙. This is done by
setting all higher coefficients 𝑏 𝑛 = 𝑙 +1 equal to zero, 𝑏 𝑛 = 𝑙 +1 = 0.
𝑏 𝑛 = 𝑙 +1 = −
(2𝑙 + 1)2
− (2𝜅−1
𝜇 𝐸)2
8𝑀(𝑙 + 1)
𝑏𝑙 = 0
(21.4)
The consequence of this termination is already expressed in (17.11) and we can re-write (21.1) as
𝑥2
𝑄 ′′2(𝑙) − 2𝑀𝑥2
𝑄 ′2(𝑙) + 𝑥𝑄 ′2(𝑙) − 𝑀𝑥𝑄2(𝑙) =
(2𝑙 + 1)2
4
𝑄2(𝑙)
(21.5)
given with a terminated series solution
𝑄2(𝑙)(𝑥) =
1
√ 𝑥
∑
𝑏 𝑛
𝑥 𝑛
𝑙
𝑛=0
(21.6)
9. 𝑥𝑄2(2)
′
= −
1
2
𝑏0
√ 𝑥
−
9
2𝑀
𝑏0
√𝑥3
−
15
2𝑀2
𝑏0
√𝑥5
(23.8)
𝑥2
𝑄2(2)
′
= −
1
2
𝑏0√ 𝑥 −
9
2𝑀
𝑏0
√ 𝑥
−
15
2𝑀2
𝑏0
√𝑥3
(23.9)
𝑥2
𝑄2(2)
′′
=
3
4
𝑏0
√ 𝑥
+
45
4𝑀
𝑏0
√𝑥3
+
105
4𝑀2
𝑏0
√𝑥5
(23.10)
For this case (𝑙 = 2) the differential equation to be satisfied is given by
𝑥2
𝑄 ′′2(2) − 2𝑀𝑥2
𝑄 ′2(2) + 𝑥𝑄 ′2(2) − 𝑀𝑥𝑄2(2) =
25
4
𝑄2(2)
(23.11)
Massless Case
Continuing to the massless case, we take (17.12) for the differential equation of 𝑄1(𝑙) that is given for
𝑀 = 0.
𝑥2
𝑄 ′′1(𝑙) + 𝑥𝑄 ′1(𝑙) =
(2𝑙 + 1)2
4
𝑄1(𝑙)
(24.1)
In (24.1), we apply (17.13) and see what constraint for the coefficients 𝑎 𝑛 we would have in the massless
case. Using (17.13) we obtain the following results
𝑥𝑄 ′1(𝑙) = ∑ − (𝑛 +
1
2
)
𝑙
𝑛=0
𝑎 𝑛 𝑥
−(𝑛+
1
2
)
(24.1.1)
𝑥2
𝑄 ′′1(𝑙) = ∑ (𝑛 +
1
2
)
𝑙
𝑛=0
(𝑛 +
3
2
) 𝑎 𝑛 𝑥−(𝑛+
1
2
)
(24.1.2)
and write (24.1) as
∑((2𝑛 + 1)2
− (2𝑙 + 1)2)
𝑙
𝑛=0
𝑎 𝑛 𝑥−(𝑛+
1
2
)
= 0
(24.2)
This resulting equation (see (15.3) above) is satisfied with a constraint that there is only one 𝑎 𝑛=𝑙
coefficient that is non-zero (𝑎 𝑛=𝑙 ≠ 0), while all coefficients 𝑎 𝑛≠𝑙 vanish. That is, ∀𝑎 𝑛≠𝑙 = 0. Thus, in
the massless case the series solution consists only of one term that corresponds to the non-vanishing
coefficient.
