En un post anterior, publique un artículo de Albert Einstein, que me resultó muy revelador en cuanto a como se crean las teorías científicas, dado su interés, quise ver como interpretaba una IA, unos conceptos que a mi me parecen complejos, para saber el alcance de estas nuevas herramientas...
Aquí compartiré el resultado de la interpretación de la IA, pero ante de que ustedes lectores se aventuren a leer la interpretación de la IA, quisiera que leyeran el artículo de Einstein (aunque sea traducido automáticamente si es necesario) y saquen sus propias conclusiones. De esta manera, van a poder comparar sus conclusiones con las de la IA y determinar si les aporta algo, si lo interpretó como ustedes, si les ayudó a comprender lo expuesto por Einstein, o si les distorsionó de forma negativa su comprensión del escrito de Einstein.
Cuidado con las traducciones automáticas, que son una forma de interpretación y además son los fundamentos por los cuales se rigen los Chats de IA actuales https://arxiv.org/pdf/1506.05869v3
A continuación el "prompt" y el resultado en Perplexity:
analizarme el artículo publicado en: [https://www.scientificamerican.com/article/on-the-generalized-theory-of-gravit/](https://www.scientificamerican.com/article/on-the-generalized-theory-of-gravit/)
El artículo al que te refieres es el ensayo que Albert Einstein publicó en Scientific American en abril de 1950, titulado “On the Generalized Theory of Gravitation”, donde expone de forma semi‑divulgativa su intento de unificar la gravitación con el resto de los campos físicos mediante una teoría de campo unificado basada en una generalización de la relatividad general.scientificamerican+1
A continuación tienes un análisis estructurado: tesis, argumentos, rigor y limitaciones.
1. Tesis principal del artículo
La tesis central puede resumirse así:
La relatividad general es solo un primer paso hacia una teoría de campo más general (“total field theory”) que describa de manera unificada la gravitación y los demás campos físicos (en particular, el electromagnético). Esa teoría debe construirse extendiendo el marco geométrico y de covarianza general de la relatividad general, reemplazando el campo gravitatorio métrico simétrico por un campo tensorial más general (no simétrico), escogido por criterios de simplicidad lógica y consistencia matemática.
Einstein no pretende “vender” la teoría como válida empíricamente; insiste en que se trata de un programa matemático y conceptual prometedor, pero todavía sin confirmación experimental y sin predicciones concretas derivadas hasta el final.
2. Estructura y argumentos clave
2.1. Motivación filosófica y epistemológica
Einstein abre con una reflexión sobre por qué se formulan nuevas teorías si ya “sabemos” tanta física clásica (óptica geométrica, termodinámica, etc.):
Defiende que la ciencia nace de una “pasión por comprender”, similar a la pasión por la música.
Subraya que las grandes ideas teóricas (como el atomismo) no se deducen lógicamente de la experiencia ni son puramente empíricas: requieren un acto creativo.
Pero tampoco son metafísica pura: surgen motivadas por problemas concretos y se sostienen mientras no conduzcan a contradicciones con la experiencia.
Ejemplifica esto con el atomismo (Leucipo, Bernoulli): cómo la idea de átomos y movimiento térmico permite entender la presión de los gases y el calor, aunque nadie “vea” los átomos.
Con esto prepara el terreno para presentar su teoría generalizada como una nueva construcción conceptual que no se impone por la experiencia actual, sino por exigencias internas de simplicidad y coherencia.
2.2. De partículas a campos: Maxwell, Lorentz y la relatividad especial
Después recorre la transición histórica:
Newton: el concepto básico es el punto material; las interacciones se describen mediante fuerzas, incluso a distancia.
Faraday y Maxwell: introducción del campo como entidad física autónoma que llena el espacio; las ecuaciones de Maxwell unifican electricidad, magnetismo y óptica en un marco de ondas electromagnéticas.
Esto elimina la acción a distancia inmediata: las interacciones se median por un campo que obedece ecuaciones diferenciales locales.
Las ecuaciones de Maxwell son covariantes bajo las transformaciones de Lorentz, lo que lleva de forma natural al principio de relatividad especial y al abandono de la simultaneidad absoluta.
Aquí Einstein muestra cómo un cambio de concepto básico (de partículas a campos) llevó a una gran unificación, preparando el argumento de que ahora se necesita un paso similar para la gravitación y el resto de la física.[en.wikipedia]
2.3. Principio de equivalencia y relatividad general
Einstein revisita su camino hacia la relatividad general:
La igualdad de masa inercial y gravitatoria sugiere el principio de equivalencia: un campo gravitatorio homogéneo es indistinguible de un sistema acelerado.
Esto obliga a abandonar la limitación a sistemas inerciales y a admitir todas las transformaciones continuas de coordenadas: nace el principio de covarianza general.
La gravitación se describe por un campo tensorial simétrico gik (el tensor métrico); las ecuaciones de campo de Einstein relacionan la geometría del espaciotiempo con la distribución de materia/energía.
La inercia deja de ser algo definido “respecto al espacio absoluto”: el propio campo gravitatorio/geométrico fija qué trayectorias son geodésicas.
Subraya dos puntos clave:
Sin el principio de relatividad general, es prácticamente imposible “adivinar” las ecuaciones correctas de la gravitación solo recogiendo datos.
Por tanto, cualquier intento serio de profundizar en los fundamentos de la física debe partir de este principio, no de la relatividad especial como marco último.
