What If 𝑐 Changed? Revisiting Relativity with a Faster Speed of Light

PHYSICXION:The speed of light is often considered a fundamental limit in the universe, and this is rooted in the principles of relativity established

What If 𝑐 Changed? Revisiting Relativity with a Faster Speed of Light

The speed of light is often considered a fundamental limit in the universe, and this is rooted in the principles of relativity established by Albert Einstein.

Evidence of c as the maximum speed of the observable universe.

Maximum Speed: According to Einstein's theory of relativity, the speed of light in a vacuum (approximately 299,792,458 meters per second or which is) is the maximum speed at which all energy, matter, and information in the universe can travel. No object with mass can reach this speed. c=3×10^8m/s

Relativity of Simultaneity: As objects approach the speed of light, time dilation occurs. For an observer moving at a significant fraction of the speed of light, time would pass more slowly relative to an observer at rest. This affects how events are perceived in different frames of reference.

Energy Requirements: To accelerate an object with mass to the speed of light would require an infinite amount of energy, which is physically impossible. This further enforces the idea that no massive object can achieve or exceed this speed.

Cosmic Implications: The finite speed of light has profound implications for our understanding of the universe, including causality. It means that there is a limit to how quickly information and signals can travel, affecting everything from communications to the way we observe distant celestial events.

Quantum Mechanics: While the speed of light is a limit for classical objects, in quantum mechanics, phenomena such as quantum entanglement can appear to allow instantaneous interactions. However, this does not violate the speed of light limit in terms of classical information transfer.

The special theory of relativity (STR)

Massive particles and consequences

To demonstrate mathematically that the speed of light is the maximum speed using the concepts of length contraction and time dilation, we can derive the relevant equations from Einstein's theory of special relativity. Below, we explore both phenomena and show how they imply that the speed of light is the ultimate speed limit.

1. Time Dilation

Time Dilation Formula: The time dilation formula relates the proper time

Δ t_{0}

(the time interval measured by an observer at rest relative to the event) and the dilated time

Δ t

(the time interval measured by an observer moving relative to the event):

\Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}}

Where:

$Δ t$ is the dilated time.
$Δ t_{0}$ is the proper time.
$v$ is the relative velocity of the observer.
$c$ is the speed of light.

Interpretation: As the velocity

v

approaches the speed of light

c

, the denominator

\sqrt{1 - \frac{v^2}{c^2}}

approaches zero, leading to

Δ t

approaching infinity. This means that time effectively "stops" for an object moving at the speed of light, implying that nothing with mass can reach this speed.

Infinite Time as

v \to c

\frac{v^2}{c^2} \to 1

Δ t^{'} \to \infty

2. Length Contraction

Length Contraction Formula: The length contraction formula relates the proper length

L_{0}

(the length measured by an observer at rest relative to the object) and the contracted length

L

(the length measured by an observer moving relative to the object):

L = L_{0} \sqrt{1 - \frac{v^{2}}{c^{2}}}

Where:

$L$ is the contracted length.
$L_{0}$ is the proper length.
$v$ is the relative velocity of the observer.
$c$ is the speed of light.

Interpretation: As

v

approaches

c

, the factor

\sqrt{1 - \frac{v^2}{c^2}}

approaches zero, leading to

L

approaching zero. This indicates that, from the perspective of a moving observer, lengths along the direction of motion contract, effectively "squeezing" objects as they approach the speed of light.

v \to c

L' \to 0

3. Energy and Speed: Increasing Without Bound

The total energy of an object moving at velocity v is given by:

E = \frac{m_{0} c^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}

Interpretation: As the velocity v approaches the speed of light c:

v \to c \Rightarrow \frac{v^{2}}{c^{2}} \to 1

Thus, the denominator of the equation becomes very small as v approaches c, and the energy E increases without bound:

E \to \infty

4. Combined Implications

Both time dilation and length contraction reveal that as an object approaches the speed of light, several significant consequences arise:

Infinite Energy Requirement: To accelerate a mass to speed $c$ , as shown in the time dilation formula, you would require infinite energy, making it impossible for massive objects to reach the speed of light.
Speed of Light as a Limit: When considering the transformations for observers moving at different velocities, the equations inherently constrain the maximum speed to $c$ . No matter the relative speed, the results of time and space transformations preserve the invariance of the speed of light.

5. Causality and the Speed of Light

Special relativity preserves causality, the principle that cause must precede effect in all reference frames. If information, energy, or objects could travel faster than light, it would allow for causality violations. Specifically, events could happen in reverse order depending on the observer’s frame of reference.
For example, if a signal or object traveled faster than light, an observer in one reference frame might see an event happen before its cause, which leads to paradoxes such as the grandfather paradox.
The speed of light,

c

, acts as the ultimate limit to preserve causality in all reference frames. No information, object, or particle can exceed this speed without violating the fundamental order of cause and effect.

