Global Footprint Network

 

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Global_Footprint_Network 

National ecological surplus or deficit, measured as a country's biocapacity per person (in global hectares) minus its ecological footprint per person (also in global hectares). Data from 2013.

x ≤ -9   -9 < x ≤ -8   -8 < x ≤ -7   -7 < x ≤ -6   -6 < x ≤ -5

-5 < x ≤ -4   -4 < x ≤ -3   -3 < x ≤ -2   -2 < x ≤ -1   -1 < x < 0

0 ≤ x < 2   2 ≤ x < 4   4 ≤ x < 6   6 ≤ x < 8   8 ≤ x   Data unavailable

The Global Footprint Network was founded in 2003 and is an independent think tank originally based in the United States, Belgium and Switzerland. It was established as a charitable not-for-profit organization in each of those three countries. Its aim is to develop and promote tools for advancing sustainability, including the ecological footprint and biocapacity, which measure the amount of resources we use and how much we have. These tools aim at bringing ecological limits to the center of decision-making.

Work

Global Footprint Network's goal is to create a future where all humans can live well, within the means of one planet Earth. The organization is headquartered in Oakland, California. The Network brings together over 70 partner organizations, including WWF International, ICLEI, Bank Sarasin, The Pictet Group, the New Economics Foundation, Pronatura México, and the Environment Agency Abu Dhabi.

National Footprint and Biocapacity Accounts

Every year, Global Footprint Network produced a new edition of its National Footprint and Biocapacity Accounts, which calculate Ecological Footprint and biocapacity of more than 200 countries and territories from 1961 to the present. Based on up to 15,000 data points per country per year, these data have been used to influence policy in more than a dozen countries, including Ecuador, France, Germany, Japan, Korea, the Philippines, Russia, Switzerland, and the United Arab Emirates. Since 2019, the National Footprint and Biocapacity Accounts are produced in collaboration between Global Footprint Network, York University, and Footprint Data Foundation.

The 2022 Edition of the National Footprint and Biocapacity Accounts cover 1961-2018 (latest UN data available), and incorporate data from the Food and Agriculture Organization, the UN Comtrade database, the International Energy Agency, and over 20 other sources.

Ecological Footprint Explorer

In April 2017, Global Footprint Network launched the Ecological Footprint Explorer, an open data platform for the National Footprint and Biocapacity Accounts. The website provides ecological footprint results for over 200 countries and territories, and encourages researchers, analysts, and decision-makers to visualize and download data.

Earth Overshoot Day

Previously known as Ecological Debt Day, Earth Overshoot Day is the day when humanity has exhausted nature's budget for the year. For the rest of the year, society operates in ecological overshoot by drawing down local resource stocks and accumulating carbon dioxide in the atmosphere. The first Earth Overshoot Day was December 19, 1987. In 2014, Earth Overshoot Day was August 19. The Earth Overshoot Day in 2015 was on August 13 and on August 8 in 2016. In 2017, Earth Overshoot Day landed on August 2, and in 2020 on August 22.

Founding

In 2003, Mathis Wackernagel, PhD, and Susan Burns founded Global Footprint Network, an international think-tank headquartered in Oakland, California, with offices in Geneva and Brussels. Wackernagel received an honorary doctorate in December 2007 from the University of Bern in Switzerland.

Leadership

• Co-founder & President: Dr. Mathis Wackernagel
• Co-founder: Susan Burns
• Chief Executive Officer: Laurel Hanscom
• Chief Science Officer: Dr. David Lin
• Director, Marketing & Communications: Amanda Diep
• Director, Mediterranean and MENA Regions: Dr. Alessandro Galli
• Board Chair: Keith Tuffley
• Honorary Chair: Swiss entrepreneur and investor André Hoffmann

Awards and honours

World Sustainability Award 2018

International Association for Impact Assessment's Global Environment Award 2015

RECYCLAPOLIS National Sustainability Award 2015

ISSP Sustainability Hall of Fame Inductees Susan Burns and Mathis Wackernagel 2014

Prix Nature Swisscanto 2013

Global Footprint Network was recognized as one of the top 100 NGOs worldwide by the Global Journal in 2012 and 2013

Blue Planet Prize 2012

Boulding Award 2012

Binding Prize 2012

Ecological overshoot

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Ecological_overshoot

Ecological overshoot expressed in terms of how many Earths equivalent of natural resources are consumed by humanity each year.