2.4. Insuficiencias de la relatividad general “pura”
Einstein reconoce que, aunque la relatividad general describe muy bien la gravitación, tiene limitaciones profundas:
La materia aparece en las ecuaciones como singularidades del campo gik (masas puntuales o concentradas), no como parte del campo mismo.
El campo gravitatorio por sí solo no incorpora de forma unificada el electromagnetismo ni las demás interacciones conocidas.
Desde el punto de vista de programa unificador, esto es insatisfactorio: se querría una teoría en la que “materia” y “campo” sean manifestaciones de una sola entidad.
Este diagnóstico motiva la búsqueda de un “total field theory”: una teoría de campo que describa a la vez gravitación, electromagnetismo y, en principio, toda la física de campos.
2.5. La generalización: tensor no simétrico y nuevas ecuaciones de campo
Aquí entra la parte más técnica y específica del artículo:
En la relatividad general estándar, el campo está representado por un tensor simétricogik=gki.
En la teoría generalizada, Einstein propone reemplazarlo por un tensor no simétrico, cuyos componentes se pueden descomponer en:
parte simétrica (asociada a algo parecido al campo gravitatorio geométrico),
parte antisimétrica (que podría vincularse de forma natural con un campo tipo electromagnético).
Solo con el principio de relatividad general no basta para fijar las ecuaciones de este campo no simétrico: se necesitan requisitos adicionales.
Introduce la exigencia de que las ecuaciones:
sean derivables de un principio variacional (lo que ayuda a garantizar compatibilidad y conservación),
tengan una estructura “rígida”: que la teoría esté tan fuertemente determinada que no admita ajustes ad hoc.
En su trabajo técnico, Einstein encuentra:
Dos sistemas diferentes de ecuaciones de campo posibles (los llama esquemáticamente E y E∗).
Uno de ellos resulta más “fuerte” (más restrictivo, menos arbitrario) y mejor ajustado a las exigencias de rigidez y consistencia, y es el que propone como base de la teoría.
Sin entrar en detalles matemáticos en el artículo de Scientific American, se limita a explicar el criterio de selección: equilibrio entre covarianza general, derivabilidad variacional y rigidez.
2.6. Carácter especulativo y dificultad de contrastación
Einstein es muy explícito sobre el estado del programa:
Reconoce que el procedimiento para obtener consecuencias empíricamente contrastables es extremadamente difícil en esta teoría generalizada.
Admite que, hasta el momento de escribir el artículo, no se ha obtenido ninguna predicción concreta que pueda someterse a prueba experimental.
Defiende la teoría sobre todo por:
la simplicidad relativa de sus premisas,
su coherencia interna,
y la rigidez (o se acepta completa o se rechaza, no permite “tuneos”).
Termina señalando que se ha logrado, al menos, formular una pregunta bien definida: “¿corresponde esta teoría rígida y lógicamente simple a la naturaleza o no?”. Pero admite que responderla requerirá mucho trabajo matemático adicional.
3. Nivel de rigor científico del artículo
El artículo es híbrido: no es un paper técnico completo, pero tampoco es divulgación superficial.
Rigor conceptual: muy alto. La argumentación sobre:
el papel de la covarianza general,
el principio de equivalencia,
la relación entre campo y materia,
el papel de los principios variacionales y la compatibilidad de las ecuaciones,
es conceptualmente muy precisa.
Rigor matemático explícito: limitado en el texto tal como aparece en Scientific American.
Apenas hay fórmulas explícitas; se mencionan objetos como el tensor gik, el intervalo ds2, el número de ecuaciones, pero no se escriben las ecuaciones completas.
Las derivaciones matemáticas detalladas están en artículos técnicos y monografías, no en este ensayo.
Rigor empírico:
Einstein insiste en que no hay aún contraste experimental y que, por tanto, no se justifica un “exceso de detalle” en la exposición para el gran público.
La teoría está en la fase de “arquitectura matemática” y todavía no ha producido resultados cuantitativos observables.
En resumen, es riguroso como manifiesto programático de un enfoque unificador, pero no como presentación completa de una teoría ya madura y contrastada.
4. Posibles críticas y limitaciones
4.1. Limitaciones reconocidas por el propio Einstein
El propio texto deja claras varias limitaciones:
Ausencia de predicciones concretas calculadas.
Complejidad extrema para obtener soluciones y ligarlas con fenómenos medibles.
Distancia creciente entre los conceptos básicos de la teoría y la experiencia inmediata (“renuncia” a la cercanía intuitiva para ganar simplicidad lógica).
Einstein admite que esto va “más lejos que teorías anteriores” en alejar los conceptos fundamentales de la experiencia directa, lo que dificulta tanto la comprensión como la falsación.
4.2. Críticas metodológicas
Desde la perspectiva actual y también de muchos contemporáneos, pueden señalarse críticas metodológicas:
Sobrepeso de criterios estéticos:
Se priorizan la simplicidad lógica y la elegancia matemática (covarianza general, rigidez, variacionales) frente a la orientación empírica.
El peligro es construir marcos matemáticos muy bonitos pero desconectados de la fenomenología real.
Suposición de que la covarianza general es la clave para todo:
Einsten concluye que cualquier teoría fundamental debe partir ya de la relatividad general, y desconfía de enfoques que partan de la relatividad especial y luego intenten “encajar” la gravitación.
Esto pudo cerrarle a otras vías de unificación que hoy consideramos fructíferas (teorías gauge relativistas, modelos cuánticos de campos, etc.).
