Relativistic Bullet Striking a Bell: Frame Analysis

Events in the bell’s rest frame :

• : bullet strikes bell at .
• : bell begins emitting a pressure (sound) wave at , with a (possibly tiny) response time.

Consequences: For timelike-separated events, all inertial observers agree on the order. Thus occurs before in every frame—no frame can render them simultaneous.

Observer 1 (on the bell; frame )

She is at rest with the bell, so:

Not simultaneous. The strike precedes the onset of emission by .

Observer 2 (receding from the bell along the bullet’s direction)

Let Observer 2 move at constant speed in . Their inertial frame is with Lorentz factor . For the emission event (which has in ):

Thus the time gap between strike and start of emission is even longer in (time dilation). Still not simultaneous.

When does Observer 2 actually hear the sound?

Let sound propagate in the air rest frame at speed . Suppose at Observer 2 is at and recedes at speed . The sound front, launched at , obeys:

The reception time solves :

• If : is finite and positive ⇒ Observer 2 eventually hears the bell.
• If : the sound never catches up ⇒ Observer 2 never hears the bell (in the medium’s rest frame).

The reception event is also timelike relative to (any sub- signal yields ). Hence the ordering is invariant across frames. In , the elapsed time from strike to reception is:

You can insert the expression for to see explicitly how , , , and shape the delay in .

Key Takeaways

• Never simultaneous in any frame: (strike) precedes (emission start) for all observers since the separation is timelike.
• Moving observer sees a larger gap: the interval between strike and emission start is in the receding frame.
• Hearing depends on outrunning sound: if , the receding observer never hears the bell.
• Causal order is invariant: strike emit (possibly) receive is preserved in all inertial frames; only the measured time intervals differ.

The post Striking of a bell and the sound heard – as per two observers appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Deutsch’s Computational Circuits with Negative Time Delays

1) Context: Computation with Closed Timelike Curves (CTCs)

Deutsch (1991) proposed a quantum–mechanical model of computation in which circuits can include
closed timelike curves (CTCs) – effectively, wires that feed outputs back to earlier inputs.
Consistency is imposed by a quantum fixed–point condition so that “time travel” does not yield contradictions.

2) Negative Time Delays (Backwards-in-Time Wires)

In a normal circuit, gates act in time order. Deutsch allows a wire with a negative time delay that sends an output at time to an earlier input at time with :

Let be the CTC system’s state and the chronology–respecting (forward–going) system’s state.
If is the joint unitary that couples them, Deutsch’s self–consistency condition is

In words: the state that emerges from the CTC interaction must equal the state that (earlier) entered the past.

3) Classical Grandfather Paradox vs. Quantum Resolution

Classically, a one–bit circuit that flips its own past value via a NOT gate has no self–consistent assignment:
input implies output (contradiction) and vice versa. In the quantum model, mixed states can resolve this:
the maximally mixed qubit

is invariant under a Pauli–X (NOT) and can satisfy the fixed–point equation—so no contradiction arises.

4) Computational Power with CTC Access

Because the output must be a fixed point of a global, nonlinear map induced by the CTC interaction,
the model can “jump” to self–consistent solutions. Subsequent results show that classical or quantum
computers with CTCs can decide exactly PSPACE in polynomial time.

5) Time–Travel Paradox Analogues

• Grandfather paradox → resolved by fixed–point mixed states.
• Bootstrap/information paradox → information appears “from nowhere,” stabilized by self–consistency.
• Decision paradoxes → solutions are fixed points of a global noncausal map.

6) Summary Table

7) Explicit Circuit Example (NOT on a Negative-Delay Wire)

7.1 Circuit Sketch (ASCII)

CR:   |0⟩ ── H ──■──────── H ── (trace out CR)
                 │
CTC:  ρ_in ◄─────X────── X ─────►  ρ_out
         ^       (CNOT)   (NOT)         |
         |_______________________________|
                 negative time delay

Here the chronology–respecting (CR) line flows forward; the CTC line loops back to the past (negative delay).
We apply a CNOT with CR as control and CTC as target, followed by a NOT on the CTC line before it loops back.

7.2 Unitary and Induced Map

Let the CR system be initialized to with
.
Define

Deutsch’s condition becomes

A direct calculation shows that tracing out the CR induces the affine map

7.3 Solving the Fixed-Point Equation

Write a general qubit state

Conjugation by yields

Therefore

Self–consistency implies and . Thus the fixed–point family is

Deutsch’s maximum–entropy rule selects the unique maximally mixed solution

7.4 Interpretation

• The NOT on the CTC wire encodes “I flip my own past state.”
• The loop enforces on the CTC system.
• The induced map has self–consistent fixed points; picking resolves the paradox.