Ecological overshoot is the phenomenon which occurs when the demands made on a natural ecosystem exceed its regenerative capacity. Global ecological overshoot occurs when the demands made by humanity exceed what the biosphere of Earth can provide through its capacity for renewal.

Record of global ecological overshoot

To determine whether ecological overshoot is happening requires the collection of global and nation-specific data regarding the availability of natural resources, the capability of the ecosystems to renew any natural resources that were consumed, and the rate at which the resources are being consumed, usually assessed for each calendar year.

This data collection, and analysis is typically done by scientific and conservation organisations, such as the Global Footprint Network, which aggregates data to assess the ecological footprint of each country and the global community.

These ecological resource accounts reveal that the global community has been exceeding the regenerative capacity of the Earth since 1970, which was the year when the consumption capacity of humanity first exceeded the biocapacity the Earth. Each year since 1970 humanity has witnessed global ecological overshoot.

Earth Overshoot Day

This problem is highlighted each year on Earth Overshoot Day, an illustrative calendar date obtained through calculation, on which day humanity's resource consumption for the year is considered to have exceeded the Earth’s capacity to regenerate those resources for that year.

Global ecological debt

This ecological debt is often referred to as our global 'ecological overshoot'. The data from the Global Footprint Network has been used to create the graph below, it shows that since the 1970s the global population is increasingly compromising the Earth's ecosystem. The red section of the graph indicates that the global population have been accruing a global ecological overshoot since 1970. This means that the rate at which we are using natural resources exceeds the time required by the ecosystems to regenerate the resources and absorb the waste products that are involved.

The continued over-exploitation of natural resources results in ever more severe damage to global ecosystems over time, this has destabilised many micro ecosystems causing increasing extinction rates and the macro ecosystems are coming under increasing pressure. In this way humans are currently exceeding the carrying capacity of Earth as we increase the ecological overshoot each year. The IPAT equation attempts to quantify the environmental impact ("I") of the human population ("P"), their affluence ("A") and technology ("T"). Furthermore the Jevons paradox warns us that increasing our efficiency using technology will usually result in increased ecological damage.

Causes

The majority of the world currently follow an economic paradigm that seeks to grow all three of the IPAT parameters: population size, affluence and use of technology. These behaviour patterns are causing escalating environmental damage and there is evidence for growing risk of ecological collapse.

The outcomes from various possible human behaviour scenarios have been explored in a demographic model developed by Prof Chris Bystroff. According to the Bystroff predictions, continuing with the growth economic paradigm will result in a rapid decrease in population numbers halving global population by 2040. The Bystroff predictions are echoed in further research by Dr William E. Rees, who originally developed the concept of Ecological Footprint. This research states that to reduce ecological overshoot it is necessary to reduce economic consumption drastically to stop growing the economy and to repay the accrued ecological debt by restoration and rewilding back to the one planet level or less. A recent review of the World 3 demographic model by KPMG also concludes that humans need to rethink their pursuit of economic growth or anticipate collapse by 2040. For countries that have already achieved social affluence, although their social performance and resource utilization levels are high, the ecological overshoot brought about by these developments is still maintaining a continuously increasing trend. On the other hand, many low-income countries tried increasing their per capita wealth through economic activities to improve their social shortfalls. However, their social development is slower than the resulting increase in ecological overshoot. In this case, the ecological environment will be more overwhelmed.