Desatención a la mecánica cuántica:
La teoría de campo unificado de Einstein, incluida esta versión generalizada, prácticamente ignora la estructura cuántica de la materia y los campos.
A mediados del siglo XX ya estaba claro que cualquier teoría fundamental debía incorporar cuántica de forma esencial.
4.3. Limitaciones históricas y científicas a la luz de la física actual
Visto desde hoy, el programa de Einstein terminó siendo un callejón sin salida en términos de progreso empírico:
Su enfoque basado en un tensor métrico generalizado no condujo a una teoría aceptada de las interacciones ni a una síntesis con la teoría cuántica de campos.
La unificación “ganadora” en la segunda mitad del siglo XX se produjo vía:
incorporación sistemática de la relatividad especial y la cuántica en el formalismo de campos, dejando la gravitación relativista, en buena medida, aparte.
La gravitación cuántica moderna (supergravedad, teoría de cuerdas, gravedad cuántica de bucles, etc.) sigue otra ruta muy distinta de la de la métrica no simétrica.
No obstante, muchas de las reflexiones de Einstein sobre la necesidad de una teoría de campo sin singularidades, la insatisfacción con la separación nítida entre “materia” y “campo”, y la crítica a la acción a distancia han influido filosóficamente en desarrollos posteriores.
5. Valor e interés del artículo
Aunque la teoría concreta que Einstein describe en este ensayo no se haya consolidado como teoría física viable, el artículo tiene un gran valor:
Histórico: muestra el último gran programa de Einstein en sus propios términos, y cómo él entendía la relación entre matemática, experiencia y metafísica en física.[scientificamerican]
Filosófico: es una pieza clave para entender su postura frente a:
el papel de los principios heurísticos (equivalencia, covarianza),
la creatividad teórica,
y la prioridad de la coherencia conceptual sobre el ajuste inmediato a los datos.
Pedagógico avanzado: para lectores con base en relatividad, ayuda a ver qué problemas Einstein consideraba aún abiertos (singularidades, papel de la materia, límites de la teoría) y por qué pensaba que la relatividad general no era el punto final.
En síntesis: el artículo es una defensa clara y honesta de un programa de unificación geométrica profundamente elegante, pero extremadamente difícil de confrontar con la realidad y, a la larga, superado por otros enfoques. Sirve más como ventana al pensamiento tardío de Einstein que como pilar de la física actual.
An
account of the newly published extension of the general theory of
relativity against its historical and philosophical background
By Albert Einstein:
The editors of Scientific American
have asked me to write about my recent work which has just been
published. It is a mathematical investigation concerning the foundations
of field physics.
Some
readers may be puzzled: Didn’t we learn all about the foundations of
physics when we were still at school? The answer is “yes” or “no,”
depending on the interpretation. We have become acquainted with concepts
and general relations that enable us to comprehend an immense range of
experiences and make them accessible to mathematical treatment. In a
certain sense these concepts and relations are probably even final. This
is true, for example, of the laws of light refraction, of the relations
of classical thermodynamics as far as it is based on the concepts of
pressure, volume, temperature, heat and work, and of the hypothesis of
the non-existence of a perpetual motion machine.
What,
then, impels us to devise theory after theory? Why do we devise
theories at all? The answer to the latter question is simply: Because we
enjoy “comprehending,” i.e., reducing phenomena by the process
of logic to something already known or (apparently) evident. New
theories are first of all necessary when we encounter new facts which
cannot be “explained” by existing theories. But this motivation for
setting up new theories is, so to speak, trivial, imposed from without.
There is another, more subtle motive of no less importance. This is the
striving toward unification and simplification of the premises of the
theory as a whole (i.e., Mach’s principle of economy, interpreted as a logical principle).
There
exists a passion for comprehension, just as there exists a passion for
music. That passion is rather common in children, but gets lost in most
people later on. Without this passion, there would be neither
mathematics nor natural science. Time and again the passion for
understanding has led to the illusion that man is able to comprehend the
objective world rationally, by pure thought, without any empirical
foundations—in short, by metaphysics. I believe that every true theorist
is a kind of tamed metaphysicist, no matter how pure a “positivist” he
may fancy himself. The metaphysicist believes that the logically simple
is also the real. The tamed metaphysicist believes that not all that is
logically simple is embodied in experienced reality, but that the
totality of all sensory experience can be “comprehended” on the basis of
a conceptual system built on premises of great simplicity. The skeptic
will say that this is a “miracle creed.” Admittedly so, but it is a
miracle creed which has been borne out to an amazing extent by the
development of science.
The
rise of atomism is a good example. How may Leucippus have conceived
this bold idea? When water freezes and becomes ice—apparently something
entirely different from water—why is it that the thawing of the ice
forms something which seems indistinguishable from the original water?
Leucippus is puzzled and looks for an “explanation.” He is driven to the
conclusion that in these transitions the “essence” of the thing has not
changed at all. Maybe the thing consists of immutable particles and the
change is only a change in their spatial arrangement. Could it not be
that the same is true of all material objects which emerge again and
again with nearly identical qualities?
This
idea is not entirely lost during the long hibernation of occidental
thought. Two thousand years after Leucippus, Bernoulli wonders why gas
exerts pressure on the walls of a container. Should this be “explained”
by mutual repulsion of the parts of the gas, in the sense of Newtonian
mechanics? This hypothesis appears absurd, for the gas pressure depends
on the temperature, all other things being equal. To assume that the
Newtonian forces of interaction depend on temperature is contrary to the
spirit of Newtonian mechanics. Since Bernoulli is aware of the concept
of atomism, he is bound to conclude that the atoms (or molecules)
collide with the walls of the container and in doing so exert pressure.
