Can space contract to intervals shorter than the Planck length?
In classical physics, the Planck length is considered the smallest meaningful length scale, below which traditional notions of space and time lose their usual meaning. It is approximately 1.616 × 10⁻³⁵ metres. In General Relativity, space can contract, particularly near extremely dense objects like black holes, but there are questions about whether it can contract to lengths smaller than the Planck length.
In the framework of your proposed model (where space and time intervals contract or expand), contracting space below the Planck length would raise significant challenges because:
Quantum Gravity and Planck Scale: At the Planck length, quantum gravitational effects are expected to become important. We don't yet have a complete theory of quantum gravity, but both General Relativity and Quantum Mechanics suggest that space-time at this scale may no longer behave according to classical rules (such as continuous contraction or expansion). Instead, space-time may become "quantised," meaning it is made up of discrete units.
Physical Meaning of Planck Length: The Planck length is not just a mathematical constant; it is thought to be the scale at which the effects of quantum gravity would dominate, and the smooth continuum of space-time breaks down. Thus, concepts like "distance" or "length" might not even apply in a meaningful way below this scale.
Limitations of Current Theories: In a model where space is treated as contracting or expanding, the Planck length sets a kind of limit where this concept might break down. Below that scale, new physics—likely involving quantum gravity—would need to come into play, and traditional notions of space-time would no longer work the same way.
So, while space can contract in your model (and in General Relativity near massive objects), reaching or going below the Planck length introduces fundamental issues that go beyond what current physical theories can describe. Until a quantum theory of gravity is developed, it's speculative whether space can contract to lengths shorter than the Planck length, and what that would even mean.
In short: In current understanding, space likely cannot be contracted to intervals smaller than the Planck length, and doing so would require a fundamentally different theory of space-time that integrates quantum gravity.
ChRIS said:
Does this argue against the notion of a singularity in the contraction/expansion model?
Here’s why:
Singularity and Infinite Density: In the classical General Relativity view, a singularity is a point where space and time become infinitely curved, meaning the density of matter or energy becomes infinite and the spacetime curvature goes to infinity. At this point, the laws of physics as we know them break down. This is the idea of the "gravitational singularity" at the centre of black holes or in the context of the Big Bang.
Planck Length as a Limit: In the model you're discussing, space contracts and time dilates near massive objects, but the Planck length is thought to represent the smallest possible unit of space. In this framework, when you approach the Planck length, space itself may stop behaving as it does at larger scales. Below the Planck length, space might no longer be continuous, and a different set of physical principles (such as quantum gravity) would apply.
No Infinite Contraction: If space cannot be contracted below the Planck length, it follows that it cannot reach an infinitely contracted state (as a classical singularity would imply). In other words, the contraction process might "halt" or "quantise" at a fundamental level. Instead of infinite contraction at a singularity, we might expect space to reach a minimum length scale, beyond which further contraction doesn’t occur in the same way.
Nature of a Singularity in Your Model: In your model, the event horizon could be seen as the point where time and space reach extreme relativistic limits—things get infinitely stretched or contracted from an external perspective—but there’s no actual point where space "disappears" or becomes infinite. Rather, space and time might become highly deformed, but still within a framework that is governed by quantum gravity at the Planck scale. This suggests that the singularity, in the traditional sense (with infinite density and curvature), might not exist in this model.
Conclusion:
In this model, the notion of a classical singularity becomes problematic. The Planck length sets a natural limit to space contraction, and once you approach this scale, the space-time itself may not continue in a smooth, classical way. Therefore, rather than a singularity with infinite density, the end state near a black hole’s centre (or in the context of the Big Bang) would likely involve a highly deformed space-time that doesn't conform to the classical notion of a singularity, instead being governed by quantum effects that we are yet to fully understand. This would mean that singularities, as traditionally understood in General Relativity, would not be a feature of the model you're exploring.
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