It is important to bear in mind that the data collected by the Global Footprint Network (GFN) makes the assumption that the whole biocapacity of the Earth is entirely at the disposal of humanity. However it is evident that we need biodiversity in order to survive, therefore unless we reserve some of the global biocapacity for other species we cannot survive. Several organisations argue that to reinstate biodiversity to levels comparable to those preceding the high extinction rates associated with the ongoing Holocene extinction event, at least 50% of the Earths biocapacity would need to be protected as nature reserve areas which are kept free from human intervention. This suggestion was presented in the book titled Half Earth. Global Footprint Network data shows that for over 50 years humanity has been stressing the ecosystems on the planet beyond their ability to recover.

A crisis of human behaviour (the Human Behavioural Crisis) has been highlighted as the driver of anthropogenic ecological overshoot in a peer-reviewed World Scientists' Warning paper led by Joseph J. Merz and co-authored by William E. Rees, Phoebe Barnard et al.

Effects

The most well known symptom of ecological overshoot is the rising extinction rate. Pandemics of zoonotic diseases, like COVID-19 also become increasingly likely with overpopulation and global travel because we encroach on wildlife habitats and accelerate the spread. Biocapacity is measured by calculating the amount of biologically productive land and sea area available to provide the resources a population consumes and to absorb its wastes, given the prevailing technology and management practices. Countries differ in the productivity of their ecosystems, and this is reflected in the National Footprint and Biocapacity Accounts kept by York University, FoDaFo and Global Footprint Network. A country has an ecological reserve if its Ecological footprint is smaller than its biocapacity; otherwise it is operating with an ecological overshoot. The former are often referred to as ecological creditors, and the latter as ecological debtors. Today, most countries, and the world as a whole, are in ecological overshoot. Over 85% of the world population lives in countries operating with an ecological overshoot.

Solving the problem of Ecological Overshoot

Further information: Human population planning

 

The pursuit of growth economics relies on continual increase in our numbers and our consumption. Several economists have been challenging the wisdom of this prevailing discipline for many years. Those suggesting a new economic paradigm can be considered collectively as advocates for degrowth.

Novel ecosystem

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Novel_ecosystem

Novel ecosystems are human-built, modified, or engineered niches of the Anthropocene. They exist in places that have been altered in structure and function by human agency. Novel ecosystems are part of the human environment and niche (including urban, suburban, and rural), they lack natural analogs, and they have extended an influence that has converted more than three-quarters of wild Earth . These anthropogenic biomes include technoecosystems that are fuelled by powerful energy sources (fossil and nuclear) including ecosystems populated with technodiversity, such as roads and unique combinations of soils called technosols. Vegetation associations on old buildings or along field boundary stone walls in old agricultural landscapes are examples of sites where research into novel ecosystem ecology is developing.

Overview

Human society has transformed the planet to such an extent that we may have ushered in a new epoch known as the anthropocene. The ecological niche of the anthropocene contains entirely novel ecosystems that include technosols, technodiversity, anthromes, and the technosphere. These terms describe the human ecological phenomena marking this unique turn in the evolution of Earth's history. The total human ecosystem (or anthrome) describes the relationship of the industrial technosphere to the ecosphere.

Technoecosystems interface with natural life-supporting ecosystems in competitive and parasitic ways. Odum (2001) attributes this term to a 1982 publication by Zev Naveh: "Current urban-industrial society not only impacts natural life-support ecosystems, but also has created entirely new arrangements that we can call techno-ecosystems, a term believed to be first suggested by Zev Neveh (1982). These new systems involve new, powerful energy sources (fossil and atomic fuels), technology, money, and cities that have little or no parallels in nature." The term technoecosystem, however, appears earliest in print in a 1976 technical report and also appears in a book chapter (see in Lamberton and Thomas (1982) written by Kenneth E. Boulding).

Novel Ecosystems

A novel ecosystem is one that has been heavily influenced by humans but is not under human management. A working tree plantation doesn't qualify; one abandoned decades ago would.

Marris 2009

Novel ecosystems "differ in composition and/or function from present and past systems". Novel ecosystems are the hallmark of the recently proposed anthropocene epoch. They have no natural analogs due to human alterations on global climate systems, invasive species, a global mass extinction, and disruption of the global nitrogen cycle. Novel ecosystems are creating many different kinds of dilemmas for terrestrial and marine conservation biologists. On a more local scale, abandoned lots, agricultural land, old buildings, field boundary stone walls or residential gardens provide study sites on the history and dynamics of ecology in novel ecosystems.