After all, one has to assume that atoms are in motion; how else can one
account for the varying temperature of gases?
A simple mechanical
consideration shows that this pressure depends only on the kinetic
energy of the particles and on their density in space. This should have
led the physicists of that age to the conclusion that heat consists in
random motion of the atoms. Had they taken this consideration as
seriously as it deserved to be taken, the development of the theory of
heat—in particular the discovery of the equivalence of heat and
mechanical energy—would have been considerably facilitated. This example
is meant to illustrate two things. The theoretical idea (atomism in
this case) does not arise apart from and independent of experience; nor
can it be derived from experience by a purely logical procedure. It is
produced by a creative act. Once a theoretical idea has been acquired,
one does well to hold fast to it until it leads to an untenable
conclusion.
As
for my latest theoretical work, I do not feel justified in giving a
detailed account of it before a wide group of readers interested in
science. That should be done only with theories which have been
adequately confirmed by experience. So far it is primarily the
simplicity of its premises and its intimate connection with what is
already known (viz., the laws of the pure gravitational field)
that speak in favor of the theory to be discussed here. It may, however,
be of interest to a wide group of readers to become acquainted with the
train of thought which can lead to endeavors of such an extremely
speculative nature. Moreover, it will be shown what kinds of
difficulties are encountered and in what sense they have been overcome.
In
Newtonian physics the elementary theoretical concept on which the
theoretical description of material bodies is based is the material
point, or particle. Thus matter is considered a priori to be
discontinuous. This makes it necessary to consider the action of
material points on one another as “action at a distance.” Since the
latter concept seems quite contrary to everyday experience, it is only
natural that the contemporaries of Newton—and indeed Newton
himself—found it difficult to accept. Owing to the almost miraculous
success of the Newtonian system, however, the succeeding generations of
physicists became used to the idea of action at a distance. Any doubt
was buried for a long time to come.
But
when, in the second half of the 19th century, the laws of
electrodynamics became known, it turned out that these laws could not be
satisfactorily incorporated into the Newtonian system. It is
fascinating to muse: Would Faraday have discovered the law of
electromagnetic induction if he had received a regular college
education? Unencumbered by the traditional way of thinking, he felt that
the introduction of the “field” as an independent element of reality
helped him to coordinate the experimental facts. It was Maxwell who
fully comprehended the significance of the field concept; he made the
fundamental discovery that the laws of electrodynamics found their
natural expression in the differential equations for the electric and
magnetic fields. These equations implied the existence of waves, whose
properties corresponded to those of light as far as they were known at
that time.
This
incorporation of optics into the theory of electromagnetism represents
one of the greatest triumphs in the striving toward unification of the
foundations of physics; Maxwell achieved this unification by purely
theoretical arguments, long before it was corroborated by Hertz’
experimental work. The new insight made it possible to dispense with the
hypothesis of action at a distance, at least in the realm of
electromagnetic phenomena; the intermediary field now appeared as the
only carrier of electromagnetic interaction between bodies, and the
field’s behavior was completely determined by contiguous processes,
expressed by differential equations.
Now
a question arose: Since the field exists even in a vacuum, should one
conceive of the field as a state of a “carrier,” or should it rather be
endowed with an independent existence not reducible to anything else? In
other words, is there an “ether” which carries the field; the ether
being considered in the undulatory state, for example, when it carries
light waves?
The question has a natural
answer: Because one cannot dispense with the field concept, it is
preferable not to introduce in addition a carrier with hypothetical
properties. However, the pathfinders who first recognized the
indispensability of the field concept were still too strongly imbued
with the mechanistic tradition of thought to accept unhesitatingly this
simple point of view. But in the course of the following decades this
view imperceptibly took hold.
The
introduction of the field as an elementary concept gave rise to an
inconsistency of the theory as a whole. Maxwell’s theory, although
adequately describing the behavior of electrically charged particles in
their interaction with one another, does not explain the behavior of
electrical densities, i.e., it does not provide a theory of the
particles themselves. They must therefore be treated as mass points on
the basis of the old theory. The combination of the idea of a continuous
field with that of material points discontinuous in space appears
inconsistent. A consistent field theory requires continuity of all
elements of the theory, not only in time but also in space, and in all
points of space. Hence the material particle has no place as a
fundamental concept in a field theory. Thus even apart from the fact
that gravitation is not included, Maxwell’s electrodynamics cannot be
considered a complete theory.
Maxwell’s
equations for empty space remain unchanged if the spatial coordinates
and the time are subjected to a particular kind of linear
transformations—the Lorentz transformations (“covariance” with respect
to Lorentz transformations). Covariance also holds, of course, for a
transformation which is composed of two or more such transformations;
this is called the “group” property of Lorentz transformations.
Maxwell’s
equations imply the “Lorentz group,” but the Lorentz group does not
imply Maxwell’s equations. The Lorentz group may indeed be defined
independently of Maxwell’s equations as a group of linear
transformations which leave a particular value of the velocity —the
velocity of light—invariant. These transformations hold for the
transition from one “inertial system” to another which is in uniform
motion relative to the first. The most conspicuous novel property of
this transformation group is that it does away with the absolute
character of the concept of simultaneity of events distant from each
other in space. On this account it is to be expected that all equations
of physics are covariant with respect to Lorentz transformations
(special theory of relativity). Thus it came about that Maxwell’s
equations led to a heuristic principle valid far beyond the range of the
applicability or even validity of the equations themselves.