𝑄1(𝑙) =
𝑎 𝑙
√𝑥2𝑙 + 1
(24.3)
Test for 𝑙 = 0:
𝑄1(0)(𝑥) =
𝑎 0
√ 𝑥
(24.4.1)
𝑥𝑄 ′1(0) = −
1
2
𝑎 0
√ 𝑥
(24.4.2)
𝑥2
𝑄 ′′1(0) =
3
4
𝑎 0
√ 𝑥
` (24.4.3)
These satisfy their given differential equation
10. 𝑥2
𝑄 ′′1(0) + 𝑥𝑄 ′1(0) =
1
4
𝑄1(0)
(24.4.4)
Test for 𝑙 = 1:
𝑄1(1)(𝑥) =
𝑎 1
√𝑥3
(24.5.1)
𝑥𝑄1(1)
′
= −
3
2
𝑎 1
√𝑥3
(24.5.2)
𝑥2
𝑄1(1)
′′
=
15
4
𝑎 1
√𝑥3
(24.5.3)
Their corresponding differential equation is given by
𝑥2
𝑄 ′′1(1) + 𝑥𝑄 ′1(1) =
9
4
𝑄1(1)
(24.5.4)
Test for 𝑙 = 2:
𝑄1(2) =
𝑎 2
√𝑥5
(24.6.1)
𝑥𝑄1(2)
′
= −
5
2
𝑎 2
√𝑥5
(24.6.2)
𝑥2
𝑄1(2)
′′
=
35
4
𝑎 2
√𝑥5
(24.6.3)
It is straightforward to check that these satisfy the following differential equation
𝑥2
𝑄 ′′1(2) + 𝑥𝑄 ′1(2) =
25
4
𝑄1(2)
(24.6.4)
To be noted that this constraint in the massless case will also lead to (17.11) if we are to reflect on (17.7)
upon the setting of 𝑀 = 0 to give us
𝑥2
𝑄 ′′1(𝑙) + 𝑥𝑄 ′1(𝑙) = (𝜇 𝐸 𝜅−1)2
𝑄1(𝑙)
(24.7)
For the second part of the solution in the massless case we continue from (21.5) with 𝑀 = 0. So here we
write that as
𝑥2
𝑄 ′′2(𝑙) + 𝑥𝑄 ′2(𝑙) =
(2𝑙 + 1)2
4
𝑄2(𝑙)
(24.8)
Not surprisingly, this is identical to (24.1).
With (21.6) at hand we get the following results
𝑥𝑄 ′2(𝑙) = ∑ − (𝑛 +
1
2
)
𝑙
𝑛=0
𝑏 𝑛 𝑥−(𝑛+
1
2
)
(24.9.1)
and
11. 𝑥2
𝑄 ′′2(𝑙) = ∑ (𝑛 +
1
2
)
𝑙
𝑛=0
(𝑛 +
3
2
) 𝑏 𝑛 𝑥−(𝑛+
1
2
)
(24.9.2)
Given these results along with (21.5), we write (24.8) as
1
4
∑((2𝑛 + 1)2
− (2𝑙 + 1)2)
𝑙
𝑛=0
𝑏 𝑛 𝑥−(𝑛+
1
2
)
= 0
(24.9.3)
from which we infer constraint regarding our coefficients 𝑏 𝑛 and it follows that the only non-zero
coefficients are those at 𝑛 = 𝑙 that is, 𝑏 𝑛 = 𝑙 ≠ 0, while those at 𝑛 ≠ 𝑙 must vanish, 𝑏 𝑛 ≠ 𝑙 = 0. As a
consequence our series solution in the massless case would only consist of a single term containing this
non-vanishing coefficient.
𝑄2(𝑙) =
𝑏 𝑙
√𝑥2𝑙 + 1
(24.10)
Test for 𝑙 = 0:
𝑄2(0)(𝑥) =
𝑏0
√ 𝑥
(24.11.1)
𝑥𝑄 ′2(0) = −
1
2
𝑏0
√ 𝑥
(24.11.2)
𝑥2
𝑄 ′′2(0) =
3
4
𝑏0
√ 𝑥
` (24.11.3)
The differential equation of these is given by
𝑥2
𝑄 ′′2(0) + 𝑥𝑄 ′2(0) =
1
4
𝑄2(0)
(24.11.4)
Test for 𝑙 = 1:
𝑄2(1)(𝑥) =
𝑏1
√𝑥3
(24.12.1)
𝑥𝑄2(1)
′
= −
3
2
𝑏1
√𝑥3
(24.12.2)
𝑥2
𝑄2(1)
′′
=
15
4
𝑏1
√𝑥3
(24.12.3)
Their corresponding differential equation is given by
𝑥2
𝑄 ′′2(1) + 𝑥𝑄 ′2(1) =
9
4
𝑄2(1)
(24.12.4)
Test for 𝑙 = 2:
𝑄2(2) =
𝑏2
√𝑥5
(24.13.1)
12. 𝑥𝑄2(2)
′
= −
5
2
𝑏2
√𝑥5
(24.13.2)
𝑥2
𝑄2(2)
′′
=
35
4
𝑏2
√𝑥5
(24.13.3)
It is straightforward to check that these satisfy the following differential equation
𝑥2
𝑄 ′′2(2) + 𝑥𝑄 ′2(2) =
25
4
𝑄2(2)
(24.13.4)
[To be continued…]
References
[1] J. Foster, J. D. Nightingale, A SHORT COURSE IN GENERAL RELATIVITY, 2nd
edition copyright
1995, Springer-Verlag, New York, Inc.,
[2] P. K.Townsend, Blackholes , Lecture Notes, http://xxx.lanl.gov/abs/gr-qc/9707012
[3] S. M. Carroll, Lecture Notes on General Relativity, arXiv:gr-qc/9712019
[4] Kaluza-Klein Theory, http://faculty.physics.tamu.edu/pope/ihplec.pdf