Anthropogenic biomes

See also: Total Human Ecosystem

Anthropogenic biomes tell a completely different story, one of “human systems, with natural ecosystems embedded within them”. This is no minor change in the story we tell our children and each other. Yet it is necessary for sustainable management of the biosphere in the 21st century.

Ellis (2008) identifies twenty-one different kinds of anthropogenic biomes that sort into the following groups: 1) dense settlements, 2) villages, 3) croplands, 4) rangeland, 5) forested, and 6) wildlands. These anthropogenic biomes (or anthromes for short) create the technosphere that surrounds us and are populated with diverse technologies (or technodiversity for short). Within these anthromes the human species (one species out of billions) appropriates 23.8% of the global net primary production. "This is a remarkable impact on the biosphere caused by just one species."

Noosphere

Main article: Noosphere

Noosphere (sometimes noösphere) is the "sphere of human thought". The word is derived from the Greek νοῦς (nous "mind") + σφαῖρα (sphaira "sphere"), in lexical analogy to "atmosphere" and "biosphere". Introduced by Pierre Teilhard de Chardin 1922 in his Cosmogenesis. Another possibility is the first use of the term by Édouard Le Roy, who together with Chardin was listening to lectures of Vladimir Vernadsky at Sorbonne. In 1936 Vernadsky presented on the idea of the Noosphere in a letter to Boris Leonidovich Lichkov (though, he states that the concept derives from Le Roy).

Technosphere

The technosphere is the part of the environment on Earth where technodiversity extends its influence into the biosphere. "For the development of suitable restoration strategies, a clear distinction has to be made between different functional classes of natural and cultural solar-powered biosphere and fossil-powered technosphere landscapes, according to their inputs and throughputs of energy and materials, their organisms, their control by natural or human information, their internal self-organization and their regenerative capacities." The weight of Earth's technosphere has been suggested to be 30 trillion tons, a mass greater than 50 kilos for every square metre of the planet's surface.

Technoecosystems

The concept of technoecosystems has been pioneered by ecologists Howard T. Odum and Zev Naveh. Technoecosystems interfere with and compete against natural systems. They have advanced technology (or technodiversity) money-based market economies and have a large ecological footprints. Technoecosystems have far greater energy requirements than natural ecosystems, excessive water consumption, and release toxic and eutrophicating chemicals. Other ecologists have defined the extensive global network of road systems as a type of technoecosystem.

Technoecotypes

"Bio-agro- and techno-ecotopes are spatially integrated in larger, regional landscape units, but they are not structurally and functionally integrated in the ecosphere. Because of the adverse impacts of the latter and the great human pressures on bio-ecotopes, they are even antagonistically related and therefore cannot function together as a coherent, sustainable ecological system."

Technosols

Main article: Technosols

Technosols are a new form of ground group in the World Reference Base for Soil Resources (WRB). Technosols are "mainly characterised by anthropogenic parent material of organic and mineral nature and which origin can be either natural or technogenic."

Technodiversity

Technodiversity refers to the varied diversity of technological artifacts that exist in technoecosystems.

Individual

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Individual

An individual is one that exists as a distinct entity. Individuality (or self-hood) is the state or quality of living as an individual; particularly (in the case of humans) as a person unique from other people and possessing one's own needs or goals, rights and responsibilities. The concept of an individual features in many fields, including biology, law, and philosophy. Every individual contributes significantly to the growth of a civilization. Society is a multifaceted concept that is shaped and influenced by a wide range of different things, including human behaviors, attitudes, and ideas. The culture, morals, and beliefs of others as well as the general direction and trajectory of the society can all be influenced and shaped by an individual's activities.