References

• Deutsch, D. (1991). Quantum mechanics near closed timelike lines. Phys. Rev. D 44, 3197–3217.
• Aaronson, S., & Watrous, J. (2009). Closed timelike curves make quantum and classical computing equivalent. Proc. R. Soc. A 465, 631–647.

Rendering tip: If your environment does not auto-render LaTeX, keep the
equations exactly between (inline) and

   

(display) as shown,
or include MathJax (already linked above).

The post Negative Time Delays and Time Travel Paradox Computational Circuits appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Lorentz Boost of a Spin-Singlet: A Worked Example with Wigner Rotations

We show explicitly how a Lorentz boost acts on a two–spin-1/2 singlet state by inducing Wigner rotations that depend on the particle momenta.
The key takeaway: the full bipartite state remains entangled in every inertial frame, but the distribution of entanglement between spin and momentum,
and therefore the spin-only correlations accessible to a given observer, can change across frames.

1) Setup in the Center-of-Momentum (CM) Frame

Consider two spin-1/2 particles with equal and opposite momenta along the \(z\)-axis in the CM frame:
\(\mathbf{p}_A = +p\,\hat{\mathbf{z}}\), \(\mathbf{p}_B = -p\,\hat{\mathbf{z}}\).
Let the spin state be the singlet:

Assume (for clarity) sharply peaked momenta (delta-like wavepackets) so that we can track definite momenta through the boost.

2) Apply a Pure Lorentz Boost Along x

Boost the whole system by rapidity \(\eta\) along the \(x\)-axis, i.e., a boost \(\Lambda(\eta\,\hat{\mathbf{x}})\).
For a particle with 4-momentum \(p^\mu\), a pure boost followed by the standard momentum alignment induces a Wigner rotation
on its spin. Because the two particles carry opposite momenta in the CM frame, they experience opposite Wigner rotation angles.

Let \(\xi\) be the rapidity corresponding to the particle’s CM-frame momentum magnitude \(p\) and mass \(m\):

For this geometry (boost along \(x\), momenta along \(\pm z\)), the induced Wi

The post Boost of a Singlet Entangled State appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Relativistic Treatment of Entangled Particles

Also read – Boost of a singlet entangled state

1. Lorentz Invariance of Entanglement

Entanglement is a property of the quantum state as a whole. If two particles are entangled in one inertial frame,
they are entangled in all inertial frames.

• Lorentz transformations (boosts and rotations) are unitary transformations on the Hilbert space.
• Unitary transformations cannot convert an entangled state into a separable one (or vice versa).

Thus, the fact of entanglement is invariant.

\[
|\Psi^-\rangle = \frac{1}{\sqrt{2}}\Big(|\uparrow\rangle_A|\downarrow\rangle_B – |\downarrow\rangle_A|\uparrow\rangle_B\Big)
\]

A Lorentz boost maps spin states to spin states (through Wigner rotations), but does not turn the state into a separable product.

2. What Does Change With Reference Frame?

While entanglement as such is invariant, the manifestation of correlations can change:

• Spin and momentum entanglement trade-off: In relativistic treatments, spin and momentum are not independent. A Lorentz boost can “mix” these degrees of freedom (via Wigner rotations).
• Observable correlations: Measurement outcomes depend on the orientation of detectors. In a boosted frame, the effective measurement axis is rotated relative to the particle’s momentum, so the correlations can appear altered.
• Accessibility of entanglement: If you “trace out” momentum degrees of freedom (because your detectors don’t resolve them), then in some frames you may see reduced spin entanglement—even though the full state is still entangled.

The full quantum state remains entangled, but how much entanglement you observe in a particular subsystem depends on the frame.

3. Relativity of Simultaneity

• Entanglement correlations are nonlocal, but they don’t imply causal signaling.
• Different frames disagree on the time ordering of spacelike-separated measurements, but all observers predict the same joint statistics.
• This is why the no-communication theorem survives relativity.

4. Research Frontiers

• Relativistic quantum information: Studies how entanglement is affected by boosts, acceleration, and curved spacetime.
• Unruh effect: Accelerated observers perceive the vacuum as a thermal bath, which can degrade entanglement.
• Gravitational backgrounds: In curved spacetime, entanglement entropy can change depending on observer horizons (e.g., black holes, Rindler observers).