Special
relativity has this in common with Newtonian mechanics: The laws of
both theories are supposed to hold only with respect to certain
coordinate systems: those known as “inertial systems.” An inertial
system is a system in a state of motion such that “force-free” material
points within it are not accelerated with respect to the coordinate
system. However, this definition is empty if there is no independent
means for recognizing the absence of forces. But such a means of
recognition does not exist if gravitation is considered as a “field.”
Let
A be a system uniformly accelerated with respect to an “inertial
system” I. Material points, not accelerated with respect to I, are
accelerated with respect to A, the acceleration of all the points being
equal in magnitude and direction. They behave as if a gravitational
field exists with respect to A, for it is a characteristic property of
the gravitational field that the acceleration is independent of the
particular nature of the body. There is no reason to exclude the
possibility of interpreting this behavior as the effect of a “true”
gravitational field (principle of equivalence). This
interpretation implies that A is an “inertial system,” even though it is
accelerated with respect to another inertial system. (It is essential
for this argument that the introduction of independent gravitational
fields is considered justified even though no masses generating the
field are defined. Therefore, to Newton such an argument would not have
appeared convincing.) Thus the concepts of inertial system, the law of
inertia and the law of motion are deprived of their concrete meaning—not
only in classical mechanics but also in special relativity. Moreover,
following up this train of thought, it turns out that with respect to A
time cannot be measured by identical clocks; indeed, even the immediate
physical significance of coordinate differences is generally lost In
view of all these difficulties, should one not try, after all, to hold
on to the concept of the inertial system, relinquishing the attempt to
explain the fundamental character of the gravitational phenomena which
manifest themselves in the Newtonian system as the equivalence of inert
and gravitational mass? Those who trust in the comprehensibility of
nature must answer: No.
This
is the gist of the principle of equivalence: In order to account for
the equality of inert and gravitational mass within the theory it is
necessary to admit non-linear transformations of the four coordinates.
That is, the group of Lorentz transformations and hence the set of the
“permissible” coordinate systems has to be extended.
What
group of coordinate transformations can then be substituted for the
group of Lorentz transformations? Mathematics suggests an answer which
is based on the fundamental investigations of Gauss and Riemann: namely,
that the appropriate substitute is the group of all continuous
(analytical) transformations of the coordinates. Under these
transformations the only thing that remains invariant is the fact that
neighboring points have nearly the same coordinates; the coordinate
system expresses only the topological order of the points in space
(including its four-dimensional character). The equations expressing the
laws of nature must be covariant with respect to all continuous
transformations of the coordinates. This is the principle of general
relativity.
The procedure
just described overcomes a deficiency in the foundations of mechanics
which had already been noticed by Newton and was criticized by Leibnitz
and, two centuries later, by Mach: Inertia resists acceleration, but
acceleration relative to what? Within the frame of classical mechanics
the only answer is: Inertia resists acceleration relative to space.
This is a physical property of space—space acts on objects, but objects
do not act on space. Such is probably the deeper meaning of Newton’s
assertion spatium est absolutum (space is absolute). But the idea
disturbed some, in particular Leibnitz, who did not ascribe an
independent existence to space but considered it merely a property of
“things” (contiguity of physical objects). Had his justified doubts won
out at that time, it hardly would have been a boon to physics, for the
empirical and theoretical foundations necessary to follow up his idea
were not available in the 17th century.
According
to general relativity, the concept of space detached from any physical
content does not exist. The physical reality of space is represented by a
field whose components are continuous functions of four independent
variables—the coordinates of space and time. It is just this particular
kind of dependence that expresses the spatial character of physical
reality.
Since the theory of general relativity implies the representation of physical reality by a continuous
field, the concept of particles or material points cannot play a
fundamental part, nor can the concept of motion. The particle can only
appear as a limited region in space in which the field strength or the
energy density are particularly high.
A
relativistic theory has to answer two questions: 1) What is the
mathematical character of the field? 2) What equations hold for this
field?
Concerning the first
question: From the mathematical point of view the field is essentially
characterized by the way its components transform if a coordinate
transformation is applied. Concerning the second question: The equations
must determine the field to a sufficient extent while satisfying
the postulates of general relativity. Whether or not this requirement
can be satisfied depends on the choice of the field-type.
The
attempt to comprehend the correlations among the empirical data on the
basis of such a highly abstract program may at first appear almost
hopeless. The procedure amounts, in fact, to putting the question: What
most simple property can be required from what most simple object
(field) while preserving the principle of general relativity? Viewed
from the standpoint of formal logic, the dual character of the question
appears calamitous, quite apart from the vagueness of the concept
“simple.” Moreover, from the standpoint of physics there is nothing to
warrant the assumption that a theory which is “logically simple” should
also be “true.”
Yet every theory is speculative. When the basic concepts of a theory are comparatively “close to experience” (e.g.,
the concepts of force, pressure, mass), its speculative character is
not so easily discernible. If, however, a theory is such as to require
the application of complicated logical processes in order to reach
conclusions from the premises that can be confronted with observation,
everybody becomes conscious of the speculative nature of the theory. In
such a case an almost irresistible feeling of aversion arises in people
who are inexperienced in epistemological analysis and who are unaware of
the precarious nature of theoretical thinking in those fields with
which they are familiar.
