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Today in History – 3 June

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Toad in History

Toad in History

1818

Maratha wars between British & Maratha Confederacy in India ended.

1864

Some 7,000 Union troops were killed within 30 minutes during the Battle of Cold Harbor in Virginia.

1888

The classic baseball poem “Casey at the Bat,” written by Ernest L. Thayer, was published in the San Francisco Examiner.

1890

Khan Abdul Gafar Khan, follower of Mahatma Gandhi, was born.

1901

Mahakavi G.Sankara Kurup, Malayalam poet, was born. He was the first winner of the conveted Gyaanpeeth Award in 1965 for his poems “”Odakkuzhal”” in Malayala.

1915

Sir’ knighthood was granted to Gurudev Ravindranath Tagore by the British Government.

1916

Smt. Nathibai Damodar Thakarsey Women’s University was established.

1940

The German Luftwaffe hit Paris with 1,100 bombs.

1947

British Government issued the partition plan, which was worked out by Lord Louis Mountbatten in New Delhi and was accepted by both Muslim League and Congress.

1947

Evening newspaper ‘Jay Hind’ published.

1972

First British designed Indian built modern warship of the Leander class, ‘INS Nilgiri’ commissioned by Prime Minister Indira Gandhi, was completely made in Mazgaon Dock, Mumbai.

1985

The week comprising of five working days was implemented in Central Government Offices for its staff.

1993

K N Venkatasubramanian,IOC Chairman, suspended for leaking information about oil purchase plans to international dealers.

1998

The naval version of the surface-to-air missile, Trishul, was test-fired in Kochi.

1999

Flight Lieutenant K. Nachiketa was handed over by the International Committee of the Red Cross to the Indian High Commissioner inside the High Commission premises.

Also Read:

Today in History – 2 June

Today in History – 1 June

Today in History – 31 May

Today in History- 30 May

Tensor Calculus Primer: Foundations of Geometry and Physics

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tensor calculus primer

Table of Contents

  1. Introduction
  2. Motivation and Applications of Tensor Calculus
  3. Scalars, Vectors, and Tensors: A Unified Language
  4. Index Notation and Einstein Summation Convention
  5. Covariant and Contravariant Components
  6. Transformation Laws for Tensors
  7. The Metric Tensor and Index Manipulation
  8. Tensor Operations: Addition, Outer Product, and Contraction
  9. Covariant Derivative and Connection Coefficients
  10. Christoffel Symbols and Geodesics
  11. Riemann Curvature Tensor
  12. Ricci Tensor and Ricci Scalar
  13. Tensor Densities and Volume Elements
  14. Applications in Physics
  15. Conclusion

1. Introduction

Tensor calculus is the language of modern physics and differential geometry. It allows us to express laws of nature — such as general relativity — in a coordinate-independent way. Tensors generalize scalars and vectors, and tensor operations provide a consistent framework for dealing with curved spaces, multiple dimensions, and complex physical interactions.


2. Motivation and Applications of Tensor Calculus

Why do we need tensor calculus?

  • Describes physical laws in any coordinate system
  • Used in general relativity, continuum mechanics, fluid dynamics, and electromagnetism
  • Necessary for studying curved manifolds like spacetime
  • Enables rigorous and compact mathematical formulation

3. Scalars, Vectors, and Tensors: A Unified Language

  • Scalars: Single-valued, coordinate-independent quantities (e.g., temperature \( T \))
  • Vectors: Ordered set of numbers transforming linearly under coordinate changes
  • Tensors: Generalizations of vectors; a rank-\((m, n)\) tensor has \(m\) contravariant and \(n\) covariant indices

A second-rank tensor \( T^\mu_{\ \nu} \) transforms as:

\[
T’^\mu_{\ \nu} = \frac{\partial x’^\mu}{\partial x^\alpha} \frac{\partial x^\beta}{\partial x’^\nu} T^\alpha_{\ \beta}
\]


4. Index Notation and Einstein Summation Convention

Einstein notation simplifies tensor expressions:

  • Repeated indices imply summation
  • \( A^i B_i = \sum_i A^i B_i \)

Free indices (those not repeated) must match on both sides of an equation.