6. Example Calculation

Let’s illustrate this with an example using time dilation:

Suppose a spaceship moves at $v = 0.8 c$ .
The proper time (time experienced by someone on the spaceship) for a journey is $10$ years.

Using the time dilation formula:

\Delta t = \frac{10 \text{ years}}{\sqrt{1 - (0.8)^2}} = \frac{10 \text{ years}}{\sqrt{1 - 0.64}} = \frac{10 \text{ years}}{\sqrt{0.36}} = \frac{10 \text{ years}}{0.6} \approx 16.67 \text{ years}

Thus, while the travelers experience

10

years, observers on Earth will perceive the journey taking about

16.67

years due to time dilation. If the spaceship were to reach the speed of light, time dilation would mean that time does not pass at all from the spaceship's frame, demonstrating that no object with mass can achieve light speed.

Massless Particles and the Speed of Light

According to special relativity, only particles with zero rest mass (such as photons) can travel at the speed of light. These massless particles always travel at exactly

c

in a vacuum. The reason they can do so is that they require no energy to accelerate to this speed — they are born traveling at c.
For any particle with mass, however, it is impossible to reach the speed of light because it would require infinite energy, as discussed above.

Problems if we imagine a universe Taking v=2c for any massive particle

In the framework of Einstein's theory of special relativity, an object cannot move faster than the speed of light (

c

). However, let's explore what happens if we hypothetically consider a velocity of

2 c

(twice the speed of light) using the equations for time dilation and length contraction.

1. Time Dilation with $v = 2 c:$

The time dilation formula is:

\Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{v^2}{c^2}}}

If we substitute

v = 2 c

\Delta t = \frac{\Delta t_0}{\sqrt{1 - \frac{(2c)^2}{c^2}}} = \frac{\Delta t_0}{\sqrt{1 - 4}} = \frac{\Delta t_0}{\sqrt{-3}}

Interpretation:

When considering particles moving faster than light, one must confront the implications for causality:

Imaginary Time: The square root of a negative number leads to an imaginary number. In physics, this indicates that the scenario is not physically meaningful within the context of special relativity. In other words, the equations break down, showing that the concept of traveling at $2 c$ does not exist in our current understanding of physics.
Temporal Paradoxes: A particle moving at $2 c$ could theoretically allow for backward time travel or cause effects to precede their causes, leading to paradoxes. This suggests that time itself would become ill-defined or imaginary at such speeds.

2. Length Contraction with $v = 2 c$

Using the length contraction formula:

L = L_{0} \sqrt{1 - \frac{v^{2}}{c^{2​}}}

Substituting

v = 2 c

L = L_0 \sqrt{1 - \frac{(2c)^2}{c^2}} = L_0 \sqrt{1 - 4} = L_0 \sqrt{-3}

Interpretation:

Imaginary Length: Similar to time dilation, the length contraction also results in an imaginary value. This means that lengths, as measured by an observer moving at $2 c$ , do not have a real, physical meaning within the framework of special relativity.

3. Velocity Addition Formula

The relativistic velocity addition formula determines how velocities combine when observed from different inertial frames. The formula is given by:

u' = \frac{u + v}{1 + \frac{uv}{c^2}}

Where:

$u$ is the velocity of one object (e.g., $2 c$ ),
$v$ is the velocity of another object (e.g., a stationary observer),
$u^{'}$ is the resultant velocity observed.

If we substitute

u = 2 c

and

v = 0

u' = \frac{2c + 0}{1 + 0} = 2c

Even from a stationary observer’s perspective, the speed is still

2 c

. However, this is an invalid case within the framework of special relativity, which states that no object with mass can reach or exceed

c

4. Imaginary Mass and Energy

As explored earlier, when substituting

2 c

into the relativistic mass and energy equations:

Mass:
$m = \frac{m_{0}}{\sqrt{1 - \frac{(2 c)^{2}}{c^{2}}}} = \frac{m_{0}}{\sqrt{- 3}} ⟹ Imaginary mass$

Where:

$m_0$ is the rest mass (invariant mass) of the particle,
$v$ is the velocity of the particle,
$c$ is the speed of light.

Energy:
$E = mc^2 = \frac{m_0 c^2}{\sqrt{-3}} \implies \text{Imaginary energy}$

Again, the denominator becomes imaginary:

E = \frac{m_0 c^2}{i\sqrt{3}} = \frac{m_0 c^2}{\sqrt{3}} i

Thus, the energy also becomes imaginary

Relativistic Energy:

E = m_{0} c^{2} + K . E . = m_{0} c^{2} + \frac{1}{2} m v^{2} (non-relativistic, not valid at high speeds)

A more general expression for the total energy E in terms of relativistic mass is:

E = mc^2

The emergence of imaginary values indicates that such conditions lead to unphysical scenarios.