On
the other hand, it must be conceded that a theory has an important
advantage if its basic concepts and fundamental hypotheses are “close to
experience,” and greater confidence in such a theory is certainly
justified. There is less danger of going completely astray, particularly
since it takes so much less time and effort to disprove such theories
by experience. Yet more and more, as the depth of our knowledge
increases, we must give up this advantage in our quest for logical
simplicity and uniformity in the foundations of physical theory. It has
to be admitted that general relativity has gone further than previous
physical theories in relinquishing “closeness to experience” of
fundamental concepts in order to attain logical simplicity. This holds
already for the theory of gravitation, and it is even more true of the
new generalization, which is an attempt to comprise the properties of
the total field. In the generalized theory the procedure of deriving
from the premises of the theory conclusions that can be confronted with
empirical data is so difficult that so far no such result has been
obtained. In favor of this theory are, at this point, its logical
simplicity and its “rigidity.” Rigidity means here that the theory is
either true or false, but not modifiable.
The
greatest inner difficulty impeding the development of the theory of
relativity is the dual nature of the problem, indicated by the two
questions we have asked. This duality is the reason why the development
of the theory has taken place in two steps so widely separated in time.
The first of these steps, the theory of gravitation, is based on the
principle of equivalence discussed above and rests on the following
consideration: According to the theory of special relativity, light has a
constant velocity of propagation. If a light ray in a vacuum starts
from a point, designated by the coordinates x1, x2 and x3 in a three dimensional coordinate system, at the time x4, it spreads as a spherical wave and reaches a neighboring point (x1 +dx1, x2 + dx2, x3 + dx3) at the time x4 + dx4. Introducing the velocity of light, c, we write the expression:
This can also be written in the form:
dx12 + dx22 + dx32 – c2 dx42 = 0
This
expression represents an objective relation between neighboring
spacetime points in four dimensions, and it holds for all inertial
systems, provided the coordinate transformations are restricted to those
of special relativity. The relation loses this form, however, if
arbitrary continuous transformations of the coordinates are admitted in
accordance with the principle of general relativity. The relation then
assumes the more general form:
The gik
are certain functions of the coordinates which transform in a definite
way if a continuous coordinate transformation is applied. According to
the principle of equivalence, these gik functions describe a
particular kind of gravitational field: a field which can be obtained by
transformation of “field-free” space. The gik satisfy a
particular law of transformation. Mathematically speaking, they are the
components of a “tensor” with a property of symmetry which is preserved
in all transformations; the symmetrical property is expressed as
follows:
gik = gki
The
idea suggests itself: May we not ascribe objective meaning to such a
symmetrical tensor, even though the field cannot be obtained from the
empty space of special relativity by a mere coordinate transformation?
Although we cannot expect that such a symmetrical tensor will describe
the most general field, it may well describe the particular case of the
“pure gravitational field.” Thus it is evident what kind of field, at
least for a special case, general relativity has to postulate: a
symmetrical tensor field.
Hence
only the second question is left: What kind of general covariant field
law can be postulated for a symmetrical tensor field?
This
question has not been difficult to answer in our time, since the
necessary mathematical conceptions were already at hand in the form of
the metric theory of surfaces, created a century ago by Gauss and
extended by Riemann to manifolds of an arbitrary number of dimensions.
The result of this purely formal investigation has been amazing in many
respects. The differential equations which can be postulated as field
law for gik cannot be of lower than second order, i.e., they must at least contain the second derivatives of the gik
with respect to the coordinates. Assuming that no higher than second
derivatives appear in the field law, it is mathematically determined by
the principle of general relativity. The system of equations can be
written in the form:
Rik = 0
The Rik transform in the same manner as the gik, i.e., they too form a symmetrical tensor.
These
differential equations completely replace the Newtonian theory of the
motion of celestial bodies provided the masses are represented as
singularities of the field. In other words, they contain the law of
force as well as the law of motion while eliminating “inertial
systems.”
The fact that the masses appear as singularities indicates that these masses themselves cannot be explained by symmetrical gik
fields, or “gravitational fields.” Not even the fact that only positive
gravitating masses exist can be deduced from this theory. Evidently a
complete relativistic field theory must be based on a field of more
complex nature, that is, a generalization of the symmetrical tensor
field.
Before
considering such a generalization, two remarks pertaining to
gravitational theory are essential for the explanation to follow. The
first observation is that the principle of general relativity imposes
exceedingly strong restrictions on the theoretical possibilities.
Without this restrictive principle it would be practically impossible
for anybody to hit on the gravitational equations, not even by using the
principle of special relativity, even though one knows that the field
has to be described by a symmetrical tensor. No amount of collection of
facts could lead to these equations unless the principle of general
relativity were used. This is the reason why all attempts to obtain a
deeper knowledge of the foundations of physics seem doomed to me unless
the basic concepts are in accordance with general relativity from the
beginning. This situation makes it difficult to use our empirical
knowledge, however comprehensive, in looking for the fundamental
concepts and relations of physics, and it forces us to apply free
speculation to a much greater extent than is presently assumed by most
physicists. I do not see any reason to assume that the heuristic
significance of the principle of general relativity is restricted to
gravitation and that the rest of physics can be dealt with separately on
the basis of special relativity, with the hope that later on the whole
may be fitted consistently into a general relativistic scheme. I do not
think that such an attitude, although historically understandable, can
be objectively justified. The comparative smallness of what we know
today as gravitational effects is not a conclusive reason for ignoring
the principle of general relativity in theoretical investigations of a
fundamental character. In other words, I do not believe that it is
justifiable to ask: What would physics look like without gravitation?
The
second point we must note is that the equations of gravitation are 10
differential equations for the 10 components of the symmetrical tensor gik.