5. Covariant and Contravariant Components

  • Contravariant: \( A^i \) — transforms opposite to coordinate basis
  • Covariant: \( A_i \) — transforms like basis covectors (dual basis)

This distinction is crucial in general geometry.


6. Transformation Laws for Tensors

For a vector:

\[
A’^i = \frac{\partial x’^i}{\partial x^j} A^j
\]

For a covector:

\[
A’_i = \frac{\partial x^j}{\partial x’^i} A_j
\]

This ensures tensorial quantities behave consistently across frames.


7. The Metric Tensor and Index Manipulation

The metric tensor \( g_{\mu\nu} \) defines the inner product of vectors and distances in spacetime.

  • It allows lowering and raising indices:

\[
A_\mu = g_{\mu\nu} A^\nu, \quad A^\mu = g^{\mu\nu} A_\nu
\]

  • It encodes the geometry (flat, curved, etc.)

8. Tensor Operations: Addition, Outer Product, and Contraction

  • Addition: Only tensors of same rank and type
  • Outer product: Combines tensors into higher-rank ones
  • Contraction: Sum over a pair of one upper and one lower index:

\[
T^\mu_{\ \mu} = \text{Trace}
\]

This operation reduces the rank by 2.


9. Covariant Derivative and Connection Coefficients

The covariant derivative generalizes partial derivatives to curved spaces.

For a vector \( V^\mu \):

\[
\nabla_\nu V^\mu = \partial_\nu V^\mu + \Gamma^\mu_{\nu\sigma} V^\sigma
\]

Where \( \Gamma^\mu_{\nu\sigma} \) are Christoffel symbols — not tensors.


10. Christoffel Symbols and Geodesics

Christoffel symbols are defined as:

\[
\Gamma^\rho_{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} – \partial_\sigma g_{\mu\nu} \right)
\]

They appear in the geodesic equation:

\[
\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0
\]

This describes free-fall in curved spacetime.


11. Riemann Curvature Tensor

The Riemann tensor measures how much vectors are rotated by parallel transport:

\[
R^\rho_{\ \sigma\mu\nu} = \partial_\mu \Gamma^\rho_{\nu\sigma} – \partial_\nu \Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} – \Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma}
\]

If \( R^\rho_{\ \sigma\mu\nu} = 0 \), the space is flat.


12. Ricci Tensor and Ricci Scalar

Contracting the Riemann tensor:

  • Ricci tensor: \( R_{\mu\nu} = R^\alpha_{\ \mu\alpha\nu} \)
  • Ricci scalar: \( R = g^{\mu\nu} R_{\mu\nu} \)

These appear in Einstein’s equations and determine how curvature sources gravitational effects.


13. Tensor Densities and Volume Elements

The volume element in curved space involves the determinant of the metric:

\[
dV = \sqrt{|g|} \, d^4x
\]

Tensor densities adjust for coordinate volume scaling and are used in action integrals and field theory.


14. Applications in Physics

  • General relativity: Field equations use Ricci tensor and scalar curvature
  • Electromagnetism: Expressed using the antisymmetric electromagnetic field tensor
  • Fluid dynamics: Stress-energy tensors encode momentum and pressure
  • Quantum field theory: Tensor fields represent interactions in spacetime

15. Conclusion

Tensor calculus is the backbone of modern theoretical physics. From describing gravitational waves to the dynamics of the early universe, it offers a consistent and powerful framework for expressing physical laws in any coordinate system.

Understanding tensors, their transformations, and operations like covariant differentiation and curvature is essential for anyone venturing into relativity, cosmology, and high-energy physics.


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Today in History – 2 June

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Today in History - 2 JuneA

Today in History - 2 JuneA

1756

Fort William, the English stronghold in Bengal, was surrendered to Siraj-ud-Daulah.

1934

Nissan Motor Company founded.

1908

Sri Aurobindo, freedom fighter, was arrested in Manik Tala Bomb explosion case.

1953

Mountaineer Edmund Hillary of New Zealand and his Nepalese Sherpa guide Tensing became the first men to conquer Mount Everest, the world’s tallest mountain. The two reached the pinnacle of Everest, more than 29,000 feet above sea level, at 11.30 a.m.

1965

The second of 2 cyclones in less than a month kills 35,000 (Ganges R India).

