Question

In one of his lectures, L Susskind stated that he cannot make sense of a metric with more than one timelike dimension. I also have trouble imagining it, but is there a good mathematical or physical reason why this is not possible?

Let us assume all extra timelike dimensions are compactified, so we cannot directly observe them. What kind of trouble or unphysical problems would this create? I am not asking for non-standard physics here, but just a more formal reason than intuition as to why we assume the extra dimensions are necessarily spatial (I have a basic understanding of why string theory needs extra spatial dimensions, but I do not understand if it forbides extra time dimensions). The question is not limited to string theory though.

Answer 1

Multiple timelike dimensions would violate causality and conservation laws, leading to paradoxes and inconsistencies in our understanding of physics.

Step-by-step explanation:

In physics, there is a good mathematical and physical reason why we assume that there are no more than one timelike dimension. One reason is that having more than one timelike dimension would violate causality, where events can influence each other in a cause-and-effect relationship. If there are multiple timelike dimensions, it would mean that an event can have multiple pasts and multiple futures, which would lead to paradoxes and inconsistencies in our understanding of cause and effect.

Another reason is that having more than one timelike dimension would lead to a breakdown of conservation laws, such as conservation of energy and conservation of momentum. These laws are essential in physics and are derived from the symmetries of spacetime. If there are multiple timelike dimensions, these symmetries would be violated and our understanding of fundamental physics would be fundamentally altered.

Overall, the assumption of one timelike dimension in our universe is supported by both mathematical consistency and physical observations and experiments.