In the case of a non-general relativistic theory, a system is
ordinarily not overdetermined if the number of equations is equal to the
number of unknown functions. The manifold of solutions is such that
within the general solution a certain number of functions of three
variables can be chosen arbitrarily. For a general relativistic theory
this cannot be expected as a matter of course. Free choice with respect
to the coordinate system implies that out of the 10 functions of a
solution, or components of the field, four can be made to assume
prescribed values by a suitable choice of the coordinate system. In
other words, the principle of general relativity implies that the number
of functions to be determined by differential equations is not 10 but
10 – 4 = 6. For these six functions only six independent differential
equations may be postulated. Only six out of the 10 differential
equations of the gravitational field ought to be independent of each
other, while the remaining four must be connected to those six by means
of four relations (identities). And indeed there exist among the
left-hand sides, Rik, of the 10 gravitational equations four identities—“Bianchi’s identities”—which assure their “compatibility.”
In
a case like this—when the number of field variables is equal to the
number of differential equations—compatibility is always assured if the
equations can be obtained from a variational principle. This is indeed
the case for the gravitational equations.
However,
the 10 differential equations cannot be entirely replaced by six. The
system of equations is indeed “overdetermined,” but due to the existence
of the identities it is overdetermined in such a way that its
compatibility is not lost, i.e., the manifold of solutions is not
critically restricted. The fact that the equations of gravitation imply
the law of motion for the masses is intimately connected with this
(permissible) overdetermination.
After
this preparation it is now easy to understand the nature of the present
investigation without entering into the details of its mathematics. The
problem is to set up a relativistic theory for the total field. The
most important clue to its solution is that there exists already the
solution for the special case of the pure gravitational field. The
theory we are looking for must therefore be a generalization of the
theory of the gravitational field. The first question is: What is the
natural generalization of the symmetrical tensor field?
This
question cannot be answered by itself, but only in connection with the
other question: What generalization of the field is going to provide the
most natural theoretical system? The answer on which the theory under
discussion is based is that the symmetrical tensor field must be
replaced by a non-symmetrical one. This means that the condition gik = gki for the field components must be dropped. In that case the field has 16 instead of 10 independent components.
There
remains the task of setting up the relativistic differential equations
for a non-symmetrical tensor field. In the attempt to solve this problem
one meets with a difficulty which does not arise in the case of the
symmetrical field. The principle of general relativity does not suffice
to determine completely the field equations, mainly because the
transformation law of the symmetrical part of the field alone does not
involve the components of the antisymmetrical part or vice versa.
Probably this is the reason why this kind of generalization of the
field has hardly ever been tried before. The combination of the two
parts of the field can only be shown to be a natural procedure if in the
formalism of the theory only the total field plays a role, and not the
symmetrical and antisymmetrical parts separately.
It
turned out that this requirement can indeed be satisfied in a natural
way. But even this requirement, together with the principle of general
relativity, is still not sufficient to determine uniquely the field
equations. Let us remember that the system of equations must satisfy a
further condition: the equations must be compatible. It has been
mentioned above that this condition is satisfied if the equations can be
derived from a variational principle.
This
has indeed been achieved, although not in so natural a way as in the
case of the symmetrical field. It has been disturbing to find that it
can be achieved in two different ways. These variational principles
furnished two systems of equations—let us denote them by E1 and E2—which
were different from each other (although only slightly so), each of
them exhibiting specific imperfections. Consequently even the condition
of compatibility was insufficient to determine the system of equations
uniquely.
It was, in fact, the formal defects of the systems E1 and E2 that indicated a possible way out. There exists a third system of equations, E3, which is free of the formal defects of the systems E1 and E2 and represents a combination of them in the sense that every solution of E3 is a solution of E1 as well as of E2. This suggests that E3 may be the system we have been looking for. Why not postulate E3, then, as the system of equations? Such a procedure is not justified without further analysis, since the compatibility of E1 and that of E2 do not imply compatibility of the stronger system E3, where the number of equations exceeds the number of field components by four.
An independent consideration shows that irrespective of the question of compatibility the stronger system, E3, is the only really natural generalization of the equations of gravitation.
But E3 is not a compatible system in the same sense as are the systems E1 and E2,
whose compatibility is assured by a sufficient number of identities,
which means that every field that satisfies the equations for a definite
value of the time has a continuous extension representing a solution in
four-dimensional space. The system E3, however, is not
extensible in the same way. Using the language of classical mechanics we
might say: In the case of the system E3 the “initial
condition” cannot be freely chosen. What really matters is the answer to
the question: Is the manifold of solutions for the system E3 as extensive as must be required for a physical theory? This purely mathematical problem is as yet unsolved.
The
skeptic will say: “It may well be true that this system of equations is
reasonable from a logical standpoint. But this does not prove that it
corresponds to nature.” You are right, dear skeptic. Experience alone
can decide on truth. Yet we have achieved something if we have succeeded
in formulating a meaningful and precise question. Affirmation or
refutation will not be easy, in spite of an abundance of known empirical
facts. The derivation, from the equations, of conclusions which can be
confronted with experience will require painstaking efforts and probably
new mathematical methods.
Tabaco, cambio climático, pesticidas… Nunca antes el conocimiento científico había parecido tan vasto, detallado y compartido. Y sin embargo, parece estar cada vez más cuestionado.
