Energy–momentum Bookkeeping On A Self-intersecting Timelike World-line

Introduction

In the realm of general relativity, the concept of energy-momentum conservation plays a crucial role in understanding the behavior of objects in curved spacetime. However, when dealing with closed timelike curves (CTCs), the traditional notion of energy-momentum conservation is put to the test. In this article, we will delve into the intricacies of energy-momentum bookkeeping on a self-intersecting timelike world-line, exploring the implications of CTCs on energy conservation and the stress-energy momentum tensor.

Closed Timelike Curves and Energy Conservation

A closed timelike curve (CTC) is a world-line that returns to its starting point in spacetime, intersecting itself in the process. This phenomenon is often associated with the concept of time travel, where an object or observer can revisit a point in spacetime that they have already visited. However, the existence of CTCs raises fundamental questions about energy conservation, as the traditional notion of energy-momentum conservation is based on the idea that energy is conserved along a timelike world-line.

The Stress-Energy Momentum Tensor

The stress-energy momentum tensor (SEM tensor) is a fundamental object in general relativity, describing the distribution of energy and momentum in spacetime. The SEM tensor is defined as:

$$T^{\mu\nu} = \frac{2}{\sqrt{-g}} \frac{\delta S}{\delta g_{\mu\nu}}$$

where $S$ is the action, $g$ is the determinant of the metric tensor, and $g_{\mu\nu}$ is the metric tensor itself. The SEM tensor encodes information about the energy and momentum of matter and radiation in spacetime.

Energy-Momentum Bookkeeping on a Self-Intersecting Timelike World-Line

Consider a classical, finite-mass traveller in 3+1-dimensional general relativity who follows a smooth closed timelike curve (CTC). After a proper time $\Delta\tau$ the world-line returns to its starting point, intersecting itself in the process. To understand the implications of this scenario on energy-momentum conservation, we need to examine the SEM tensor along the world-line.

The SEM Tensor Along the World-Line

Let's consider a point $P$ on the world-line, where the traveller has a 4-velocity $u^\mu$ and a 4-momentum $p^\mu$. The SEM tensor at point $P$ is given by:

$$T^{\mu\nu}(P) = \frac{2}{\sqrt{-g(P)}} \frac{\delta S}{\delta g_{\mu\nu}(P)}$$

where $g(P)$ is the determinant of the metric tensor at point $P$. To evaluate the SEM tensor along the world-line, we need to consider the variation of the action $S$ with respect to the metric tensor $g_{\mu\nu}$.

Variation of the Action

The action $S$ in general relativity is given by:

$$S = \int d^4x \sqrt{-g} R$$

where $R$ is the Ricci scalar. To evaluate the variation of action with respect to the metric tensor, we need to consider the following:

$$\delta S = \int d^4x \left( \frac{\delta \sqrt{-g}}{\delta g_{\mu\nu}} R + \sqrt{-g} \frac{\delta R}{\delta g_{\mu\nu}} \right)$$

where $\delta g_{\mu\nu}$ is an infinitesimal variation of the metric tensor.

Energy-Momentum Conservation

To examine the implications of the SEM tensor on energy-momentum conservation, we need to consider the following:

$$\nabla_\mu T^{\mu\nu} = 0$$

where $\nabla_\mu$ is the covariant derivative. This equation represents the conservation of energy-momentum along a timelike world-line.

Causality and the SEM Tensor

The SEM tensor plays a crucial role in understanding the causal structure of spacetime. In particular, the SEM tensor is used to determine the causal relationship between different points in spacetime. However, when dealing with CTCs, the traditional notion of causality is put to the test.

Closed Timelike Curves and Causality

A closed timelike curve (CTC) is a world-line that returns to its starting point in spacetime, intersecting itself in the process. This phenomenon raises fundamental questions about causality, as the traditional notion of causality is based on the idea that cause precedes effect.

The Novikov Self-Consistency Principle

The Novikov self-consistency principle is a proposal for resolving the paradoxes associated with closed timelike curves. According to this principle, any events that occur through time travel have already occurred and are therefore predetermined. This means that any attempts to create paradoxes through time travel are self-consistent and do not violate causality.

Conclusion

In conclusion, energy-momentum bookkeeping on a self-intersecting timelike world-line is a complex and challenging problem in general relativity. The existence of closed timelike curves raises fundamental questions about energy conservation and the stress-energy momentum tensor. While the SEM tensor provides a framework for understanding energy-momentum conservation, the traditional notion of causality is put to the test when dealing with CTCs. The Novikov self-consistency principle offers a proposal for resolving the paradoxes associated with closed timelike curves, but further research is needed to fully understand the implications of CTCs on energy-momentum conservation and causality.