1975

India’s First Sponge Iron project inaugurated at Vijaywada.

1984

Army took control in Punjab as 22 people were killed in Sikh struggle for autonomy. The state was declared a restricted area under the Foreigners’ Act.

1988

Raj Kapoor, famous film actor, director and Dadasaheb Phalke awardee, passed away.

1989

10,000 Chinese soldiers were blocked by 100,000 citizens protecting students demonstrating for democracy in Tiananmen Square, Beijing.

1992

The government accepted all the 15 recommendations of the Janakiraman Committee which unearthed massive collusion of banks in the security scam.

1997

Saudi Boeing 747 Jumbo with 348 people on board missed the Meenambakkam international airport, Chennai and forcelands at Tambaram Air Force strip.

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Today in History – 1 June

Today in History – 31 May

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Today in History – 29 May

Curved Spacetime and Geodesics: Motion and Geometry in General Relativity

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curved time space

Table of Contents

  1. Introduction
  2. Flat vs Curved Spacetime
  3. Curvature in General Relativity
  4. The Metric Tensor and Line Element
  5. Geodesics: The Straightest Paths
  6. Deriving the Geodesic Equation
  7. Christoffel Symbols and Covariant Derivatives
  8. Geodesics in Schwarzschild Geometry
  9. Types of Geodesics: Timelike, Null, Spacelike
  10. Gravitational Lensing and Geodesic Bending
  11. Free Fall as Geodesic Motion
  12. Geodesic Deviation and Tidal Forces
  13. The Twin Paradox in Curved Spacetime
  14. Applications and Observations
  15. Conclusion

1. Introduction

In general relativity, spacetime is not flat. Instead, it bends in response to mass and energy. Objects under the influence of gravity are not pulled by a force but follow paths called geodesics, which are the closest analogs to straight lines in a curved geometry.

Understanding geodesics allows us to describe planetary orbits, black hole accretion, and even light bending due to gravity.


2. Flat vs Curved Spacetime

In special relativity, spacetime is flat and described by the Minkowski metric:

\[
ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2
\]

In general relativity, the presence of mass-energy curves spacetime, and distances are measured via:

\[
ds^2 = g_{\mu\nu} dx^\mu dx^\nu
\]

Where \( g_{\mu\nu} \) is the metric tensor of spacetime.


3. Curvature in General Relativity

Spacetime curvature is described using:

  • Riemann curvature tensor \( R^\rho_{\ \sigma\mu\nu} \)
  • Ricci tensor \( R_{\mu\nu} \)
  • Ricci scalar \( R \)

These quantities appear in Einstein’s field equations:

\[
R_{\mu\nu} – \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}
\]

Curvature determines how geodesics behave and deviate.


4. The Metric Tensor and Line Element

The metric tensor \( g_{\mu\nu} \) encodes the geometry of spacetime.

The line element:

\[
ds^2 = g_{\mu\nu} dx^\mu dx^\nu
\]

This is used to compute intervals, proper time, and distances in curved spacetime. It generalizes the Pythagorean theorem to curved, 4D manifolds.


5. Geodesics: The Straightest Paths

A geodesic is the generalization of a straight line to curved spacetime. It is defined as the path of extremal proper time (for timelike curves) or extremal path length (for spacelike curves).

Freely falling particles follow timelike geodesics. Light rays follow null geodesics.


6. Deriving the Geodesic Equation

We derive the geodesic equation by extremizing the action:

\[
S = \int ds = \int \sqrt{-g_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}} \, d\lambda
\]

Applying the Euler–Lagrange equation leads to:

\[
\frac{d^2 x^\rho}{d\lambda^2} + \Gamma^\rho_{\mu\nu} \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda} = 0
\]

Where \( \Gamma^\rho_{\mu\nu} \) are Christoffel symbols.


7. Christoffel Symbols and Covariant Derivatives

Christoffel symbols describe how coordinates and vectors change in curved space:

\[
\Gamma^\rho_{\mu\nu} = \frac{1}{2} g^{\rho\sigma} \left( \partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} – \partial_\sigma g_{\mu\nu} \right)
\]

They appear in covariant derivatives, which replace ordinary derivatives in curved spacetime.