Etymology

From the 15th century and earlier (and also today within the fields of statistics and metaphysics) individual meant "indivisible", typically describing any numerically singular thing, but sometimes meaning "a person". From the 17th century on, an individual has indicated separateness, as in individualism.

Biology

In biology, the question of the individual is related to the definition of an organism, which is an important question in biology and the philosophy of biology, despite there having been little work devoted explicitly to this question. An individual organism is not the only kind of individual that is considered as a "unit of selection". Genes, genomes, or groups may function as individual units.

Asexual reproduction occurs in some colonial organisms so that the individuals are genetically identical. Such a colony is called a genet, and an individual in such a population is referred to as a ramet. The colony, rather than the individual, functions as a unit of selection. In other colonial organisms, individuals may be closely related to one another but may differ as a result of sexual reproduction.

Law

Although individuality and individualism are commonly considered to mature with age/time and experience/wealth, a sane adult human being is usually considered by the state as an "individual person" in law, even if the person denies individual culpability ("I followed instructions").

An individual person is accountable for their actions/decisions/instructions, subject to prosecution in both national and international law, from the time that they have reached the age of majority, often though not always more or less coinciding with the granting of voting rights, responsibility for paying tax, military duties, and the individual right to bear arms (protected only under certain constitutions).

Philosophy

Individuals may stand out from the crowd, or may blend in with it.

Buddhism

In Buddhism, the concept of the individual lies in anatman, or "no-self." According to anatman, the individual is really a series of interconnected processes that, working together, give the appearance of being a single, separated whole. In this way, anatman, together with anicca, resembles a kind of bundle theory. Instead of an atomic, indivisible self distinct from reality, the individual in Buddhism is understood as an interrelated part of an ever-changing, impermanent universe (see Interdependence, Nondualism, Reciprocity).

Empiricism

Empiricists such as Ibn Tufail in early 12th century Islamic Spain and John Locke in late 17th century England viewed the individual as a tabula rasa ("blank slate"), shaped from birth by experience and education. This ties into the idea of the liberty and rights of the individual, society as a social contract between rational individuals, and the beginnings of individualism as a doctrine.

Hegel

Georg Wilhelm Friedrich Hegel regarded history as the gradual evolution of the Mind as it tests its own concepts against the external world. Each time the mind applies its concepts to the world, the concept is revealed to be only partly true, within a certain context; thus the mind continually revises these incomplete concepts so as to reflect a fuller reality (commonly known as the process of thesis, antithesis, and synthesis). The individual comes to rise above their own particular viewpoint, and grasps that they are a part of a greater whole insofar as they are bound to family, a social context, and/or a political order.

Existentialism

With the rise of existentialism, Søren Kierkegaard rejected Hegel's notion of the individual as subordinated to the forces of history. Instead, he elevated the individual's subjectivity and capacity to choose their own fate. Later Existentialists built upon this notion. Friedrich Nietzsche, for example, examines the individual's need to define his/her own self and circumstances in his concept of the will to power and the heroic ideal of the Übermensch. The individual is also central to Sartre's philosophy, which emphasizes individual authenticity, responsibility, and free will. In both Sartre and Nietzsche (and in Nikolai Berdyaev), the individual is called upon to create their own values, rather than rely on external, socially imposed codes of morality.

Objectivism

Ayn Rand's Objectivism regards every human as an independent, sovereign entity that possesses an inalienable right to their own life, a right derived from their nature as a rational being. Individualism and Objectivism hold that a civilized society, or any form of association, cooperation or peaceful coexistence among humans, can be achieved only on the basis of the recognition of individual rights — and that a group, as such, has no rights other than the individual rights of its members. The principle of individual rights is the only moral base of all groups or associations. Since only an individual man or woman can possess rights, the expression "individual rights" is a redundancy (which one has to use for purposes of clarification in today's intellectual chaos), but the expression "collective rights" is a contradiction in terms. Individual rights are not subject to a public vote; a majority has no right to vote away the rights of a minority; the political function of rights is precisely to protect minorities from oppression by majorities (and the smallest minority on earth is the individual).