References

• Novikov, I. D. (1980). The River of Time. Moscow: Nauka.
• Hawking, S. W., & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge University Press.
• Wald, R. M. (1984). General Relativity. University of Chicago Press.
  Energy–momentum bookkeeping on a self-intersecting timelike world-line: Q&A =====================================================================

Q: What is a closed timelike curve (CTC)?

A: A closed timelike curve (CTC) is a world-line that returns to its starting point in spacetime, intersecting itself in the process. This phenomenon is often associated with the concept of time travel, where an object or observer can revisit a point in spacetime that they have already visited.

Q: What are the implications of CTCs on energy conservation?

A: The existence of CTCs raises fundamental questions about energy conservation, as the traditional notion of energy-momentum conservation is based on the idea that energy is conserved along a timelike world-line. In the presence of CTCs, energy-momentum conservation is no longer guaranteed, and the stress-energy momentum tensor (SEM tensor) plays a crucial role in understanding the implications of CTCs on energy conservation.

Q: What is the stress-energy momentum tensor (SEM tensor)?

A: The stress-energy momentum tensor (SEM tensor) is a fundamental object in general relativity, describing the distribution of energy and momentum in spacetime. The SEM tensor is defined as:

$$T^{\mu\nu} = \frac{2}{\sqrt{-g}} \frac{\delta S}{\delta g_{\mu\nu}}$$

where $S$ is the action, $g$ is the determinant of the metric tensor, and $g_{\mu\nu}$ is the metric tensor itself.

Q: How does the SEM tensor relate to energy-momentum conservation?

A: The SEM tensor plays a crucial role in understanding energy-momentum conservation along a timelike world-line. The conservation of energy-momentum is represented by the following equation:

$$\nabla_\mu T^{\mu\nu} = 0$$

where $\nabla_\mu$ is the covariant derivative.

Q: What is the Novikov self-consistency principle?

A: The Novikov self-consistency principle is a proposal for resolving the paradoxes associated with closed timelike curves. According to this principle, any events that occur through time travel have already occurred and are therefore predetermined. This means that any attempts to create paradoxes through time travel are self-consistent and do not violate causality.

Q: Can CTCs be used for time travel?

A: The possibility of using CTCs for time travel is still a topic of debate among physicists. While CTCs do allow for the possibility of time travel, the Novikov self-consistency principle suggests that any events that occur through time travel are predetermined and do not create paradoxes.

Q: What are the implications of CTCs on causality?

A: The existence of CTCs raises fundamental questions about causality, as the traditional notion of causality is based on the idea that cause precedes effect. In the presence of CTCs, causality is no longer guaranteed, and the SEM tensor plays a crucial role in understanding the implications of CTCs on causality.

Q: Can CTCs be used to create paradox?

A: The possibility of creating paradoxes through CTCs is still a topic of debate among physicists. While CTCs do allow for the possibility of creating paradoxes, the Novikov self-consistency principle suggests that any events that occur through time travel are predetermined and do not create paradoxes.

Q: What is the current understanding of CTCs in general relativity?

A: The current understanding of CTCs in general relativity is still evolving, and much research is needed to fully understand the implications of CTCs on energy-momentum conservation and causality. However, the Novikov self-consistency principle provides a framework for resolving the paradoxes associated with CTCs and understanding the implications of CTCs on causality.

Q: What are the potential applications of CTCs in physics?

A: The potential applications of CTCs in physics are still being explored, but some possible applications include:

• Quantum gravity: CTCs may play a crucial role in understanding the behavior of matter and energy in the presence of strong gravitational fields.
• Black hole physics: CTCs may be used to study the behavior of matter and energy in the vicinity of black holes.
• Cosmology: CTCs may be used to study the evolution of the universe on large scales.

Q: What are the challenges associated with studying CTCs?

A: The challenges associated with studying CTCs include:

• Mathematical complexity: CTCs are a complex and challenging topic in general relativity, requiring advanced mathematical tools to study.
• Physical interpretation: The physical interpretation of CTCs is still evolving, and much research is needed to fully understand the implications of CTCs on energy-momentum conservation and causality.
• Experimental verification: The experimental verification of CTCs is still a topic of debate, and much research is needed to develop experimental techniques for detecting CTCs.