8. Geodesics in Schwarzschild Geometry

The Schwarzschild metric (outside a static spherical mass):

\[
ds^2 = -\left(1 – \frac{2GM}{rc^2}\right) c^2 dt^2 + \left(1 – \frac{2GM}{rc^2}\right)^{-1} dr^2 + r^2 d\Omega^2
\]

This describes black holes and stars. Geodesics in this geometry yield:

  • Planetary orbits (including perihelion precession)
  • Light bending (gravitational lensing)

9. Types of Geodesics: Timelike, Null, Spacelike

  • Timelike: \( ds^2 < 0 \) — paths of massive particles
  • Null: \( ds^2 = 0 \) — paths of light
  • Spacelike: \( ds^2 > 0 \) — hypothetical non-causal paths

These determine causal structure in spacetime.


10. Gravitational Lensing and Geodesic Bending

Massive objects bend light by curving spacetime. Light follows null geodesics, which curve around stars and galaxies.

This leads to:

  • Einstein rings
  • Multiple images of quasars
  • Maps of dark matter via lensing

11. Free Fall as Geodesic Motion

Objects in free fall are not accelerating in their local frame — they move along geodesics.

This resolves the apparent paradox of why astronauts feel weightless — they are simply following spacetime’s geometry.


12. Geodesic Deviation and Tidal Forces

Geodesic deviation equation:

\[
\frac{D^2 \xi^\mu}{d\tau^2} = -R^\mu_{\ \nu\alpha\beta} u^\nu \xi^\alpha u^\beta
\]

Where \( \xi^\mu \) is the separation vector. This explains:

  • Tidal forces
  • Stretching and compression in gravitational fields

13. The Twin Paradox in Curved Spacetime

In GR, proper time depends on the entire path in spacetime.

The twin who travels and returns follows a geodesic with less proper time due to acceleration and gravity — explaining the paradox without contradiction.


14. Applications and Observations

Geodesics play central roles in:

  • GPS satellite calibration
  • Black hole simulations
  • Cosmology (light paths from distant galaxies)
  • Predicting gravitational wave signatures

15. Conclusion

Geodesics and curved spacetime lie at the heart of general relativity. They explain how matter and light move in the presence of gravity — not as a force, but as a manifestation of geometry.

From black holes to cosmological expansion, understanding geodesics gives us a profound map of how the universe bends, twists, and evolves.


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Today in History – 1 June

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Today in History - 1 June

Today in History - 1 June

1774

The British government ordered the port of Boston to be closed.

1786

Madras General Post Office started functioning.

1790

A trilateral treaty was signed between Marathas, Britishers and Nizam.

1819

William Carey, Ward and Marshman established the Serampore College in Bengal.

1842

Satyendranath Tagore, Bengali officer and litterateur, was born.

1874

East India Company was dissolved

1904

Four Language panel were used for Rangoon. This panel had languages namely Burmese, Urdu, Tamil and Chinese.

1916

Lokmanya Tilak roared ”Freedom is by birthright” at Ahmednagar.

1921

A race riot erupted in Tulsa, Oklahoma, killing 85 people.

1930

First Indian luxurious fast train ‘Deccan Queen’ started from Victoria Terminus (CST) to Pune.

1933

Fortnightly ‘Bal Sanmitra’ was published.

1941

The German Army completed the capture of Crete as the Allied evacuation ended.

1945

Tata Fundamental Research Institute was established.

1959

The Chinese occupy Tibet and Dalai Lama fled to India.

1959

National Aeronautical Research Laboratory (NARL) set up under CSIR with temporary offices in New Delhi. Dr. P. Nilakantan was appointed as NARL’s first Director.

1959

N.G. Ranga resigned from the Indian National Congress and took up the leadership of the newly formed Swatantra Party.

1964

Nava Paisa, established in 1947, was renamed as ‘Paisa’.

1972

Space Commission and Department of Space was set up and the Indian Space Research Organisation (ISRO) was brought under Department of Space (DOS).

1992

Dr. Babasaheb Ambedkar Academy was established  at Satara.

1997

Ghulam Rasool Wani, National Conference leader, kidnapped and shot dead by militants at Kaskot in Doda district, Jammu.

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Today in History – 31 May

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