Gravity train

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Gravity_train

Ceres gravity train concept. Mining the asteroid belt could use gravity trains to haul raw material to a central refining point and launch point / space elevator

A gravity train is a theoretical means of transportation for purposes of commuting between two points on the surface of a sphere, by following a straight tunnel connecting the two points through the interior of the sphere.

In a large body such as a planet, this train could be left to accelerate using just the force of gravity, since during the first half of the trip (from the point of departure until the middle), the downward pull towards the center of gravity would pull it towards the destination. During the second half of the trip, the acceleration would be in the opposite direction relative to the trajectory, but, ignoring the effects of friction, the speed acquired before would overcome this deceleration, and as a result, the train's speed would reach zero at approximately the moment the train reached its destination.

Origin of the concept

In the 17th century, British scientist Robert Hooke presented the idea of an object accelerating inside a planet in a letter to Isaac Newton. A gravity train project was seriously presented to the French Academy of Sciences in the 19th century. The same idea was proposed, without calculation, by Lewis Carroll in 1893 in Sylvie and Bruno Concluded. The idea was rediscovered in the 1960s when physicist Paul Cooper published a paper in the American Journal of Physics suggesting that gravity trains be considered for a future transportation project.

Mathematical considerations

Under the assumption of a spherical planet with uniform density, and ignoring relativistic effects as well as friction, a gravity train has the following properties:

• The duration of a trip depends only on the density of the planet and the gravitational constant, but not on the diameter of the planet.
• The maximum speed is reached at the middle point of the trajectory.

For gravity trains between points which are not the antipodes of each other, the following hold:

• The shortest time tunnel through a homogeneous earth is a hypocycloid; in the special case of two antipodal points, the hypocycloid degenerates to a straight line.
• All straight-line gravity trains on a given planet take exactly the same amount of time to complete a journey (that is, no matter where on the surface the two endpoints of its trajectory are located).

On the planet Earth specifically, since a gravity train's movement is the projection of a very-low-orbit satellite's movement onto a line, it has the following parameters:

• The travel time equals 2530.30 seconds (nearly 42.2 minutes, half the period of a low Earth orbit satellite), assuming Earth were a perfect sphere of uniform density.
• By taking into account the realistic density distribution inside the Earth, as known from the preliminary reference Earth model, the expected fall-through time is reduced from 42 to 38 minutes.

To put some numbers in perspective, the deepest current bore hole is the Kola Superdeep Borehole with a true depth of 12,262 meters; covering the distance between London and Paris (350 km) via a hypocycloidical path would require the creation of a hole 111,408 metres deep. Not only is such a depth nine times as great, but it would also necessitate a tunnel that passes through the Earth's mantle.

Mathematical derivation

Using the approximations that the Earth is perfectly spherical and of uniform density ${\displaystyle \rho }$, and the fact that within a uniform hollow sphere there is no gravity, the gravitational acceleration ${\displaystyle a}$ experienced by a body within the Earth is proportional to the ratio of the distance from the center ${\displaystyle r}$ to the Earth's radius ${\displaystyle R}$. This is because underground at distance ${\displaystyle r}$ from the center is like being on the surface of a planet of radius ${\displaystyle r}$, within a hollow sphere which contributes nothing.

${\displaystyle a=\frac{GM}{r^2}=\frac{G\rho V}{r^2}=\frac{G\rho \frac{4}{3}\pi r^3}{r^2}=G\rho \frac{4}{3}\pi r}$

On the surface, ${\displaystyle r=R}$, so the gravitational acceleration is ${\displaystyle g=G\rho \frac{4}{3}\pi R}$. Hence, the gravitational acceleration at ${\displaystyle r}$ is

${\displaystyle a=\frac{r}{R}g}$

Diametric path to antipodes

In the case of a straight line through the center of the Earth, the acceleration of the body is equal to that of gravity: it is falling freely straight down. We start falling at the surface, so at time ${\displaystyle t}$ (treating acceleration and velocity as positive downwards):

${\displaystyle r_t=R-{\int }_0^t{v}_{t}dt=R-{\int }_0^t{\int }_0^t{a}_{t}dtdt}$

Differentiating twice:

${\displaystyle \frac{d^{2}r}{d{t}^2}=-a_t=-\frac{r}{R}g=-{\omega }^{2}r}$

where ${\displaystyle \omega =\sqrt{\frac{g}{R}}}$. This class of problems, where there is a restoring force proportional to the displacement away from zero, has general solutions of the form ${\displaystyle r=k\cos (\omega t+\phi )}$, and describes simple harmonic motion such as in a spring or pendulum.

