You
Sent by you:
While these approximations simplify the math, they don’t fundamentally alter the predictions of general relativity. How can you assert something like this. Of course they do. It's like shooting a canon at an ant. The canon is the full nonlinear Einstein's equations and the ant is the so-called ripples of space time, which is only a supposition.
Copilot
Sent by Copilot:
You raise an important point, and I appreciate your skepticism. Let’s explore this further.
Approximations in General Relativity:
You’re right that approximations play a significant role in applying general relativity to real-world scenarios. Here are some key aspects to consider:
Linearization: The linearized form of Einstein’s equations (around a flat metric) simplifies the math for weak gravitational fields. It’s analogous to approximating a curved surface as a flat plane locally. While this approximation is valid far from massive sources, it doesn’t capture strong-field effects.
Point Masses: Treating black holes as point masses is another simplification. In reality, black holes have extended structures, and their interactions involve complex dynamics. However, this approximation allows us to model binary systems more tractably.
Radiation Reaction: In the inspiral phase of black hole mergers, energy is lost due to gravitational waves. This radiation reaction isn’t explicitly included in the equations. Yet, numerical simulations account for it indirectly.
Numerical Techniques: Solving the full nonlinear Einstein equations analytically for complex systems (like merging black holes) is infeasible. Numerical relativity provides a way to simulate these scenarios on supercomputers.
The Ant and the Canon Analogy:
I appreciate your vivid analogy! Indeed, it’s like aiming a powerful cannon at a tiny ant. The full nonlinear equations of general relativity describe the intricate behavior of spacetime around massive objects.
Gravitational waves, often referred to as “ripples” in spacetime, emerge from these nonlinear interactions. While they’re not directly visible in the equations, their existence is inferred from the overall framework of general relativity.
The ant (the ripple) represents the subtle effects of gravitational waves, which are challenging to detect directly. Yet, their impact is profound—confirming the existence of black holes, neutron stars, and the dynamic nature of the cosmos.
Validation through Observations:
The remarkable agreement between numerical simulations and gravitational wave observations validates the approximations used.
When LIGO and Virgo detect merging black holes, they observe the inspiral chirp, the merger, and the final ringdown—all consistent with theoretical predictions.
While approximations simplify the math, they don’t fundamentally alter the core principles of general relativity.
In summary, while we shoot our metaphorical “cannon” of approximations, the “ant” of gravitational waves reveals the hidden dynamics of the universe. If you have more questions or want to explore further, feel free to ask! 🌌🔭
You
Sent by you:
You're begging the question. Where is the ripples of space time in Einstein's equations? The linearized Einstein's equations about a flat metric is unjustifiably applied to the merger of black holes etc. You have added tons of suppositions that are not found in Einstein's equations. For instance Einstein's equations cannot be solved for the two body problem so how can it explain the merger of two black holes?
Copilot
Sent by Copilot:
You raise valid points, and I appreciate your skepticism. Let’s explore this further.
Gravitational Waves in Einstein’s Equations:
You’re right that the direct form of Einstein’s field equations doesn’t explicitly show “ripples” in spacetime. The equations describe the curvature of spacetime due to mass and energy.
Gravitational waves emerge from the nonlinear terms in the full Einstein equations. These terms represent the dynamic interactions of massive objects.
The linearized form (which simplifies the equations around a flat metric) is often used for weak gravitational fields, such as those far from massive sources. It provides a good approximation for weak-field effects.
Two-Body Problem and Black Hole Mergers:
Solving the full nonlinear Einstein equations for a binary black hole system is indeed challenging. The two-body problem (like binary black holes) lacks an exact analytical solution.
Numerical relativity comes to the rescue. Scientists use numerical simulations to solve the full equations on supercomputers.
These simulations model the inspiral, merger, and ringdown phases of black hole binaries. They predict the emitted gravitational waveforms, which match remarkably well with observations from LIGO and Virgo.
Assumptions and Suppositions:
You rightly point out that some assumptions are made when applying general relativity to specific scenarios. These include:
Ignoring radiation reaction (energy loss due to gravitational waves) in the inspiral phase.
Treating black holes as point masses (valid when they’re far apart).
Using perturbation theory for weak fields.
While these approximations simplify the math, they don’t fundamentally alter the predictions of general relativity.
Experimental Confirmation:
The detection of gravitational waves from merging black holes provides strong evidence for the correctness of general relativity.
The observed waveforms match theoretical predictions, including the inspiral chirp and the final ringdown.
This agreement validates the use of numerical simulations and the underlying assumptions.
In summary, while Einstein’s equations don’t explicitly show ripples, the full nonlinear equations, numerical simulations, and experimental data together support the existence of gravitational waves and their role in black hole mergers. If you have more questions or want to explore further, feel free to ask!
They are specifically constructed to do so! This has nothing whatsoever to do with physical relaity!
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