Answer in one line

Entanglement itself is Lorentz invariant—if two particles are entangled in one frame, they remain entangled in all—
but the type of entanglement (e.g., spin vs momentum) and the measurable correlations can appear different
to observers in different reference frames.

The post Entangled States and Reference Frames appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Quantum Particle in an Accelerating Box — How to Treat the Problem

There isn’t a single “one-size” answer because what you do depends on how the box moves. Below are practical regimes from easiest to most realistic, with LaTeX-compatible equations.

1) Box at constant relativistic velocity (no acceleration)

If the box moves inertially at speed relative to the lab, go to the box’s rest frame, solve the usual problem, then transform back.

• In the box rest frame (proper length ):

Non-relativistic particle (valid when ):

Relativistic particle (use Klein–Gordon/Dirac dispersion with the same ):

• Transform back to the lab: the box length is with ; momenta and frequencies Lorentz-transform. Operationally, compute in the comoving frame and transform observables at the end.

2) Slow/weak acceleration: instantaneous comoving frame (adiabatic NR QM)

If the acceleration is small enough that levels don’t mix appreciably over a level-spacing time, work in the instantaneous rest frame of the box. Keep the walls at fixed (assuming Born rigidity; see §4). The Schrödinger equation acquires an inertial “gravity” term:

with infinite walls at .

• A uniform acceleration adds a linear potential , shifting energies and skewing eigenfunctions.
• Level shifts remain small if (with a local level spacing).
• Across the box, clocks redshift by (equivalence principle).
• If the wall separation changes in time , map to a fixed domain via and use scaling/Lewis–Riesenfeld invariants. Generic causes mode mixing and non-adiabatic transitions.

3) Relativistic acceleration: Rindler frame and field-theoretic treatment

For appreciable accelerations (or long proper times “approaching ” during the run-up), use a relativistic description in accelerating coordinates (Rindler).

Rindler coordinates for constant proper acceleration

Metric:

A Born-rigid cavity of proper length spans with . Different points have different proper accelerations

Quantize the appropriate relativistic field (Klein–Gordon or Dirac) inside the cavity with boundary conditions at . Stationary modes are defined with respect to Rindler time ; their frequencies vary across the cavity due to gravitational redshift.

Phenomena beyond NR QM

• Unruh effect: an accelerated observer sees a thermal bath at
  

     

  
  Coupling to this bath induces excitations and decoherence in the accelerated frame.

• Dynamical Casimir / moving-mirror particle creation: time-dependent boundaries mix positive and negative frequency modes (Bogoliubov coefficients), creating quanta in the cavity.
• No global stationary state during non-uniform acceleration; evolve the state in time and track mode mixing.

Computation steps (relativistic regime)

1. Choose KG/Dirac as appropriate.
2. Specify the cavity worldlines (Born-rigid hyperbolae are standard).
3. Expand the field in instantaneous cavity modes; compute Bogoliubov coefficients as the walls accelerate.
4. Evolve the state; extract observables in the Rindler frame or transform back to Minkowski.

4) Note on Born rigidity

Special relativity forbids perfectly rigid bodies with uniform acceleration. To keep the proper length fixed in the box’s momentary rest frame while speeding up, the rear must have slightly larger proper acceleration than the front. To leading order, the required difference scales as

Ignoring this turns the problem into a moving-wall problem with changing , which is different physics.

5) Quick decision tree

• Constant : solve in rest frame; Lorentz-transform observables.
• Weak/slow acceleration, NR particle: instantaneous comoving box with linear inertial potential ; adiabatic if .
• Strong/relativistic acceleration or long boxes: Rindler cavity + relativistic field theory; expect redshifts, Unruh temperature, and possible particle creation from moving boundaries.

If you share your intended regime (, particle mass, the scale of , and whether you want NR vs. relativistic accuracy), I can write the explicit mode basis and, if needed, the Bogoliubov evolution for the accelerating cavity.