In this case ${\displaystyle r_t=R\cos \sqrt{\frac{g}{R}}t}$ so that ${\displaystyle r_0=R}$, we begin at the surface at time zero, and oscillate back and forth forever.

The travel time to the antipodes is half of one cycle of this oscillator, that is the time for the argument to ${\displaystyle \cos \sqrt{\frac{g}{R}}t}$ to sweep out ${\displaystyle \pi }$ radians. Using simple approximations of ${\displaystyle g=10\text{ m}/{\text{s}}^2,R=6500\text{ km}}$ that time is

${\displaystyle T=\frac{\pi }{\omega }=\frac{\pi }{\sqrt{\frac{g}{R}}}\approx \frac{3.1415926}{\sqrt{\frac{10}{6500000}}}\approx 2532\text{ s}}$

Straight path between two arbitrary points

Path of gravity train

For the more general case of the straight line path between any two points on the surface of a sphere we calculate the acceleration of the body as it moves frictionlessly along its straight path.

The body travels along AOB, O being the midpoint of the path, and the closest point to the center of the Earth on this path. At distance ${\displaystyle r}$ along this path, the force of gravity depends on distance ${\displaystyle x}$ to the center of the Earth as above. Using the shorthand ${\displaystyle b=R\sin \theta }$ for length OC:

${\displaystyle g_r=\frac{x}{R}g=\frac{\sqrt{r^2+b^2}}{R}g}$

The resulting acceleration on the body, because is it on a frictionless inclined surface, is ${\displaystyle g_r\cos \phi }$:

Diagram of forces on a gravity train on non-diametrical straight line path

${\displaystyle a_r=g_r\cos \phi =\frac{\sqrt{r^2+b^2}}{R}g\cos \phi }$

But ${\displaystyle \cos \phi }$ is ${\displaystyle r/x=\frac{r}{\sqrt{r^2+b^2}}}$, so substituting:

${\displaystyle a_r=\frac{\sqrt{r^2+b^2}}{R}g\frac{r}{\sqrt{r^2+b^2}}=\frac{r}{R}g}$

which is exactly the same for this new ${\displaystyle r}$, distance along AOB away from O, as for the ${\displaystyle r}$ in the diametric case along ACD. So the remaining analysis is the same, accommodating the initial condition that the maximal ${\displaystyle r}$ is ${\displaystyle R\cos \theta =AO}$ the complete equation of motion is

${\displaystyle r_t=R\cos \theta \cos \sqrt{\frac{g}{R}}t}$

The time constant ${\displaystyle \omega =\sqrt{\frac{g}{R}}}$ is the same as in the diametric case so the journey time is still 42 minutes; it's just that all the distances and speeds are scaled by the constant ${\displaystyle \cos \theta }$.

Dependence on radius of planet

The time constant ${\displaystyle \omega }$ depends only on ${\displaystyle \frac{g}{R}}$ so if we expand that we get

${\displaystyle \frac{g}{R}=\frac{GM/R^2}{R}=\frac{GM}{R^3}=\frac{G\rho V}{R^3}=\frac{G\rho \frac{4}{3}\pi R^3}{R^3}=G\rho \frac{4}{3}\pi }$

which depends only on the gravitational constant and ${\displaystyle \rho }$ the density of the planet. The size of the planet is immaterial; the journey time is the same if the density is the same.

In fiction

In the 2012 movie Total Recall, a gravity train called "The Fall" goes through the center of the Earth to commute between Western Europe and Australia.