Ya no sorprende ver cómo ciertas empresas privadas ponen en marcha estrategias para confundir el debate público y paralizar la toma de decisiones políticas. Saturados de información, ¿cómo podemos, como ciudadanos, distinguir lo verdadero de lo falso?
Una a una, esta película desmonta las mecánicas de esta hábil maniobra cuyo objetivo es volcar la ciencia contra sí misma.
Gracias a archivos desclasificados, animaciones gráficas y testimonios de expertos, lobistas y políticos, esta investigación nos sumerge en la ciencia de la duda.
De la mano de un equipo de especialistas (filósofos, economistas, científicos cognitivos, políticos e incluso agnotólogos), exploramos ejemplos concretos de fabricación de la duda y tratamos de entender su funcionamiento y lo que está en juego.
Documentario: Manufacturing Ignorance (2021)
Dirigido por: François Cuveillier & Pascal Vasselin
Producción: ZED
Según Sócrates, los estupidos nos gobiernan, la evolución favorece a los mediocres y los inteligentes son sometidos por la cultura en la que viven.
Ya que dijo:
"La gente INTELIGENTE aprende de TODO y de TODOS, la gente NORMAL aprende de sus EXPERIENCIAS, la gente ESTÚPIDA ya lo SABE TODO."
El video habla de un futuro donde cada hogar tenga computadoras conectadas a vastas bibliotecas. Esta configuración permite a cualquier persona hacer preguntas y recibir respuestas o materiales de referencia sobre cualquier tema de su interés desde una edad temprana. Se enfatiza el aprendizaje personalizado y a su propio ritmo en casa, haciendo que la educación sea agradable para todos.
El ponente critica la educación moderna por su enfoque uniforme, donde todos los estudiantes se ven obligados a aprender el mismo material al mismo ritmo, lo cual no se adapta a las necesidades individuales de cada uno. Sugiere complementar la educación tradicional con oportunidades para desarrollar intereses y estilos de aprendizaje personales desde el principio. La conversación aborda posteriormente la preocupación sobre si las máquinas y los ordenadores podrían deshumanizar el proceso de aprendizaje.
El ponente analiza cómo una máquina permite ahora una conexión directa e individualizada entre la fuente de información y el consumidor. Compara esto con la educación personalizada tradicional, donde los tutores adaptaban las lecciones a los intereses y capacidades de cada niño, destacando el potencial de las experiencias de aprendizaje a medida.
Este segmento explica el desafío histórico de educar a los niños cuando pocos podían costear la educación privada. Analiza la necesidad de educar a todos y cómo se abordó esta necesidad al contar con un solo docente para impartir clases a muchos estudiantes mediante un currículo estandarizado. También destaca que no todos los docentes tienen la misma cualificación, ya que el número de docentes supera al de buenos docentes.
El segmento analiza la transición de las relaciones tradicionales de enseñanza individualizada a un modelo donde todos pueden acceder al conocimiento humano colectivo a través de bibliotecas y computadoras conectadas en casa. Destaca cómo este acceso permite a las personas profundizar en sus intereses personales, como aprender sobre béisbol, lo que puede generar curiosidad en campos relacionados como las matemáticas. El ponente señala que este aprendizaje autodirigido puede ser transversal, como por ejemplo, alguien interesado en las matemáticas que se fascina por la física de lanzar una bola curva.
El orador analiza el desafío de proporcionar computadoras de calidad a todos los niños, incluidos aquellos de entornos de bajos recursos, comparándolo con la lucha histórica por el suministro universal de agua potable. Explica cómo algunas sociedades dependían del alcohol para eliminar gérmenes debido a la falta de agua potable, pero las mejoras han hecho que el agua potable sea accesible para la mayoría de las personas en países desarrollados como Estados Unidos. De igual manera, si bien puede que no sea factible de inmediato garantizar que todos los niños tengan computadoras perfectas o acceso a la tecnología, el objetivo debería ser lograr una accesibilidad generalizada con el tiempo.
El ponente reflexiona sobre cómo tecnologías antes poco comunes, como los automóviles y los teléfonos, ahora son comunes e imagina una transformación similar en la educación. Cada estudiante podría tener una experiencia de aprendizaje personalizada donde controlara lo que estudia, aunque la educación tradicional para los conocimientos fundamentales y la interacción social sigue siendo esencial. La idea de la educación personalizada se presenta como una revolución más amplia que no se limita a los estudiantes jóvenes, sino que es relevante para todas las edades.
El orador explica cómo terminar la escuela suele considerarse un rito de paso a la edad adulta, donde los niños sienten que deben dejar de aprender y de preguntarse para ser considerados adultos. Abandonar la escuela se percibe erróneamente como convertirse en un adulto prematuro. Esta mentalidad lleva a las personas a avergonzarse de retomar el aprendizaje más adelante en la vida y contribuye a una cultura donde el aprendizaje es algo que se termina en lugar de disfrutarlo continuamente.
El orador argumenta que el aprendizaje debería ser una actividad que se realice durante toda la vida por placer, similar a los pasatiempos o las actividades físicas que las personas continúan hasta la vejez. La mayoría de las personas no disfrutan del aprendizaje debido a sus circunstancias, pero si fuera placentero, seguirían haciéndolo. El ejemplo de Oliver Wendell Holmes, quien a sus más de 90 años aún estudiaba activamente gramática griega, ilustra el valor de la dedicación intelectual continua.
Oliver Wendell Holmes le explicó al presidente Roosevelt que estudió gramática griega para mejorar su mente, demostrando la importancia del aprendizaje permanente y del ejercicio mental incluso a una edad avanzada.