The post Quantum Particle in an Accelerating Box appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Entanglement, Extra Dimensions, and Faster-Than-Light Communication

The question of whether extra dimensions (spatial or temporal) accessible to entangled systems
could allow faster-than-light (FTL) communication has been considered in various speculative and
theoretical physics works. Short answer: standard quantum mechanics, plus relativity
(and especially the no-communication theorem), strongly argues that no, you can’t use entanglement
to send information FTL — even if one imagines extra dimensions. But there are papers exploring
related ideas, some quite recent, that try to reconcile quantum nonlocality with extra dimensions
or with higher-dimensional spacetime or hidden geometry. Below is a summary and analysis of why
it doesn’t (so far) allow real FTL signaling.

Some relevant published ideas

Why entanglement + extra dimensions ≠ FTL information (so far)

No-communication theorem

In standard QM, the no-communication (or no-signaling) theorem proves that although measurements
on entangled systems cause correlated outcomes, one party cannot control the outcome on their side
in order to send a message to the other side. This theorem is very general. Even if one embeds QM
into a larger theory (extra dimensions, hidden variables, etc.), any proposal allowing control of
outcomes so as to violate no-signaling conflicts with other well-tested principles
(relativistic causality, experimental consistency).

Entangled “state space” vs. physical spatial dimensions

The fact that the quantum state of two entangled particles lives in a Hilbert space whose
dimension is the tensor product of each subsystem’s space does not mean there are extra
spatial dimensions beyond the physical ones. Hilbert-space dimensions are labels for states, not
directions in which particles can move. Just because the full entangled wavefunction is a function
of many coordinates does not make them extra traversable dimensions.

Relativistic causality and locality

Even with extra dimensions, causal structure (light cones) must project consistently onto
4D spacetime. Allowing FTL signals would create paradoxes, e.g., closed timelike curves, unless
further constraints prohibit operational signaling.

Experimental constraints

Numerous Bell-test and quantum optics experiments confirm that entanglement behaves as predicted
by QM, with no evidence for controllable FTL messaging. Any extra-d

The post Entangled States, Extra Dimensions and FTL appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

1. Setup of Einstein’s Derivation

Einstein considers two coordinate systems:

• Stationary system : Coordinates
• Moving system : Coordinates , moving at velocity along the -axis.

He assumes that light is emitted from the moving frame’s origin, reflected at some point, and returns to the origin. He also assumes that the transformation equations must be linear.

2. Problem with Infinitesimal Analysis (Misuse of Differentials)

Einstein introduces an infinitesimally small displacement in the moving frame and relates it to time .

Einstein’s Step:

He writes:

since he assumes is independent of time.

Hannon’s Criticism:

• Incorrect Use of Differentials: Einstein treats as independent but also makes it infinitesimally small, a contradiction.
• Incorrect Assumption of Partial Derivative: If were independent of , the correct statement should be: instead of Einstein’s claimed result.

3. Contradiction in Time Transformation

Einstein applies the constancy of the speed of light and sets up the equation:

which follows from the assumption that light takes equal time to travel forward and backward.

Hannon’s Criticism:

• Inconsistent Time Dependence: The derivation implicitly makes a function of , contradicting earlier assumptions.
• Failure to Justify Function Assumptions: The assumed functional form of is not rigorously derived.

4. Flawed Interpretation of Light Signals and Synchronization

Einstein assumes that clock synchronization follows from light signal exchanges, leading to:

Hannon’s Criticism:

• Circular Reasoning: Einstein assumes part of the transformation to derive it.
• Alternative Synchronization Methods: If Einstein’s synchronization method were incorrect, his derivation would be invalid.

5. Final Conclusion from Hannon

Hannon argues that:

• Einstein’s derivation of the Lorentz transformation is invalid.
• The Lorentz transformations may still be correct but require a different derivation.
• The assumptions about differential dependence and clock synchronization need reevaluation.

Counterpoint from Mainstream Physics

Despite Hannon’s criticisms, Einstein’s derivation remains widely accepted because:

• His assumption of linearity is justified by physical symmetry.
• Alternative derivations (such as those based on group theory) confirm the Lorentz transformations.
• Experimental evidence overwhelmingly supports special relativity.

The post Hannon’s Criticism of Einstein’s original derivation appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Lorentz Transformation: The concept of Lorentz transformations, which describe how measurements of space and time by two observers in uniform relative motion are related, becomes more intuitive in Minkowski’s spacetime.

The transformations are essentially rotations in four-dimensional space, preserving the spacetime interval between events​

Unified Description of Physical Reality: According to Minkowski, the measurements of space and time are different for different observers due to their relative motion, but both sets of measurements are equally valid. They are merely different descriptions of the same physical reality.