Interpretation:

In theoretical physics, particles that travel faster than light are often called tachyons. Here are some hypothetical considerations:

Imaginary Mass: Tachyons are theorized to have imaginary mass, which would allow them to travel faster than light. This leads to different behaviors, such as decreasing energy as velocity increases.
Causality and Communication: If tachyons could exist, they could potentially allow for communication faster than light, further complicating the structure of causality.

5. Complex Momentum

If we extend the analysis to momentum, using the relativistic momentum formula:

p = mv = \frac{m_0 v}{\sqrt{1 - \frac{v^2}{c^2}}}

For

v = 2 c

p = \frac{m_0 (2c)}{\sqrt{1 - 4}} = \frac{m_0 (2c)}{\sqrt{-3}} \implies \text{Imaginary momentum}

\sqrt{-3}

Substitute p into the Energy-Momentum Relation

E^2 = \left( \frac{2 m_0 c}{\sqrt{3}} i \right)^2 c^2 + m_0^2 c^4

Simplify the Equation

E^2 = m_0^2 c^4 - \frac{4 m_0^2 c^4}{3}

Take the Square Root

E = \sqrt{-\frac{m_0^2 c^4}{3}} = \frac{m_0 c^2}{\sqrt{3}} i

Physical Interpretation of $v > c$

Causality Violation: If an object could travel faster than light, it could lead to violations of causality. For example, signals sent faster than light could theoretically arrive before they are sent, creating paradoxes.
Mass and Energy: According to relativity, as an object approaches the speed of light, its relativistic mass increases, requiring more and more energy to continue accelerating. At $c$ , the energy required becomes infinite. This concept implies that for $2 c$ , the situation is physically impossible as it would also require negative energy.
Tachyons: In some theoretical physics frameworks, hypothetical particles called tachyons are proposed to travel faster than light. However, they remain purely theoretical and have not been observed or experimentally confirmed. Tachyons, if they exist, would possess imaginary mass and could not be slowed down to the speed of light.
Inconsistency with Observational Physics: No experimental evidence supports the existence of particles traveling faster than light, making these considerations purely speculative.
Need for New Theories: The behavior of hypothetical faster-than-light particles may necessitate new theories beyond special relativity, potentially involving quantum gravity or string theory.

In the context of special relativity, discussing a particle moving at

2 c

(twice the speed of light) raises significant theoretical issues. According to Einstein's theory, nothing with mass can reach or exceed the speed of light in a vacuum. However, for the sake of understanding, we can explore the implications of hypothetical calculations involving mass and energy if we were to consider a particle moving at

2 c

Conclusion

Considering particles moving at 2c leads to a plethora of mathematical inconsistencies, including imaginary mass, energy, time, and length. While exploring these scenarios can be intellectually stimulating, they remain outside the realm of physical reality as described by established theories. The speed of light in a vacuum (c) serves as an ultimate speed limit, and no massive particle can achieve or exceed this speed, preserving the integrity of the laws of physics and causality.

PHYSICXION

What If 𝑐 Changed? Revisiting Relativity with a Faster Speed of Light

What If 𝑐 Changed? Revisiting Relativity with a Faster Speed of Light

Evidence of c as the maximum speed of the observable universe.

The special theory of relativity (STR)

Massive particles and consequences

1. Time Dilation

2. Length Contraction

3. Energy and Speed: Increasing Without Bound

4. Combined Implications

5. Causality and the Speed of Light

6. Example Calculation

Massless Particles and the Speed of Light

Problems if we imagine a universe Taking v=2c for any massive particle

1. Time Dilation with $v = 2 c:$

Interpretation:

2. Length Contraction with $v = 2 c$

Interpretation:

3. Velocity Addition Formula

4. Imaginary Mass and Energy

Interpretation:

5. Complex Momentum

6. Energy-Momentum Relation with v=2c

Physical Interpretation of $v > c$

Conclusion

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Unit-4:Chp-2: Alternating Current

What If 𝑐 Changed? Revisiting Relativity with a Faster Speed of Light

What If 𝑐 Changed? Revisiting Relativity with a Faster Speed of Light

Evidence of c as the maximum speed of the observable universe.

The special theory of relativity (STR)

Massive particles and consequences

1. Time Dilation

2. Length Contraction

3. Energy and Speed: Increasing Without Bound

4. Combined Implications

5. Causality and the Speed of Light

6. Example Calculation

Massless Particles and the Speed of Light

Problems if we imagine a universe Taking v=2c for any massive particle

1. Time Dilation with v=2c: Causality Violation

Interpretation:

2. Length Contraction with v=2c

Interpretation:

3. Velocity Addition Formula

4. Imaginary Mass and Energy

Interpretation:

5. Complex Momentum

6. Energy-Momentum Relation with v=2c

Physical Interpretation of v>c

Conclusion

Join the conversation

1. Time Dilation with $v = 2 c:$

2. Length Contraction with $v = 2 c$

Physical Interpretation of $v > c$