The post Minkowski and Lorentz Transformations appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Solution:

1. Apparatus:
   • Pulsed laser source
   • Beam splitter
   • Two mirrors (one movable and one fixed)
   • Fast photodetector
   • Oscilloscope
2. Procedure:
   • Set up the laser, beam splitter, mirrors, and photodetector in a vacuum chamber.
   • Pulse the laser and split the beam into two paths using the beam splitter.
   • Reflect the beams off the mirrors and recombine them at the photodetector.
   • Measure the time delay between the pulses using the oscilloscope.
   • Vary the distance of the movable mirror and record the corresponding time delays.
3. Calculations:
   • Use the measured time delays and the changes in the distance of the movable mirror to calculate the speed of light $c$.
4. Error Minimization:
   • Ensure precise alignment of the optical components.
   • Use high-precision timing equipment.
   • Perform measurements in a controlled environment to minimize temperature and pressure variations.

The post Measuring the Speed of light in vacuum appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

o verify that an object traveling at the speed of light $c$ in one inertial frame $S$ also travels at the speed of light in a boosted frame S′S’S′, we need to consider the Lorentz transformation equations for a boost in the $x$-direction. The Lorentz transformation equations for space and time coordinates are given by:

x′=γ(x−vt)x’ = \gamma (x – vt)x′=γ(x−vt) y′=yy’ = yy′=y z′=zz’ = zz′=z t′=γ(t−vxc2)t’ = \gamma \left( t – \frac{vx}{c^2} \right)t′=γ(t−c2vx​)

where γ=11−v2c2\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}γ=1−c2v2​​1​ is the Lorentz factor, $v$ is the relative velocity between the frames, and $c$ is the speed of light.

Assume an object is moving with speed $c$ in the $x$-direction in the $S$ frame. The velocity components of the object in the $S$ frame are vx=cv_x = cvx​=c, vy=0v_y = 0vy​=0, and vz=0v_z = 0vz​=0.

First, we need to find the velocity components of the object in the S′S’S′ frame. The velocity transformation equations for a boost in the $x$-direction are:

vx′=vx−v1−vvxc2v_x’ = \frac{v_x – v}{1 – \frac{v v_x}{c^2}}vx′​=1−c2vvx​​vx​−v​ vy′=vy1−v2c21−vvxc2v_y’ = \frac{v_y \sqrt{1 – \frac{v^2}{c^2}}}{1 – \frac{v v_x}{c^2}}vy′​=1−c2vvx​​vy​1−c2v2​​​ vz′=vz1−v2c21−vvxc2v_z’ = \frac{v_z \sqrt{1 – \frac{v^2}{c^2}}}{1 – \frac{v v_x}{c^2}}vz′​=1−c2vvx​​vz​1−c2v2​​​

Since vy=0v_y = 0vy​=0 and vz=0v_z = 0vz​=0, the transformations for vy′v_y’vy′​ and vz′v_z’vz′​ will both be zero. We are left with:

vx′=vx−v1−vvxc2v_x’ = \frac{v_x – v}{1 – \frac{v v_x}{c^2}}vx′​=1−c2vvx​​vx​−v​

Substitute vx=cv_x = cvx​=c into this equation:

vx′=c−v1−vcc2=c−v1−vcv_x’ = \frac{c – v}{1 – \frac{v c}{c^2}} = \frac{c – v}{1 – \frac{v}{c}}vx′​=1−c2vc​c−v​=1−cv​c−v​

Simplify the denominator:

vx′=c−vc−vc=c−v1−vc=c−v1−vc×cc=c(c−v)c−v=cv_x’ = \frac{c – v}{\frac{c – v}{c}} = \frac{c – v}{1 – \frac{v}{c}} = \frac{c – v}{1 – \frac{v}{c}} \times \frac{c}{c} = \frac{c(c – v)}{c – v} = cvx′​=cc−v​c−v​=1−cv​c−v​=1−cv​c−v​×cc​=c−vc(c−v)​=c

So, vx′=cv_x’ = cvx′​=c.

Therefore, in the boosted frame S′S’S′, the object still travels at the speed $c$. This verifies directly from the Lorentz transformation equations that any object traveling at the speed of light $c$ in one inertial frame travels at the speed of light $c$ in another inertial frame.

The post Verify that a boost in the x-direction that any object traveling at speed c in an inertial frame S travels at speed c in the boosted frame. appeared first on Time Travel, Quantum Entanglement and Quantum Computing.